\section{Introduction}

In the first part of this paper, we are interested in the following possible properties 
for a connected CW-space $X$.

\begin{defA}\rm
\begin{itemize}
\item[(1)] $X$ is \dfn{\cef{k}{\ell}} if $X$ is homotopy equivalent to a CW-complex without $j$-cells for $k\leq j\leq\ell$. 
\item[(2)]  $X$ is \dfn{cohomology silent  in degree $j$} if $H^j(X;R)=0$ for any ${\mathbb Z}\pi_1(X)$-module~$R$.
\item[(3)]  $X$ is \dfn{\ccs{k}{\ell}} if $X$ is cohomology silent  in degree $j$ for~$k\leq j\leq\ell$.
\end{itemize}
\end{defA}

Obviously, a CW-space which is \cef{k}{\ell} is \ccs{k}{\ell}. The converse is known to be true for simply connected CW-spaces (using a minimal cell decomposition \cite[Proposition~4C.1]{Hatcher} together the universal coefficient theorem for cohomology). In \secref{GSCSC}, we show that a connected CW-space $X$ of finite type which is \ccs{k}{\ell}
determines a \dfn{cell-dispensability obstruction} $w_k(X)\in\tilde K_0({\mathbb Z}\pi_1(X))$ which vanishes if and only if
$X$ is \cef{k}{\ell}. Any class in $\tilde K_0({\mathbb Z}\pi)$ may occur as the cell-dispensability obstruction $w_k(X)$
for a CW-space $X$ with $\pi_1(X)$ identified with $\pi$ (by somehow fixing the ($k-1$)-skeleton).

The definition of $w_k(X)$ is akin to that of the Wall finiteness obstruction, but the two obstruction are different,
since $w_k(X)$ might not vanish for a finite complex $X$.
Among the similarities is a product formula. As a consequence for example, if $X$ is \ccs{k}{\ell} ($k\geq 3$), then 
$X\times S^1$ is \cef{k+1}{\ell}.

In \secref{GScasm}, we apply the above material to smooth manifolds, using the following
\begin{defB}\rm
Let $M$ be a smooth manifold of dimension~$r\geq 2k$.
\begin{itemize}
\item[(1)] $M$ is \dfn{\as{k}} \ if $M$ admits a handle decomposition without handles of index $j$ 
for $k\leq j \leq r-k$.
\item[(2)] $M$ is \dfn{\has{k}} if it is \ccs{k}{r-k}. This definition is also used for a Poincar\'e\  complex
of formal dimension $r$. 
\end{itemize}
\end{defB}

If $M$ is \has{k}, its cell-dispensability obstruction $w_k(M)\in\tilde K_0({\mathbb Z}\pi_1(M))$ is defined.
When $M$ is closed of dimension $r\geq 8$, we prove that $w_k(M)=0$ if and only if $M$ is \as{k}. The 
obstruction $w_k(M)$ is $r$-self-dual, i.e $w_k(M)=(-1)^{r+1}w_k(M)^*$ (the involution $*$ on $\tilde K_0({\mathbb Z}\pi_(M)$
involves the orientation character $\omega\:\pi_1(M)\to\{\pm 1\}$ of $M$).
It thus defines a class 
$[w_k(M)]$ in the Tate cohomology group  $H^{r+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi_(M),\omega))$. We prove that 
$[w_k(M)]=0$ if and only if $M$ is cobordant, in an appropriate sense, to a \as{k} manifold.

A whole section (\S~\ref{Srea}) is devoted to realize an element ${\mathcal P}\in\tilde K_0({\mathbb Z}\pi)$ as the
cell-dispensability obstruction for a Poincar\'e\  space or a closed manifold with fundamental group identified with $\pi$
(via fixing the normal ($k-1$)-type). Using the technique of ends of spaces \cite{PeRa,RanEnds}, we construct for any 
${\mathcal P}\in\tilde K_0({\mathbb Z}\pi)$ a finitely dominated Poincar\'e\  space ${\mathbb P}$ of formal dimension $r\geq 6$, 
which is \has{k} with $w_k({\mathbb P})={\mathcal P}$. Its Wall finiteness obstruction ${\rm Wall}({\mathbb P})$ is equal to ${\mathcal P} + (-1)^{r}{\mathcal P}^*$. In particular, when ${\mathcal P}$ is $r$-self-dual,
${\mathbb P}$ is homotopy equivalent to a finite Poincar\'e\  complex. Using then the projective surgery theory of \cite{PeRa},
we prove that ${\mathcal P}$ may be replaced by a closed $r$-dimensional manifold when $[w_k(M)]$
belongs to the kernel of the homomorphism  $\delta_R\:H^{r+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi,\omega))\to L_{r}^h(\pi,\omega)$
occurring in the Ranicki exact sequence (see~\eqref{ranicki}). 
Examples are discussed in \S~\ref{Sexpl}

The terminology {\it antisimple} was introduced in \cite{HauNAS} for manifolds related to high dimensional knots. 
Antisimple manifolds have been studied in \cite{HauMW}, with applications in \cite{HauTWI,Hau2}. 
In \cite{DFW,WeinbHigh},  {\it cohomology antisimple manifolds} are called  
{\it antisimple manifold}.

\sk{3}\noindent{\bf Acknowledgments: }\rm The author is grateful to Ian Hambleton and Shmuel Weinberger for comments and suggestions.


\section{Cohomology silent complexes}\label{GSCSC}

\subsection{Preliminaries}\label{Snot}

\begin{ccote} Cell complexes. \rm\ 
A CW-complex is also called a \dfn{cell complex}
or just a \dfn{complex}. We denote by $X^k$ the $k$-skeleton of a complex $X$. 
The universal covering of $X$ is denoted by $\tilde X$. Maps between topological spaces are always 
supposed to be continuous.

When a cell complex $X$ is connected, we usually assume that it has only one $0$-cell: 
$X^0=\{x_0\}$ and we denote by $\pi(X)$ the fundamental group $\pi_1(X,x_0)$, often abbreviated by $\pi$
when the context is clear. The group $\pi$ acts on the left on $\tilde X$ by deck transformations.

A \dfn{CW-space} is a topological space $X$ which is homotopy equivalent to a CW-complex.
Such a CW-space is 
\begin{itemize}
 \item of \dfn{finite $k$-type} if $X$ is homotopy equivalent to a CW-complex with finite $k$-skeleton.
 \item of \dfn{finite type} if $X$ is of finite $k$-type for all $k\geq 0$. 
 \item \dfn{finitely dominated} if there exists a finite CW-complex $A$ and maps $f\:A\to X$ and $s\:X\to A$
 such that $f\kern .7pt {\scriptstyle \circ} \kern 1pt s$ is homotopic to ${\rm id}_A$ ($f$ is called a \dfn{(homotopy) domination}).  
\end{itemize} 
\end{ccote}

\begin{ccote}\label{connectivity}  $r$-connectivity and weak $r$-connectivity. \rm\ 
A map $f\:Y\to X$ between connected CW-spaces is called \dfn{$r$-connected} if $\pi_j(f)=0$ for $j\leq r$. 
When $r\geq 2$, this is equivalent to $f$ inducing an isomorphism on fundamental groups and $H_j(\tilde f)=0$
for $j\leq r$, where $\tilde f\:\tilde Y\to\tilde X$ is the induced map on the universal covers. 
A pair $(X,Y)$ of connected CW-spaces is called \dfn{$r$-connected} if the inclusion map $Y\hookrightarrow X$
is $r$-connected.

A map $f\:Y\to X$ is called \dfn{weakly $r$-connected} if $f_*\: \pi_j(Y)\to\pi_j(X)$ 
is an isomorphism for $j\leq r-1$. When this is the case, one can attach
trivial $r$-cells to $Y$, getting a cell complex $Y'$, so that $f$ extends to $f'\:Y'\to X$ which is $r$-connected.
Again, a pair $(X,Y)$ of connected CW-spaces is called \dfn{weakly $r$-connected} if the inclusion map $Y\hookrightarrow X$
is weakly $r$-connected. Note that ``weakly $r$-connected'' implies ``($r-1$)-connected''.

The following lemma, proved in \cite[Lemma 1.2]{WallFin} (see also \cite[Prop.~1.4, p.~139]{Vara}), will be useful.

\begin{Lemma}\label{r-connSkel}
Let $f\:Y\to X$ be a map between connected CW-complexes. Suppose that $f$ is $r$-connected.
Then, there exists a CW-complex $Y'$ obtained by attaching to $Y$ cells of dimension $>r$, such that 
$f$ extends to a homotopy equivalence $f'\:Y'\to X$. \hfill \ensuremath{\Box}
\end{Lemma}
\end{ccote}

\begin{ccote}\label{reference}  Reference maps. \rm\ 
Let $Y$ be a connected CW-complex of dimension $r-1$. An \dfn{$Y$-reference map} for a CW-space $X$ is a map 
$g\:X\to Y$ which is $r$-connected. The space $X$ together with the map $g$ (or the pair $(X,g)$) is called an 
\dfn{$Y$-referred space}. 

\begin{Lemma}\label{Lrefretr}
Let $g\:X\to Y$ be an $Y$-reference map. Then, there exists a CW-complex $X'$ containing $Y$ as a subcomplex 
and a homotopy equivalence $h\:X'\to X$ such that the $Y$-reference map $g\kern .7pt {\scriptstyle \circ} \kern 1pt h$ is a retraction of $X'$ onto $Y$.
The pair $(X',Y)$ is weakly $r$-connected.
\end{Lemma}

In short, a reference map may be realized up to homotopy equivalence by a {\it reference retraction}. Note that the pair $(X',Y)$ is not $r$-connected  in general (e.g. $X'=X=Y\vee S^r$). 

\begin{proof}
The reference map $g$ may be, up to homotopy equivalence, replaced by a Serre fibration $\hat g\:\hat X\to Y$, 
whose fiber is then ($r-1$)-connected. As $Y$ is of dimension $r-1$, there is no obstruction to get a section 
of $\hat g$, which provides a homotopy section $\gamma\:Y\to X$ for $g$. 
This map $\gamma$ may be, up to homotopy equivalence, 
replaced by a CW-inclusion $J\:Y \hookrightarrow X'$ and the map $g$ provides a 
reference map $g_1'\:X'\to Y$, such that
$g_1'\kern .7pt {\scriptstyle \circ} \kern 1pt J$ is homotopic to ${\rm id}_Y$. By the homotopy extension property, there is a homotopy from $g_1'$ to 
$g'\:X'\to Y$ such that $g'\kern .7pt {\scriptstyle \circ} \kern 1pt\gamma={\rm id}_Y$, namely $g$ is a retraction. Since $g$ is $r$-connected,
the pair $(X',Y)$ is weakly $r$-connected.
\end{proof}
\end{ccote}

\begin{ccote}\label{cohmod} Cohomology with local coefficients. \rm 
Let $X$ be a connected cell complex with fundamental group $\pi$. We consider the cellular chain complex 
$(C_*(\tilde X),\partial)$, with $C_k(\tilde X) = H_k(\tilde X^k,\tilde X^{k-1})$. The latter is a
is a free (left) ${\mathbb Z}\pi$-module 
with basis in bijection with the $k$-cells of $X$ (by choosing, for each $k$-cell $e$, 
an orientation and a lifting $\tilde e$ in $\tilde X$ of $e$). The \dfn{cycles} $Z_k(\tilde X)$ and 
the \dfn{boundaries} $B_{k-1}(\tilde X)$ are, as usual, defined as the ${\mathbb Z}\pi$-modules
\begin{eqnarray*}
Z_k(\tilde X) &=& \ker\big(\partial \: C_k(\tilde X)\to C_{k-1}(\tilde X)\big) \\
B_{k-1}(\tilde X) &=& {\rm Image}\big(\partial \: C_k(\tilde X)\to C_{k-1}(\tilde X)\big) \, .
\end{eqnarray*}

Modules of ${\mathbb Z}\pi$ are {\it left} modules, unless mention of the contrary.
If $R$ is a such a ${\mathbb Z}\pi$-module,
the \dfn{cohomology} $H^*(X;R)$ is the homology of the cochain complex $C^*(\tilde X;R)=\hom(C_*(\tilde X),R)$, where the coboundary
$\delta$ is defined by 
$$
\delta(\alpha) = (-1)^k \alpha\kern .7pt {\scriptstyle \circ} \kern 1pt\partial \quad  (\alpha\in C^k(\tilde X))  \, .
$$

Homology with coefficient in a ${\mathbb Z}\pi$-module $R$ will also be used, but only in the framework of manifolds or Poincar\'e\  spaces (see \ref{holoc}). 
\end{ccote}


\subsection{The cell-dispensability obstruction}\label{SCSC}

Recall from the introduction (Definition~A) that a 
connected CW-space $X$ is \dfn{cohomology silent  in degree $k$} if $H^k(X;R)=0$ for any ${\mathbb Z}\pi_1(X)$-module $R$.

\begin{Lemma}\label{LW1}
Let $X$ be a connected CW-complex and let $k\geq 1$ be an integer.
The following conditions are equivalent.
\begin{itemize}
\item[(a)] $X$ is cohomology silent  in degree $k$.
\item[(b)] The following two conditions hold true:
   \begin{itemize}
   \item[(b1)]  the inclusion $B_{k-1}(\tilde X)\hookrightarrow C_{k-1}(\tilde X)$ admits a
                retraction of ${\mathbb Z}\pi$-modules (in consequence $B_{k-1}(\tilde X)$ is projective), and
   \item[(b2)] $H_k(\tilde X)=0$.
   \end{itemize}
\end{itemize}
\end{Lemma}

\begin{proof}
We set $C_j=C_j(\tilde X)$, $Z_j=Z_j(\tilde X)$, $B_j=B_j(\tilde X)$ and $\pi=\pi_1(X)$. The proof proceeds in three steps.

\sk{1}\noindent{\it Step 1: (a) $\Rightarrow$ (b1).} 
The boundary homomorphism $\partial\:C_k\to B_{k-1}$ may be seen as an element of 
$Z^k(X;B_{k-1}(\tilde X))$. If (a) holds true, there is $\alpha\in\hom(C_{k-1},B_{k-1})$ such that
$\partial=\delta(\alpha)$, i.e.
$\partial(u)=\alpha\kern .7pt {\scriptstyle \circ} \kern 1pt\partial(u)$ for all $u\in C_k$. Thus, $\alpha$ is a retraction of $C_{k-1}$ onto $B_{k-1}$.

\sk{1}\noindent{\it Step 2: (a) $\Rightarrow$ (b2).} By Step 1 already proven, $B_{k-1}$ is projective and thus 
the exact sequence $0\to Z_k\to C_k\to B_{k-1}\to 0$ splits, implying that $Z_k$ is a direct summand of $C_k$.
Therefore, the surjective homomorphism $Z_k\to H_k(\tilde X)$ extends to a homomorphism $\alpha\:C_k\to H_k(\tilde X)$,
giving an element of $C^k(X;H_k(\tilde X))$ 
satisfying $\alpha\kern .7pt {\scriptstyle \circ} \kern 1pt \partial=0$ (i.e. $\alpha\in Z^k(X;H_k(\tilde X)))$.
By (a), $\alpha = \bar\alpha\kern .7pt {\scriptstyle \circ} \kern 1pt\partial$ for some homomorphism $\bar\alpha\:C_{k-1}\to H_k(\tilde X)$. 
The homomorphism $\bar\alpha$ is surjective since so is $\alpha$. But $\partial(Z_k)=0$, which implies that $H_k(\tilde X)=0$.

\sk{1}\noindent{\it Step 3: (b) implies (a).} Let $b\in H^k(X;R)$, represented by a homomorphism
$\beta\: C_k(\tilde X)\to R$ which is a cocycle, i.e. $\beta_{|B_k}=0$. But $B_k=Z_k$ since $H_k(\tilde X)=0$ by (b2).
Since $k\geq 1$, one has $\beta = \bar\beta\kern .7pt {\scriptstyle \circ} \kern 1pt\partial$ for some morphism of ${\mathbb Z}\pi$-module
$\bar\beta\:B_{k-1}\to R$. By (b1), $\bar\beta$ extends to
a an element $\hat\beta\in C^{k-1}(X;R)$ and thus $\beta = \delta(\hat\beta)$, which proves that $b=0$.
Therefore, $H^k(X,R)=0$ for any ${\mathbb Z}\pi_1(X)$-module $R$, which proves (a).
\end{proof}

We now prepare the definition of the obstruction $w_k(X)$ associated to a cohomology silent  cell complex in degree $k$.

\begin{Lemma}\label{LW1b}
Let $X$ be a connected CW-complex which is cohomology silent  in some degree $k\geq 3$. Let $\varphi\:K\to X$ be a $(k-1)$-conneted map,
where $K$ is a $(k-1)$-dimensional CW-complex. Then, the ${\mathbb Z}\pi$-module $\pi_k(\varphi)$ is projective.
\end{Lemma}

\begin{proof}
Up to homotopy equivalence, we may suppose that $\varphi$ is the inclusion of the $(k-1)$-skeleton $K=X^{k-1}\to X$ 
(see \lemref{r-connSkel}). 
Since $k\geq 3$, the map $\varphi$ induces an isomorphism on the fundamental groups and is covered by a
map $\tilde\varphi\:\tilde K \to \tilde X$ on the universal coverings. By the relative Hurewicz isomorphism theorem, 
one has
$$
\pi_k(\varphi) \approx H_k(\tilde\varphi) \approx H_k(\tilde X,\tilde X^{k-1}) \, .
$$
We now claim that there is an isomorphism of ${\mathbb Z}\pi$-modules between $B_{k-1}(\tilde X)$ and 
$H_k(\tilde X,\tilde X^{k-1})$. \lemref{LW1b} will then follow from \lemref{LW1}.


To establish the claim, recall that the boundary homomorphism $\partial\: C_k(\tilde X)\to C_{k-1}(\tilde X)$ 
is the composition
$$
\partial : H_k(\tilde X^{k},\tilde X^{k-1}) \to H_{k-1}(\tilde X^{k-1}) \to H_{k-1}(\tilde X^{k-1},\tilde X^{k-2})  \, .
$$
The right arrow is injective since $H_{k-1}(\tilde X^{k-2})=0$. Hence, $B_{k-1}(\tilde X)$ may be seen as a submodule of 
$H_{k-1}(\tilde X^{k-1})$:
\small
$$
B_{k-1}(\tilde X) \approx {\rm Im\,}\big(H_k(\tilde X^{k},\tilde X^{k-1})\to H_{k-1}(\tilde X^{k-1})\big) = 
\ker\big(H_{k-1}(\tilde X^{k-1})\to H_{k-1}(\tilde X^{k})\big) \, .
$$
\rm
As $H_{k-1}(\tilde X^{k})\approx H_{k-1}(\tilde X)$, one has 
$$
B_{k-1}(\tilde X) \approx
\ker\big(H_{k-1}(\tilde X^{k-1})\to H_{k-1}(\tilde X)\big) =  
{\rm Im\,}\big(H_k(\tilde X,\tilde X^{k-1}) \to H_{k-1}(\tilde X^{k-1})\big) \, .
$$
But
$$
{\rm Im\,}\big(H_k(\tilde X,\tilde X^{k-1}) \to H_{k-1}(\tilde X^{k-1})\big) \approx H_k(\tilde X,\tilde X^{k-1})
$$
since $H_k(\tilde X)=0$ by \lemref{LW1}.
\end{proof}


Let $X$ be a connected CW-complex which is cohomology silent  in some degree $k\geq 3$. 
Let $\varphi\:K^{k-1}\to X$ be a $(k-1)$-connected map. By \lemref{LW1b}, $\pi_k(\varphi)$ is a projective 
${\mathbb Z}\pi$-module. Suppose that $K$ is a finite complex and that $X$ is of finite type. By the proof of \lemref{LW1b},
the ${\mathbb Z}\pi$-module $\pi_k(\varphi) \approx H_k(\tilde\varphi)$ is finitely generated and therefore defines 
a class 
\beq{defwk}
w_k(X) = (-1)^k\,[\pi_k(\varphi)] \in \tilde K_0({\mathbb Z}\pi) \, .
\end{equation}
That the map $\varphi$ does not appear in the notation $w_k(X)$ makes sense in view of the following lemma.

\begin{Lemma}\label{WL2b}
Let $X$ be a connected CW-complex which is of finite type and cohomology silent  in degree $k\geq 3$. 
Let $\varphi_i\:K_i^{k-1}\to X$ ($i=1,2$) be two ($k-1$)-connected maps, where $K_i$ are finite complexes. Then 
$[\pi_k(\varphi_1)] = [\pi_k(\varphi_2)]$ in $\tilde K_0({\mathbb Z}\pi)$.
\end{Lemma}

\begin{Remarks}\label{WR1}\rm
(1) When $X$ has the homotopy type of a $k$-dimensional complex, then $w_k(X)$ is the 
Wall finiteness obstruction ${\rm Wall}(X)$ of $X$ \cite{WallFin}. See also \secref{SWallFO}.
\sk{1}\noindent (2)
Taking for $\varphi\:K\to X$ the inclusion of $K=X^{k-1}$ into $X$, one has $w_k(X)=(-1)^k[B_{k-1}(X)]$
(see the proof of \lemref{LW1b}).
\hfill \ensuremath{\Box}\end{Remarks}


\begin{proof}[Proof of \lemref{WL2b}] We essentially follow the proof of \cite[Lemma~3.2]{WallFin}, 
as detailed in \cite[Lemma~2.9, p.~155]{Vara}.
Up to homotopy equivalence, the map $\varphi_i$ may be replaced by a Serre fibration $\theta_i$, with fiber $F_i$.
As $\varphi_i$ is $(k-1)$-connected, one has $\pi_j(F_i)=0$ for $j<k-1$, and thus
the first obstruction to construct a section of $\theta_i$ belongs to 
$H^k(X;\pi_{k-1}(F_i))$. The latter vanishes since $X$ is cohomology silent  in degree $k$. Using obstruction theory 
\cite[Chapter VI]{WhiteheadBoo}, we can thus construct map $s^i\:X^k\to K_i$ 
such that $\varphi_i\kern .7pt {\scriptstyle \circ} \kern 1pt s^i$ is homotopic to the inclusion 
$X^k\hookrightarrow X$. This induces a morphism of ${\mathbb Z}\pi$-module 
$s^i_*\:H_{k-1}(\tilde X)\approx H_{k-1}(\tilde X^k)\to  H_{k-1}(\tilde K_i)$ which splits the exact sequence
\beq{Waeq1}
\dia{
0 \ar[r] & H_k(\tilde\varphi_i) \ar[r] & H_{k-1}(\tilde K_i)  \ar[r]^{\varphi_i}   &  H_{k-1}(\tilde X)   
\ar@/^2mm/[l]^{s^i_*}  \ar[r] & 0
}
\end{equation}
(the morphism $H_k(\tilde\varphi_i) \to H_{k-1}(\tilde K_i)$ is indeed injective since
$H_k(\tilde X)=0$ by \lemref{LW1}). 
By cellularity, one may assume that $\varphi_i(K_i)\subset X^{k-1}$. We can thus define a map 
$\psi\:K_1\to K_2$ as $\psi=s^2\kern .7pt {\scriptstyle \circ} \kern 1pt \varphi_1$.

Since $\varphi_i$ is $(k-1)$-connected and since $K_i$ is of dimension $k-1$, one has $H_j(\tilde\psi)=0$
when $j\neq k-1,k$ and the horizontal line in the following diagram is an exact sequence.
{\small
$$
\dia{ 
H_k(\tilde\psi) \ar@{>->}[r] & H_{k-1}(\tilde K_1) \ar@{>>}[dr]^{(\tilde\phi_1)_*} \ar[rr]^{\tilde\psi_*}
&& H_{k-1}(\tilde K_2) \ar@{>>}[r]& 
H_{k-1}(\tilde\psi) \\
&& H_{k-1}(\tilde X)  \ar@{>->}[ur]^{\tilde s^2_*}
}
$$
}
Therefore, using \eqref{Waeq1}, one gets isomorphisms of ${\mathbb Z}\pi$-modules
\beq{Waeq2}
H_k(\tilde\varphi_1) \approx H_k(\tilde\psi) \quad {\rm and } \quad H_k(\tilde\varphi_2) 
\approx H_{k-1}(\tilde\psi) \, .
\end{equation}

With the abbreviations $C_j=C_j(\tilde\psi)$, $Z_j=Z_j(\tilde\psi)$ and
$B_j=b_j(\tilde\psi)$, the cellular chain complex $C_*(\tilde\psi)$ gives the following exact sequences.

\sk{2}
\begin{tabular}{lcll}
$Z_1\hookrightarrow C_1 \to\kern-6.5pt\to C_0$ & $\Rightarrow$ & $B_1=Z_1$ is stably free \\[1mm]
$Z_2\hookrightarrow C_2 \to\kern-6.5pt\to B_1$ & $\Rightarrow$ & $B_2=Z_2$ is stably free, {\it etc.}\\
\end{tabular}
\sk{1}\noindent
This shows that $B_j=Z_j$ is stably free for $j < k-1$ and that $Z_{k-1}$ is stably free. Then one has
\beq{Waeq3}
B_{k-1}\hookrightarrow Z_{k-1} \to\kern-6.5pt\to H_{k-1}(\tilde\psi) \quad , \quad
Z_k\hookrightarrow C_k \to\kern-6.5pt\to B_{k-1} \quad , \quad  Z_k \fl{\approx} H_{k}(\tilde\psi) \, .
\end{equation}
Since $X$ is cohomology silent  in degree $k$, the ${\mathbb Z}\pi$-modules $H_{k-1}(\tilde\psi)$ and $H_{k-1}(\tilde\psi)$ are
projective by \lemref{LW1} and \eqref{Waeq2}. By \eqref{Waeq3}, $B_{k-1}$ and $Z_k$ are also projective and,
by \eqref{Waeq2} again,
$$
[H_k(\tilde\varphi_2)]=[H_{k-1}(\tilde\psi)]=-[B_{k-1}]=[Z_k]=
 [H_k(\tilde\psi)] = [H_k(\tilde\varphi_1)] \, .  \qedhere
$$
\end{proof}

A slight modification of the proof of \lemref{WL2b} shows that $w_k(X)$ is an invariant of the $k$-type of $X$.
More precisely, one has the following

\begin{Lemma}\label{WL2bb}
Let $X_i$ (i=1,2) be two connected CW-complexes of finite type and $k\geq 3$ be an integer.
Suppose that, for $i=1,2$, $X_i$ is cohomology silent  in degree $k$.
Let $h\:X_1\to X_2$ be a $k$-connected map. Then, $w_k(X_2)=h_*(w_k(X_1))$, where 
$h_*\:\tilde K_0({\mathbb Z}\pi_1(X_1))\fl{\approx}\tilde K_0({\mathbb Z}\pi_1(X_2))$ is the isomorphism induced by $h$.
\end{Lemma}

\begin{proof}
We assume that $h$ is cellular. We follow the proof of \lemref{WL2b} with the map $\varphi_i\:K_i\to X_i$
being the inclusion of $K_i=X_i^{k-1}$ into $X$. For $i=1,2$, we get a map $s_i\:X_i^{k}\to K_i$ 
such that $\varphi_i\kern .7pt {\scriptstyle \circ} \kern 1pt s_i$ is homotopic to the inclusion 
$X_i^k\hookrightarrow X_i$. We then just need to replace the map $\psi$ of the proof of \lemref{WL2b} by 
$\psi=s_2\kern .7pt {\scriptstyle \circ} \kern 1pt h\kern .7pt {\scriptstyle \circ} \kern 1pt \varphi_1$. 
\end{proof}

\begin{Remark}\label{RCWspace}\rm
\lemref{WL2bb} enables us to define $w_k(X)\in\tilde K_0(\pi_1(X))$ for a {\it CW-space} $X$ 
of finite $k$-type which is cohomology silent  in degree $k$.
Just choose a homotopy equivalence $f\:X'\to X$ where $X'$ is a CW-complex with finite $k$-skeleton
and define $w_k(X)=f_*(w_k(X'))$. \lemref{WL2bb} guarantees that this is undependent of the choice of $(X',f)$.   
\hfill \ensuremath{\Box}\end{Remark}

The element  $w_k(X)$ will be called the \dfn{cell-dispensability obstruction of} $X$,
thanks to the following proposition, in which the the concepts of Definitions~A of the introduction are used.

\begin{Proposition}\label{WP1} 
Let $4\leq k\leq \ell$ be integers. Let $X$ be a connected CW-space
of finite $\ell$-type. Suppose that $X$ is \ccs{k}{\ell}.
Then, the following conditions are equivalent.
\begin{itemize}
 \item[(a)]  $w_k(X)=0$.
 \item[(b)] $X$ has the homotopy type of a CW-complex with a finite $\ell$-skeleton  which is \cef{k}{\ell}
 \end{itemize}
\end{Proposition}

\begin{proof}
Using \remref{RCWspace}, we assume that $X$ is a CW-complex with finite $\ell$-skeleton.
If $X$ has no $k$-cells, then $B_{k-1}(\tilde X)=0$ and thus $w_k(X)=0$ by Part (2) of \remref{WR1}. 
Therefore, (b) implies (a).

Conversely, suppose that (a) holds. As seen in Step 1 of the proof of \lemref{LW1}, the module $B_{k-1}(\tilde X)$ is a
retract of $C_{k-1}(\tilde X)$. We may thus write $C_{k-1}(\tilde X)\approx B_{k-1}(\tilde X)\oplus T$. 
If $w_k(X)=0$, then $B_{k-1}(\tilde X)$ is a finitely generated stably free ${\mathbb Z}\pi$-module, and so is $T$.
By wedging if necessary $X$ with a finite collection of $D^{k-1}$'s (canceling pairs of $(k-2)$ and $(k-1)$-cells),
one may assume that $T$ is ${\mathbb Z}\pi$-free, with basis ${\mathcal B} = \{t_1,\dots,t_r\}$. 
As $k\geq 4$, the complexes $\tilde X^{k-1}$ and $\tilde X^{k-2}$ are $1$-connected and the 
relative Hurewicz theorem then implies that 
the ${\mathbb Z}\pi$-module  $C_{k-1}(\tilde X) = H_{k-1}(\tilde X^{k-1},\tilde X^{k-2})$ is isomorphic to 
$\pi_{k-1}(X^{k-1},X^{k-2})$. One can thus see $T$ as a free submodule of 
$\pi_{k-1}(X^{k-1},X^{k-2})$ and $t_j$ is represented by a map of pair $\tau_j\:(D^{k-1},S^{k-2})\to(X^{k-1},X^{k-2})$.
Let $K$ be the finite CW-complex of dimension $k-1$ obtained by adding $(k-1)$-cells 
$e_1,\dots,e_r$ to $X^{k-2}$, the cell $e_j$ being attached by the map $\tau_j\:S^{k-2}\to X^{k-2}$.
Thus, $C_{k-1}(\tilde K)=T$.
The inclusion $X^{k-2}\to X$ then extends to a map $f\: K\to X$, using the map $\tau_j$ on the cell~$e_j$. 
One checks that $f$ induces an isomorphism on
the fundamental groups and on the homology groups $H_i(\tilde K) \fl{\approx} H_i(\tilde X)$ for $i\leq k-1$.
But, by  \lemref{LW1}, $H_j(\tilde X)=0$ for $k\leq j\leq\ell$.
Therefore, $f$ is $\ell$-connected and, using \lemref{r-connSkel}, one has a homotopy equivalence
$$
X \simeq K \cup \{{\rm cells\ of} \dim \geq \ell+1\}  \, ,
$$
which proves (b). 
\end{proof}

\begin{Remark}\label{Rj>k}\rm
In \proref{WP1}, one may wonder that Condition (b) is just $w_k(X)=0$ and not 
$w_j(X)=0$ for $k\leq j\leq\ell$. But, actually, $w_k(X)=w_{j}(X)$ for $k\leq j\leq l$. Indeed, as $H_j(\tilde X)=0$
for $k\leq j\leq\ell$, one has $B_j=Z_j$ and thus an exact sequences $0\to B_j \to C_j \to B_{j-1} \to 0$ for 
$k\leq j\leq\ell$.
\hfill \ensuremath{\Box}\end{Remark}

We finish this section with the following lemma, which will be used in \secref{SScob}.

\begin{Lemma}\label{LXT}
Let $(X,T)$ be a ($k-1$)-connected pair of CW-spaces of finite $k$-type ($k\geq 2$). Suppose that 
$X$ and $T$ are cohomology silent  in degree $k$. Then, $H_k(\tilde T,\tilde X)$ is a finitely generated projective 
${\mathbb Z}\pi$-module ($\pi=\pi_1(X)\approx\pi_1(T)$) and
$$
w_k(T) = w_k(X) + [H_k(\tilde T,\tilde X)] \, .
$$
\end{Lemma}

\begin{proof}
We suppose that $(X,T)$ is a pair of CW-complexes with finite $k$-skeleta.
Let $K=X^{k-1}$. Then $(X,K)$ and $(T,K)$ are ($k-1$)-connected and,
by \eqref{defwk}, one has
$$
w_k(X) = [H_k(\tilde X,\tilde K)]   \ \hbox{ and } \   w_k(T) = [H_k(\tilde T,\tilde K)] \, .
$$
Let us consider the homology exact sequence of the triple $(\tilde T,\tilde X,\tilde K)$. 
One has $H_{k-1}(\tilde K,\tilde X)=0$ since $(X,K)$ is ($k-1$)-connected. The connecting homomorphism
$H_{k+1}(\tilde T,\tilde X)\to H_k(\tilde X,\tilde K)$ factors through $H_k(\tilde X)$ which vanishes 
by \lemref{LW1}. We thus get a short exact sequence
$$
0 \to H_k(\tilde X,\tilde K) \to H_k(\tilde T,\tilde K) \fl{\beta} H_k(\tilde T,\tilde X) \to 0  \ .
$$
We claim that this sequence splits, which will prove the lemma. To construct a section $\sigma$ of $\beta$, 
let us consider the commutative diagram
\beq{LcobL1-dia}
\dia{ 
H_k(\tilde T,\tilde K) \ar[d]_\beta \ar@{>->}[r] & H_{k-1}(\tilde K) \ar[d]_{\tilde\varphi_*} \ar[r] & 
H_{k-1}(\tilde T)  \ar@{<->}[d]_=
\\ 
H_k(\tilde T,\tilde X) \ar@/_3mm/@{..>}[u]_{\sigma} \ar@{>->}[r] & H_{k-1}(\tilde X) \ar[r] \ar@/_3mm/[u]_{\sigma_0}
& H_{k-1}(\tilde T)
}
\end{equation}
in which the horizontal lines are exact. The left hand horizontal arrows are injective since
$H_k(\tilde T)=0$ by \lemref{LW1}. A seen in the proof of \lemref{WL2b}, the inclusion $\varphi\:K\hookrightarrow X$,
considered as a Serre fibration, admits a section over $X^k$, whence the section $\sigma_0$ of 
$\tilde\varphi_*$. A chasing argument in Diagram~\eqref{LcobL1-dia} then produces the 
required section $\sigma$ of $\beta$.  
\end{proof}

\subsection{Realization of cell-dispensability obstructions}

We now give two propositions concerning the realization of elements of $\tilde K_0({\mathbb Z}\pi)$ as the obstruction 
$w_k(X)$ for some cohomology $k$-silent complex $X$ with an identification $\pi_1(X)\approx\pi$. This identification
will be achieved via a reference map to a fixed finite cell complex $Y$ (see \ref{reference}).

\begin{Proposition}\label{PW3}
Let $Y$ be a connected finite CW-complex of dimension \\ $k-1\geq 2$ and let $\ell> k$ be an integer.
Let ${\mathcal P}\in\tilde K_0({\mathbb Z}\pi_1(Y))$. 
Then there exists a $Y$-referred finite CW-complex $(X_k^\ell,g^\ell)$ of dimension $\ell+1$ such that 
$X_k^\ell$ is \ccs{k}{\ell} and satisfies $g^\ell_*(w_k(X))={\mathcal P}$.
\end{Proposition}

\begin{proof}
Let us represent the class $(-1)^k{\mathcal P}$ by the image $P$ of a projector ${\rm pr}_P\:F\to F$, where $F$ is a 
finitely generated free ${\mathbb Z}\pi_1(Y)$-module. Then $F$ admits a supplementary projector
${\rm pr}_Q\:F\to F$ with image $Q=\ker {\rm pr}_P$, and $F=P\oplus Q$. 
Let ${\mathcal B}=\{b_1,\dots,b_r\}$ be a basis of $F$. We introduce some precise notations which will also be used
in the proof of \proref{PW4}. Let 
\beq{PW3e15}
{\mathcal S} = S^{k-1}_1 \vee \cdots \vee S^{k-1}_r \quad , \quad  {\mathcal D} = D^{k}_1 \amalg \cdots \amalg D^{k}_r
\end{equation}
where $S^{k-1}_j$ and $D^k_j$ are copies of the sphere $S^{k-1}$ and of the disk $D^k$. 

We shall add cells to $Y$ to get skeleta $X^j$ of a cell complex $X$, together with a $Y$-reference retraction
$g^j\:X^j\to Y$, such that $(X^\ell,g^\ell)=(X_k^\ell,g^\ell)$ when $\ell>k$
(recall from \lemref{Lrefretr} that an $Y$-reference map is anyway, up to homotopy equivalence, equal to a retraction).

The $(k-1)$-skeleton of $X$ is defined to be 
$$
X^{k-1}=Y\vee {\mathcal S} \, , 
$$
with the $Y$-reference retraction $g^{k-1}\:X^{k-1}\to Y$ sending ${\mathcal S}$ to the base point.

Since $k\geq 3$, $\pi=\pi_1(Y)\approx\pi_1(X^{k-1})$ and
$\pi_{k-1}(X^{k-1},Y)\approx H_{k-1}(\tilde X^{k-1},\tilde Y)$ is identified with $F$ (the inclusion 
$S^{k-1}_j\to{\mathcal S}\to X^{k-1}$ representing the element $b_j\in F$). Since $g^{k-1}$ is a retraction, the
homotopy exact sequence of the pair
$(X^{k-1},Y)$ splits
$$
\dia{ 
0 \ar[r] & \pi_{k-1}(Y) \ar[r] & \pi_{k-1}(X^{k-1}) \ar@/^2mm/@{>>}[l]^{g^{k-1}_*}   \ar[r] & \pi_{k-1}(X^{k-1},Y) \ar@/^2mm/@{.>}[l]^{\sigma}
\ar[r] & 0
} \, .
$$
Let 
\beq{sigma}
\sigma :\pi_{k-1}(X^{k-1},Y)\to\pi_{k-1}(X^{k-1})
\end{equation}
be the section sending $\pi_{k-1}(X^{k-1},Y)$ isomorphically onto
$\ker g^{k-1}_*$. 
Form the cell complex 
$$
X^k = X^{k-1} \cup_\theta {\mathcal D}  \, .
$$
where the restriction to ${\rm Bd}(D_j^k)$ of the attaching map $\theta\:{\rm Bd} ({\mathcal D})\to X^{k-1}$  
represents $\sigma\kern .7pt {\scriptstyle \circ} \kern 1pt {\rm pr}_P(b_j)\in\pi_{k-1}(X^{k-1})$. As $g^{k-1}_*\kern .7pt {\scriptstyle \circ} \kern 1pt\sigma=0$, 
the retraction $g^{k-1}$ extends to a retraction $g^k\:X^k\to Y$. 
Again, $\pi_{k}(X^k,X^{k-1})\approx H_{k}(\tilde X^k,\tilde X^{k-1})$ is identified with $F$. One has 
a diagram whose horizontal lines are exact:
$$
\dia{
& \pi_k(Y) & \\
\pi_k(X^{k-1}) \ar[rr] \ar@{>>}[ur]^{g^{k-1}_*} && 
\pi_{k}(X^k) \ar[r] \ar@{>>}[ul]_{g^k_*}  &  \pi_{k}(X^k,X^{k-1})   \ar[r] &  \pi_{k-1}(X^{k-1}) \\
 && F   \ar@{.>}[u]_\tau   \ar[r]^{{\rm pr}_Q} & F \ar[u]_{\approx} \ar[r]^{{\rm pr}_P} & F \ar[u]_{\sigma}
}  \ .
$$
The right square being commutative, there exist homomorphisms $\tau\: Q\to \pi_{k}(X^k)$
making the left square commutative. Since $g^{k-1}_*$ is onto, there is such a $\tau$ such that $g^k_*\kern .7pt {\scriptstyle \circ} \kern 1pt\tau=0$. 
Form the cell complex 
$$
X^{k+1} = X^k \cup e_1^{k+1} \cup\cdots\cup e_r^{k+1} \, ,
$$
where the $(k+1)$-cell $e_j^{k+1}$ is attached by the element $\tau\kern .7pt {\scriptstyle \circ} \kern 1pt {\rm pr}_Q(b_j)\in\pi_k(X^k)$.
Since $g^k_*\kern .7pt {\scriptstyle \circ} \kern 1pt\tau=0$, the retraction $g^k$ extends to a retraction $g^{k+1}\:X^{k+1}\to Y$. 
One checks that $H_k(\tilde X^{k+1})=0$ (one had $H_k(\tilde X^{k})\approx Q$) and that 
\beq{PW3-e5}
C_{k-1}(\tilde X^{k+1})\approx C_{k-1}(Y)\oplus Q\oplus B_{k-1}(\tilde X^k) \approx C_{k-1}(Y)\oplus Q\oplus P \, .
\end{equation}
By \lemref{LW1}, the cell complex $X^{k+1}$ is cohomology silent  in degree $k$ and $g^{k+1}_*(w_k(X))={\mathcal P}$ by \remref{WR1}.2.
Thus, for $\ell=k+1$, we can just define $X_k^\ell = X^{k+1}$. 

Now, since $B_k(\tilde X^{k+1})\approx Q$ is projective, one has isomorphisms of ${\mathbb Z}\pi$-modules
\beq{PW4e1}
\dia{
C_{k+1}(\tilde X^{k+1}) \ar[r]^(0.37){\approx} \ar[d]^\approx  & B_{k}(\tilde X^{k+1})\oplus Z_{k+1}(\tilde X^{k+1}) 
\ar[d]^\approx     \\
F  \ar[r]^\approx &  Q\oplus P
}
\end{equation}
Exactly like above, on may attach $r$ cells of dimension $k+2$ to $X^{k+1}$ to obtain $X^{k+2}$
and the retraction $g^{k+2}$ onto $Y$, so that the boundary homomorphism 
$$
\partial :  F \approx C_{k+2}(\tilde X^{k+2})\to C_{k+1}(\tilde X^{k+1})\approx Q\oplus P
$$ 
coincides with ${\rm pr}_P$.  Therefore, 
$H^{k+1}(\tilde X^{k+2})=0$. Together with \eqref{PW4e1},
this implies by \lemref{LW1} that $X^{k+2}$ is cohomology silent  in degree $k+1$. 

The process may be repeated  one degree higher, using the diagram like \eqref{PW4e1} with $k$ replaced by $k+1$
and $P$ and $Q$ being exchanged. Carrying on this procedure, one gets the $Y$-referred pair $(X_k^\ell,g^\ell)$ for $\ell>k$ enjoying the required properties.
\end{proof}


\begin{Remark}\label{RW3} \rm
The pair $(X_k^\ell,Y)$ as constructed in the above proof is weakly ($k-1$)-connected but {\it not} 
($k-1$)-connected. Indeed, the retraction $g^\ell$ splits the homology exact sequence of $(\tilde X_k^\ell,\tilde Y)$
$$
0 \to H_j(\tilde Y) \to  H_j(\tilde X_k^\ell) \to  H_j(\tilde X_k^\ell,\tilde Y) \to 0  
$$
and $H_j(\tilde X_k^\ell)=0$ for $k\leq j < \ell$ by \lemref{LW1}. Therefore, if $H_{k-1}(\tilde X_k^\ell,\tilde Y)=0$,
the connectivity of the pair $(X_k^\ell,Y)$ would be $\ell-1\geq k$ and thus $w_k(X_k^\ell)=0$. 
\hfill \ensuremath{\Box}
\end{Remark}


Passing to the limit for $\ell\to\infty$, \proref{PW3} and its proof give rise to the following result which 
will be used in \secref{Srea}.

\begin{Proposition}\label{PW4}
Let $Y$ be a connected finite CW-complex of dimension \\ $k-1\geq 2$.
Let ${\mathcal P}\in\tilde K_0({\mathbb Z}\pi_1(Y))$. Then there exists an $Y$-referred space $(X_k,g)$ such that 
\begin{itemize}
\item[(a)] $X_k$ is of finite type.
\item[(b)] $X_k$ is cohomology silent  in degree $j$ for all integers $j\geq k$ and satisfies $g_*(w_k(X_k))={\mathcal P}$. 
\item[(c)] $X_k$ is homotopy equivalent to a countable CW-complex $L$ is of dimension~$k$.
\item[(d)] For $\ell>k$, the inclusion $X_k^\ell\hookrightarrow X_k$ is a domination. 
In particular, $X_k$ and $L$ are finitely dominated.
\end{itemize}
\end{Proposition}

\begin{proof}
Define $X_k$ as the union of $X_k^\ell$ for $\ell\geq k$, endowed with the weak topology 
(note that $X_k^\ell\subset X_k^{\ell+1}$). The retractions $g^\ell\:X_k^\ell\to Y$ induce the required $Y$-reference
retraction $g\:X_k\to Y$ and the pair $(X_k,g)$ clearly satisfies (a) and~(b).

To prove (d), we replace up to homotopy equivalence the inclusion $\psi^\ell\: X_k^\ell\hookrightarrow X_k$ 
by a Serre fibration $\hat\psi^\ell\:\hat X_k^\ell\to X_k$. 
Its fiber $F$ being ($\ell-1$)-connected, the obstructions to get a section of $\hat\psi^\ell$ belong to $H^{i}(X_k;\pi_{i-1}(F))$
for $i>\ell$.  If $\ell>k$, the latter vanishes by (b). The map $\psi^\ell$ thus admits a section and is therefore a domination. 

It remains to prove (c). We use the notations of the proof of \proref{PW3}: $F$, ${\mathcal B}$, 
${\rm pr}_P,{\rm pr}_Q\:F\to F$, ${\mathcal S}$, ${\mathcal D}$, $\sigma$, etc. Let $K=X_k$. Recall from \eqref{PW3-e5} that $K^{k-1}=Y\vee{\mathcal S}$, so that
\beq{ck-1K}
C_{k-1}(\tilde K)\approx C_{k-1}(Y)\oplus Q\oplus P
\end{equation}
with $P=B_{k-1}(\tilde K)$ and $Q=H_{k-1}(\tilde K)$.
The ($k-1$)-skeleton of $L$ will be 
$$
L^{k-1} =  
K^{k-1} \vee {\mathcal S}_1 \vee{\mathcal S}_2 \vee \cdots = Y \vee {\mathcal S}_0 \vee {\mathcal S}_1 \vee {\mathcal S}_2 \vee \cdots
$$
where ${\mathcal S}_i$ is a copy of the bouquet ${\mathcal S}$. Therefore,
\begin{eqnarray}
C_{k-1}(\tilde L) &\approx & 
C_{k-1}(\tilde K) \oplus F \oplus F \oplus \cdots \nonumber  \\ &\approx &
C_{k-1}(\tilde Y) \oplus Q \oplus P\ \oplus F \oplus F \oplus \cdots  \nonumber  \\ &\approx &
C_{k-1}(\tilde Y) \oplus Q \oplus P \oplus (Q\oplus P) \oplus (Q\oplus P)  \oplus \cdots  \label{PW4e2b}
\\ &\approx &
C_{k-1}(\tilde Y) \oplus Q \oplus (P\oplus Q) \oplus (P\oplus Q) \oplus \cdots \label{PW4e2}
\end{eqnarray}
The CW-complex $L$ is then obtained by attaching $k$-cells to $L^{k-1}$:
\beq{edefLk}
L = L^{k-1} \cup_{\theta_{0,1}} {\mathcal D}_{0,1} \cup_{\theta_{1,2}} {\mathcal D}_{1,2} \cup \cdots \, ,
\end{equation}
where ${\mathcal D}_{r,s}$ is a copy of ${\mathcal D}$, with the attaching map $\theta_{r,s}\:{\rm Bd} ({\mathcal D}_{r,s})\to L^{k-1}$
which we will now define.
One has $\pi_{k-1}(L^{k-1},Y)\approx F_0\oplus F_1\oplus\cdots$,
where $F_j$ is a copy of $F$. Since the pair $(L^{k-1},Y)$ is weakly ($k-2$)-connected, 
the homomorphism $\pi_{k-1}(L^{k-1})\to \pi_{k-1}(L^{k-1},Y)$ is onto and one can 
choose a section 
$\sigma_L\:\pi_{k-1}(L^{k-1},Y) \to \pi_{k-1}(L^{k-1})$ whose restriction to $F_j$ is denoted by $\sigma_j$.
For $\sigma_0$, we make our choice so that the following diagram
$$
\dia{
F \ar[r]^(0.28){\approx} \ar[d]^(0.44)= & \pi_{k-1}(K^{k-1},Y) \ar[r]^(0.52)\sigma \ar[d] & \pi_{k-1}(K^{k-1}) \ar[d]
\\
F_0 \ar@{>->}[r] & \pi_{k-1}(L^{k-1},Y) \ar[r]^(0.52){\sigma_0} & \pi_{k-1}(L^{k-1}) 
}
$$
is commutative, where the vertical arrows are induced by the inclusion $K^{k-1}\to L^{k-1}$ and $\sigma$ is the section 
of \eqref{sigma}. The attaching map $\theta_{r,s}\:{\rm Bd} ({\mathcal D}_{r,s})\to L^{k-1}$ is now chosen
so that its restriction to ${\rm Bd}(D_j^k)$
represents the class $\sigma_r\kern .7pt {\scriptstyle \circ} \kern 1pt {\rm pr}_P(b_j)+ \sigma_s\kern .7pt {\scriptstyle \circ} \kern 1pt {\rm pr}_Q(b_j) \in\pi_{k-1}(L^{k-1})$. 
One can thus write
\beq{PW4e4}
C_{k}(\tilde L) \approx F \oplus F \oplus \cdots \approx
(P\oplus Q) \oplus (P\oplus Q) \oplus \cdots 
\end{equation}
so that the boundary homomorphism
$\partial\: C_{k}(\tilde L)\to C_{k-1}(\tilde L)$ sends the 
$j$-th term $(P+Q)$ of \eqref{PW4e4} isomorphically, 
through ${\rm pr}_P\oplus {\rm pr}_Q$,
to the corresponding summand of \eqref{PW4e2}. This boundary homomorphism
being injective, one has $H_k(\tilde L)=0$. 

The above definitions of $K$ and $L$ guaranty the existence of a map 
$f\: L\to K=X_k$ extending the obvious identification $Y\vee{\mathcal S}_0\fl{\simeq}Y\vee{\mathcal S}$ and, for $j\geq 1$, sending
${\mathcal S}_j$ and ${\mathcal D}_{j,j+1}$ to the base point of $K$. The pairs $(X_k,Y)$ and $(L,Y)$ being weakly ($k-1$)-connected,
so is the map $f$. The homomorphism $C_{k-1}\tilde f\: C_{k-1}(\tilde L)\to C_{k-1}(\tilde K)$
sends the first summands $C_{k-1}(\tilde Y)\oplus Q$ of \eqref{PW4e2b} isomorphically those of \eqref{ck-1K}. 
But these summands provide isomorphisms $H_{k-1}(\tilde Y)\oplus Q\fl{\approx} H_{k-1}(\tilde L)$ and 
$H_{k-1}(\tilde Y)\oplus Q\fl{\approx} H_{k-1}(\tilde K)$. Therefore,
$H_{k-1}\tilde f \: H_{k-1}(\tilde L) \to H_{k-1}(\tilde K)$ is an isomorphism.
Since $H_j(\tilde L)=0=H_j(\tilde K)$ for $j\geq k$, the map $f$ is a homotopy equivalence.
\end{proof}

Here are a few remarks about \proref{PW4} and its proof. 

\begin{Remark}\label{WR7} \rm
The inclusion $\beta\:K^{k-1}\hookrightarrow L$  is ($k-1$)-connected and $K^{k-1}$ is a finite complex. 
Since $H_k(\tilde L)=0$, one has 
$$
\dia{
0 \ar[r] & H_k(\tilde\beta) \ar[r] & H_{k-1}(\tilde K^{k-1}) \ar[r] & H_{k-1}(\tilde L) \ar[r] & 0 \\
&& H_{k-1}(\tilde Y)\oplus F \ar[u]_\approx 
\ar[r]^{{\rm id}\oplus{\rm pr}_Q} & H_{k-1}(\tilde Y)\oplus Q \ar[u]_\approx \ar[r] & 0
}\ .
$$
Therefore, $\pi_k(\beta)\approx H_k(\tilde\beta)\approx P$ and, in $\tilde K_0({\mathbb Z}\pi)$, one gets
$$
w_k(L) = (1)^k[\pi_k(\beta)] = (1)^k[P] = {\mathcal P}
$$ 
as expected. 
\hfill \ensuremath{\Box}\end{Remark}

\begin{Remark}\label{RWallObs}\rm
The CW-complex $L$ (or $X_k$) fulfills the conditions for the Wall finiteness obstruction ${\rm Wall}(L)\in\tilde K_0({\mathbb Z}\pi_1(L))$
to be defined \cite[Theorem F]{WallFin}. Using \remref{WR7}, the obstruction ${\rm Wall}(L)$ is equal to ${\mathcal P}$, thus equal to $w_k(L)$, as claimed in \remref{WR1}.1. 
\hfill \ensuremath{\Box}\end{Remark}

\begin{Remark}\label{WR10}\rm
The CW-complex $L$ may be obtained as the direct limit of a nested system
\beq{WR10-dia}
Y  \hookrightarrow L_1 \hookrightarrow L_2  \hookrightarrow \cdots \hookrightarrow
L_\infty = L \fl{g} Y
\end{equation}
where
$$
L_1= Y\vee{\mathcal S}  \quad {\rm and } \quad L_{j+1} = L_j \vee {\mathcal S} \cup {\mathcal D} \ \ {\rm for}\ j\geq 1 
$$
and such that the composition of maps in \eqref{WR10-dia} is the identity of $Y$. Indeed, the inclusion chain reflects 
the very definition of $L$. 
Let $f\:L\to X_k$ be the homotopy equivalence constructed to prove Part (c) of \proref{PW4}. Then,
the composition of $f$ with the $Y$-reference map $g\: X_k\to Y$ gives 
an $Y$-reference retraction from $L$ onto $Y$, also called $g\:L\to Y$. 
Its restriction to $L_j$ provides an $Y$-reference retraction $g_j\: L_j\to Y$. 
This material is bound to be used in \secref{Srea}.
\hfill \ensuremath{\Box}\end{Remark}


\subsection{Relationship with Wall's finiteness obstruction}\label{SWallFO}

A certain relationship between the  cell-dispensability obstruction $w_k(X)$ and the
Wall's finiteness obstruction was already noticed in \remref{WR1}. The following variant of \proref{PW4}  may shed more light about this matter.

\begin{Proposition}\label{Pw-wall}
Let $X$ be a finite CW-complex of dimension $\ell$ which is \ccs{k}{\ell}. 
Then, there is a finitely dominated CW-complex $X'$ containing $X$ as a subcomplex, such that $(X',X)$ is $\ell$-connected and $X'$ is homotopy equivalent to a countable CW-complex of dimension $k$. Two such complexes 
$X'_1$ and $X'_2$ are homotopy equivalent relative $X$. Moreover, the image of $w_k(X)$ into $\tilde K_0(X')$ coincides with ${\rm Wall}(X')$. 
\end{Proposition}

\begin{proof}
Starting with with $X$ we apply the process of the proof of Propositions~\ref{PW3} and~\ref{PW4}, ending with 
the space $X'= X_k$ of \proref{PW4}, which enjoys the required properties. If $X'_1$ and $X'_2$
are two such complexes, as $X'_1$ is cohomology silent  in degree $\geq k$, there is no obstruction to extend ${\rm id}_X$ to a map $f\:X'_1\to X'_2$, which will be a homotopy equivalence, since $H_j(\tilde{X'_1})=0=H_j(\tilde{X'_2})$
for $j\geq k$ (by \lemref{LW1}). Finally, the complex $X'$ fulfills the conditions for the Wall finiteness obstruction 
${\rm Wall}(X')\in\tilde K_0({\mathbb Z}\pi_1(X'))$ to be defined \cite[Theorem F]{WallFin} and the homological algebra definitions 
of $w_k(X)$ and of ${\rm Wall}(X')$ make them to be equal modulo the identification $\pi_1(X)\approx\pi_1(X')$.  
 \end{proof}

Another consequence of the formally identical  homological algebra definitions 
of $w_k(X)$ and of ${\rm Wall}(X')$ is the following product formula. For the Wall obstruction, this is a particular
case of the product formula proven in Siebenmann's thesis \cite[Theorem~7.2]{SieTh} (see also \cite[Theorem~3.14, p.~169]{Vara}).

\begin{Proposition}[Product formula]\label{PproductFor}
Let $X$ be a connected CW-space of finite type which is \ccs{k}{\ell} for $k\geq 3$. Let $A$ be a 
connected finite CW-complex of dimension $a\leq \ell-k$. Then $X\times A$ is \ccs{k+a}{\ell}
and the equality
$$
w_{k+a}(X\times A)= \chi(A)\, i_*(w_{k}(X))
$$
holds in $\tilde K_0({\mathbb Z}(\pi_1(X\times A)))$, where $\chi(A)$ is the Euler characteristic of $A$ 
and $i_*\:\tilde K_0({\mathbb Z}\pi_1(X))\to\tilde K_0({\mathbb Z}\pi_1(X\times A))$ 
is induced by the inclusion\\ $i\:\pi_1(X)\to\pi_1(X\times A)\approx \pi_1(X)\times\pi_1(A)$ sending $u\in\pi_1(X)$
to $(u,1)$.
\end{Proposition}

\begin{proof}
The homological algebra's arguments used in \cite[pp.~168--170]{Vara} to prove the same formula for the Wall 
finiteness obstruction work as well to establish the proposition.
\end{proof}

\begin{Corollary}
 Let $X$ be a connected CW-space of finite type which is \ccs{k}{\ell} for $k\geq 3$.
 Then $X\times S^1$ is \cef{k+1}{\ell}. \hfill \ensuremath{\Box}
\end{Corollary}

For more applications of \proref{PproductFor}, see \proref{CCproductFor}


\section{Antisimplicity}\label{GScasm}

\subsection{Preliminaries}

\begin{ccote}\label{holoc} Homology with local coefficients. \rm 
While cohomology with local coefficients was defined for an arbitrary pair (see~\ref{cohmod}),
homology with local coefficients is defined for a {\it pair over $\bbr P^\infty$}. A \dfn{space over $\bbr P^\infty$} is a 
space $X$ together with a (homotopy class of) map $w_X\:X\to\bbr P^\infty$. A pair $(X,Z)$ is said to be 
\dfn{over $\bbr P^\infty$} if both $X$ and $Z$ are over $\bbr P^\infty$ with $\omega_Z$ being the restriction of 
$\omega_X$ to $Z$. 

For example, the space $BO$ is over $\bbr P^\infty$ by the classifying map
$$
\omega_{BO} : BO \to B{\pi_1(BO)} \simeq B\{\pm 1\} \simeq \bbr P^\infty \, .
$$
Thus, a stable vector bundle $\eta$ over $X$, classified by $\Phi_\eta\:X\to BO$, makes $X$ a space over $\bbr P^\infty$
by the map  $\omega_\eta=\omega_{BO}\kern .7pt {\scriptstyle \circ} \kern 1pt \Phi_\eta$, called the \dfn{orientation character of~$\eta$}.
In the case of a (smooth) manifold $M$, the orientation character of its stable normal (or tangent) bundle 
gives the \dfn{orientation character $\omega_M\:M\to\bbr P^\infty$ of~$M$}. 

When $X$ is a CW-space, the functor $\pi_1$ provides a bijection
$$
[X,\bbr P^\infty] \fl{\approx}  \hom\big(\pi_1(X),\{\pm 1\}\big)   \ , \  (f\mapsto \pi_1f) \ .    
$$
We use the same notation $\omega_X$ for 
the map $\omega_X$ itself and for its induced homomorphism $\pi_1\omega_X$. 

A group homomorphism $\omega\:\pi\to \{\pm 1\}$ produces an involution $a\mapsto\bar a$ on ${\mathbb Z}\pi$
defined by 
$$
\overline{\sum_{g\in\pi} n_g g} = \sum_{g\in\pi} \omega(g) n_g g^{-1} \ ,
$$
satisfying $\overline{a+b}=\bar a + \bar b$ and $\overline{ab}=\bar b\bar a$. Any left ${\mathbb Z}\pi$-module $R$, 
may thus be transformed into a right ${\mathbb Z}\pi$-module $\bar R$ with the action $r a=\bar a r$ for $a\in{\mathbb Z}\pi$.
Let $(X,Z)$ be a pair of CW-spaces over $\bbr P^\infty$. Set $(\pi,\omega) = (\pi_1(X),\omega_X)$ and let $R$ be a (left)
${\mathbb Z}\pi$-module.
The homology $H_*(X,{\rm Bd} Z;R)$ is defined as the homology of the 
chain complex 
$$
\overline{C_*(\tilde X,\tilde Z)}\otimes_{{\mathbb Z}\pi} R \, ,
$$
(see \cite[p.~15]{HVBoo}). 
For example, let $(M,{\rm Bd} M)$ be a compact manifold pair, with $M$ connected.
When $R={\mathbb Z}$ with trivial $\pi_1(M)$-action,
then $H_*(M,{\rm Bd} M;{\mathbb Z})$ is the ordinary homology of $(M,{\rm Bd} M)$ with integral coefficients when $M$ is orientable,
and that of the orientation cover $(M^{or},{\rm Bd} M^{or})$  otherwise.
In any case, $H_n(M,{\rm Bd} M;{\mathbb Z})\approx{\mathbb Z}$. A compact connected manifold of dimension $n$ is always supposed
to be equipped with a generator $[M]$ of $H_n(M,{\rm Bd} M;{\mathbb Z})\approx{\mathbb Z}$, called the \dfn{fundamental class} of $M$. 
\end{ccote}


\begin{ccote}\label{Pesp} Poincar\'e\  spaces. \rm 
Let $P$ be a finitely dominated CW-space over $\bbr P^\infty$, together with a class $[P]\in H_n(P;{\mathbb Z}))$.
We say that $P$ is a \dfn{Poincar\'e\  space of formal dimension $n$} if the the cap product with $[P]$ 
\beq{PeDef1}
 - \frown [P] : H^r(P,Z;R) \fl{\approx}  H_{n-r}(P;R)
\end{equation}
is an isomorphism for any ${\mathbb Z}\pi_1(P)$-module $R$.
More generally, let $(P,Q)$ be a pair over $\bbr P^\infty$ of finitely dominated CW-spaces, together with a  class 
$[P]\in H_n(P,Q))$. We say that $(P,Q)$ is a \dfn{Poincar\'e\  pair of formal dimension $n$} if the following
two conditions hold true 
\begin{itemize}
\item[(a)] the cap product with $[P]$ 
\beq{PeDef1b}
 - \frown [P] : H^r(P,Q;R) \fl{\approx}  H_{n-r}(P;R)
\end{equation}
is an isomorphism for any ${\mathbb Z}\pi_1(P)$-module $R$, and
\item[(b)] the space $Q$ together with the class $[Q]=\partial_*([P])$ is a Poincar\'e\  space of formal dimension $n-1$.
 \end{itemize}
 We may also say that $P$ is a \dfn{Poincar\'e\  space} of formal dimension $n$, {\it with boundary ${\rm Bd} P=Q$}. 
This boundary is then a Poincar\'e\  space without boundary. 

Finally, by a \dfn{Poincar\'e\  cobordism} $(P,Q_1,Q_2)$, we mean a pair $(P,Q)$ of  
finitely dominated CW-spaces with $Q=Q_1\amalg Q_2$, together with a class 
$[P]\in H_n(P,Q)$ such that (a) above is satisfied and such that $Q_i$ is a Poincar\'e\  space with fundamental class 
$[Q_i]$ such that $\partial_*([P])=[Q_1]-[Q_2]$. 

\begin{Remarks}\rm \ 
(1) A compact $n$-manifold pair $(M,{\rm Bd} M)$ is a Poincar\'e\  space of formal dimension $n$. This follows from the 
standard proof of the Poincar\'e\ -Lefschetz duality using a ${\mathcal C}^\infty$-triangulation of $M$ 
and its dual cell decomposition.

\nsk{1}
(2) By \cite[Theorem~B]{KleinQin}, Condition (a) is equivalent to

\nsk{2}\hskip 1.8mm\begin{minipage}{10cm}
\begin{itemize}
\item[(a')] the cap product with $[P]$ 
\beq{PeDef2}
 - \frown [P] : H^r(P;R) \fl{\approx}  H_{n-r}(P,Q;R)
\end{equation}
is an isomorphism for any ${\mathbb Z}\pi_1(P)$-module $R$.
\end{itemize}
\end{minipage}

Hence, if (a) and/or (a') is true, one has a sign-commutative diagram 
for the (co)homology in a ${\mathbb Z}\pi_1(P)$-module $R$: 

\sk{1}
\begin{minipage}{80mm}
\footnotesize \hskip -6mm
$
\dia{
H^{j}(P,Q) \ar[r] \ar[d]^{\frown [P]}_\approx & H^{j}(P) \ar[r] \ar[d]^{\frown [P]}_\approx 
& H^j(Q) \ar[r] \ar[d]^{\frown [Q]}  & H^{j+1}(P,Q) \ar[r] \ar[d]^{\frown [P]}_\approx  & H^{j+1}(P)  
\ar[d]^{\frown [P]}_\approx 
\\ 
H_{n-j}(P) \ar[r] & H_{n-j}(P,Q) \ar[r] & H_{n-1-j}(Q) \ar[r] &  H_{n-1-j}(P) \ar[r] & H_{n-1-j}(P,Q)
}
$
\end{minipage}\rm\nsk{2}
By the five lemma, $-\frown [Q]\: H^j(Q;R) \to H_{n-1-j}(Q;R)$ is an isomorphism for any ${\mathbb Z}\pi_1(P)$-module $R$,
but this may be not the case if $R$ is a ${\mathbb Z}\pi_1(Q)$-module (see \cite[Theorem~A]{KleinQin}). Thus, Conditions (a) and (b) are independent in general. Note however, that (a) implies (b) when $\pi_1(Q)\approx\pi_1(P)$. 
\hfill \ensuremath{\Box}
\end{Remarks}
\end{ccote}

\begin{ccote}\label{coSpivak} Spivak fibrations. \ \rm
Let $BG$ be the classifying space for stable spherical fibration. 
Let $(X,Y)$ be a pair over $BG$, i.e. a pair $(X,Y)$ together
with a map $g\:X\to BG$. Consider the graded abelian group
$$
{\mathcal H}_*(X,Y) = \pi_n^s(T(\xi),T(\xi|Q))   \quad , \quad T(\, )  = \hbox{ Thom spectrum } \, ,
$$
where $\xi$ is the pull-back by $g$ of the universal stable spherical fibration over $BG$. 
Then ${\mathcal H}_*$ is a homology theory for spaces over $BG$. 

The natural maps $BG\to\bbr P^\infty$
makes a space over $BG$ a space over $\bbr P^\infty$. Therefore, the homology $H_*(X,Y;R)$ with coefficient in a
${\mathbb Z}\pi_1(X)$-module $R$, in the sense of~\ref{holoc}, is defined. When $R={\mathbb Z}$ with trivial $\pi_1(X)$-action,
one has degree~$0$ graded homorphism 
\beq{deft}
t : {\mathcal H}_*(X,Y) \to H_*(X,Y;{\mathbb Z})
\end{equation}
obtained by the composition of the Hurewicz map with the Thom isomorphism. 

Let $(P,Q)$ be a Poincar\'e\  pair of formal dimension~$n$. 
A \dfn{Spivak fibration} for $(P,Q)$ is a stable spherical fibration $\xi$ over $P$, classified by a lifting
$P\to BG$ of $\omega_P$, such that there exists a class $(P) \in {\mathcal H}_n(X,Y)$
satisfying $t((P))=[P]$.
We call $(P)$ a \dfn{Spivak class}. In our context of (finitely dominated) Poincar\'e\  spaces,
Spivak fibrations do exist and, if $\xi_j$ ($j=1,2$) are two
such fibrations with Spivak classes $(P_j)$, there is a fibre homotopy equivalence from $\xi_1$ to $\xi_2$
sending $(P_1)$ to $(P_2)$ \cite[Corollary~3.6]{WallPoinc}. 

\begin{Remark}  \rm \
Although our definitions are inspired by those of \cite{HVBoo}, there are some differences. What we call a  Poincar\'e\  space
is called a {\it PD-space} there. A {\it Poincar\'e\ } space in \cite{HVBoo} is a PD-space thgether with a Spivak class.
The present paper is self-contained with respect to \cite{HVBoo}.
\hfill \ensuremath{\Box}
\end{Remark}
\end{ccote}


\begin{ccote}\label{MPfnormal} $\eta$-normal maps.    \ \rm
Let $Y$ be a connected CW-space and let $\eta$ be a stable vector bundle over $Y$, with characteristic map
$\Phi_\eta\: Y\to BO$.

\begin{Definitions}\rm 
(A) A map $q\:Z\to Y$, from a manifold $Z$ to $Y$ is called \dfn{$\eta$-normal} if 
$\Phi_\eta\kern .7pt {\scriptstyle \circ} \kern 1pt q$ classifies the stable normal bundle $\nu_Z$ of $Z$.

\nsk{1}
(B) If $(P,Q)$ is a Poincar\'e\  pair, 
a map $q\:P\to Y$ is called \dfn{$\eta$-normal} if the composite
$$
P \fl{q} Y \fl{\Phi_\eta} BO \to BG
$$
classifies a Spivak fibration for $(P,Q)$.
\end{Definitions}

The following result is well known.

\begin{Lemma}\label{Lnormal}
(1) A map from a compact manifold $M$ to $Y$ which is $\eta$-normal in the sense of (A) 
is $\eta$-normal in the sense of (B) for the pair $(M,{\rm Bd} M)$.
\nsk{1}
(2) Let $f\:(P',Q')\to (P,Q)$ be a map of degree 1 (i.e. $f_*([P'])=[P]$) between Poincar\'e\  pairs and let 
$q\:P\to Y$ be a map. If the map $q\kern .7pt {\scriptstyle \circ} \kern 1pt f$ is $\eta$-normal, so is the map $q$.
\nsk{1}
(3) If $q\: P\to Y$ is $\eta$-normal for the Poincar\'e\  pair $(P,Q)$, then its restriction to the Poincar\'e\  space $Q$ is 
$\eta$-normal. 
\end{Lemma}

\begin{proof}
The Thom-Pontryagin class $(M)_{TP}$ is a Spivak class for $(M,{\rm Bd} M)$, which proves (1). For (2), we note that the class $f_*((P))$ is a Spivak class for $(P,Q)$. For (3), we use that the restriction over $Q$ of a Spivak fibration for $(P,Q)$ is a Spivak fibration for $Q$, taking $(Q)=\partial_*((P))$.
\end{proof}

In most of our applications, $Y$ will be a finite complex of dimension $k-1$ and $q$ will be an $Y$-reference map.
If $q$ is $\eta$-normal, we call $q$ an \dfn{$(Y,\eta)$-reference map}.

Let $(P,Q)$ be a Poincar\'e\  pair of formal dimension $n$ and let $q\:P\to Y$ be an $\eta$-normal map. 
The Spivak fibration $\xi$ over $(P,Q)$ is thus endowed with the vector bundle reduction $q^*\eta$.
This corresponds to a surgery problem (normal map of degree one) $\alpha\:({\mathcal M},{\rm Bd}{\mathcal M})\to (P,Q)$, where ${\mathcal M}$ is a compact $n$-manifold and $q\kern .7pt {\scriptstyle \circ} \kern 1pt\alpha$ is $\eta$-normal.
Such a data determines a surgery obstruction in the Wall group $L_n^p(\pi_1(P),\pi_1(Q),\omega_{P})$,
vanishing if and only if $\alpha$ is normally cobordant to a homotopy equivalence. When $Q=\emptyset$,
We denote by $\sigma(P,q)\in L_n^p(\pi_1(Y),\omega_{\eta})$ the image by $q^*$ of this surgery obstruction.
\end{ccote}


\begin{ccote}\label{PdecNor} $\eta$-normal Poincar\'e\  decompositions. \ \rm
Let $(X_\pm,{\rm Bd} X_\pm)$ be two CW-pairs with ${\rm Bd} X_-=Z_-\amalg Z_0$ and ${\rm Bd} X_+ = Z_0\amalg Z_+$. Form the CW-pair $(X,{\rm Bd} X)$ where
\beq{decompEx}
X= X_- \cup_{Z_0} X_+  \ , \ {\rm Bd} X = Z_- \amalg Z_+ \, .
\end{equation}
For the sake of simplicity, we assume that the homomorphisms $\pi_1({\rm Bd} X_\pm)\to \pi_1(X_\pm)$
and $\pi_1(Z_0)\to\pi_1(X_\pm)$ induced by the inclusions are all isomorphisms. 

Let $Y$ be a connected CW-space and let $\eta$ be a stable vector bundle over $Y$. Let $q\:X\to Y$ be a map,
restricting to $q_\pm\:X_\pm\to Y$. These maps make $X$, ${\rm Bd} X$, $X_\pm$, etc, spaces over $BO$, $BG$ and $\bbr P^\infty$. 
Consider their homology $H_n()$ with coefficients in ${\mathbb Z}$ with trivial ${\mathbb Z}\pi_1(Y)$-action. 
Define $\beta_H\:H_n(X,{\rm Bd} X) \to H_n(X_-,{\rm Bd} X_-)\oplus H_n(X_+,{\rm Bd} X_+)$ by the following diagram
\beq{defbetah}
\dia{ 
H_n(X,{\rm Bd} X) \ar[r] \ar@/_7mm/[rr]^{\beta_H}
& H_*(X;{\rm Bd} X\amalg Z_0)  \ar@{<-}[r]^(.39)\approx_(.4){{\rm excis.}} &
 H_n(X_-,{\rm Bd} X_-)\oplus H_n(X_+,{\rm Bd} X_+)
}
\end{equation}

\begin{Definition}\label{PdecNoDef}\rm
We say that \eqref{decompEx} is an \dfn{$\eta$-normal Poincar\'e\  decomposition} if $(X,{\rm Bd} X)$ and $(X_\pm,{\rm Bd} X_\pm)$ are Poincar\'e\  pairs of formal dimension~$n$ satisfying $\beta_H([X]) = ([X_-],[X_+])$ and such that $q$ and $q_\pm$ are $\eta$-normal.
\end{Definition}

We now give results to deduce from weaker hypotheses that a decomposition like in~\eqref{decompEx} is an 
$\eta$-normal Poincar\'e\  decomposition. By excision, one has the isomorphism
\beq{deciso2}
H_*({\rm Bd} X)  \fl{\approx} H_*({\rm Bd} X\amalg Z_0,Z_0) \,.
\end{equation}
Diagram \eqref{defbetah} and isomorphism \eqref{deciso2} may be considered as well 
for the homology theory ${\mathcal H}_*$ over $BG$. Therefore, the exact sequences of the
triple $(X,{\rm Bd} X\amalg Z_0,Z_0)$ and the homomorphism  $t$ of~\eqref{deft} give rise to a morphism of exact sequences
\beq{PdecNor-dia}
\dia{
{\mathcal H}_n(X,{\rm Bd} X)  \ar[r]^(.33){\beta_{\mathcal H}} \ar[d]^t & {\mathcal H}_n(X_-,{\rm Bd} X_-)\oplus {\mathcal H}_n(X_+,{\rm Bd} X_+) 
\ar[r]^(.68){\partial_{\mathcal H}} \ar[d]^{t\oplus t} & {\mathcal H}_{n-1}(Z_0) \ar[d]^t
\\
H_n(X,{\rm Bd} X)  \ar@{>->}[r]^(.33){\beta_H}  & H_n(X_-,{\rm Bd} X_-)\oplus H_n(X_+,{\rm Bd} X_+) \ar[r]^(.68){\partial_H} & 
H_{n-1}(Z_0)
}
\end{equation}

\begin{Proposition}\label{PPdecNor}
Let $(X)\in {\mathcal H}_n(X,\partial X)$ such that $\beta_{\mathcal H}((X))=((X_-),(X_+))$.

(i) Suppose that $(X_\pm,\partial X_\pm)$ are Poincar\'e\  pairs of formal dimension $n$ with $[X_\pm]=t((X_\pm))$ (in other words, $q_\pm$ is $\eta$-normal and $(X_\pm)$ are Spivak clases). Then $(X,\partial X)$ is a Poincar\'e\  pair of formal dimension $n$ with $[X]=t((X))$. 

(ii) Suppose that $(X,{\rm Bd} X)$ and $(X_-,{\rm Bd} X_-)$ are Poincar\'e\  pairs of formal dimension~$n$ with $[X]=t((X))$
and $[X_-]=t((X_-))$. Then $(X_+,{\rm Bd} X_+)$ is a Poincar\'e\  pair of formal dimension~$n$ with $[X_+]=t((X_+))$.

Consequently, in both cases, \eqref{decompEx} is $\eta$-normal Poincar\'e\  decomposition.
\end{Proposition}

\begin{proof}
The requirements about finite dominations are guaranteed by \cite[Complement~6.6]{SieTh}.
The Poincar\'e\  duality statements  are proven in \cite[Theorem~2.1 and its addendum]{WallPoinc}.
The assertions about the Spivak classes are then deduced from the commutativity of the left square in Diagram~\eqref{PdecNor-dia}, using for~(i) that $\beta_H$ is injective. 
\end{proof}

\begin{Corollary}\label{CPdecNor}
Let $(X_\pm)\in{\mathcal H}_n(X_\pm)$ such that $ \partial_{\mathcal H}((X_-),(X_+))=0$.  
Suppose that $(X_\pm,{\rm Bd} X_\pm)$ are Poincar\'e\  pairs of formal dimension $n$ with $[X_\pm]=t((X_\pm))$.
Then $(X,{\rm Bd} X)$ is a Poincar\'e\  pair of formal dimension~$n$ and $q$ is $\eta$-normal.
Consequently, \eqref{decompEx} is $\eta$-normal Poincar\'e\  decomposition.
\end{Corollary}

\begin{proof}
Since the top line in \eqref{PdecNor-dia} is exact, there is a (possibly non-unique) class  
$(X)\in {\mathcal H}_n(X,\partial X)$ such that $\beta_{\mathcal H}((X))=((X_-),(X_+))$. The corollary then follows from
Part (i) of \proref{PPdecNor} (note that the fundamental class $[X]$ is unique since $\beta_H$ is injective).
\end{proof}
\end{ccote}

Part (ii) of \proref{PPdecNor} may be shortened in the following

\begin{Corollary}\label{CPdecNor2}
Suppose that $(X,{\rm Bd} X)$ and $(X_-,{\rm Bd} X_-)$ are Poincar\'e\  pairs of formal dimension~$n$ and that 
$q$ and $q_-$ are $\eta$-normal. Then $(X_+,{\rm Bd} X_+)$ is a Poincar\'e\  pairs of formal dimension~$n$ and 
$q_+$ is $\eta$-normal. In other words, \eqref{decompEx} is an $\eta$-normal Poincar\'e\  decomposition.
\hfill \ensuremath{\Box}
\end{Corollary}


\begin{ccote}\label{coThick} Stable $\eta$-thickenings. \ \rm
Let $Y$ be a connected finite CW-complex of dimension $n-1\geq 2$ and let $\eta$ be a stable vector bundle over $Y$.
An $(Y,\eta)$-referred manifold $(N,g)$ is called an \dfn{$\eta$-thickening of $Y$ of dimension $n$} if 
\begin{itemize}
\item $N$ is a compact connected manifold of dimension $n$ and $g$ is a simple homotopy equivalence;
\item the pair $(N,{\rm Bd} N)$ is ($n-k$)-connected (with $n-k\geq 2$). 
\end{itemize}
(Recall that $\dim Y = k-1$.) Two $\eta$-thickenings  $(N_1,g_1)$ and $(N_2,g_2)$ of $Y$ are regarded as 
equivalent if there is a degree-one diffeomorphism $h\: N_1\fl{\approx} N_2$ such that $g_2\kern .7pt {\scriptstyle \circ} \kern 1pt h$ is homotopic of $g_1$. 
In the {\it stable range} $n>2(k-1)$, one has the following 

\begin{Proposition}[Wall]\label{PWallThick}
Let $Y$ be a connected finite cell complex of dimension $k-1$ and
let $\eta$ be a stable vector bundle over $Y$. Then, if $n>2(k-1)\geq 6$, there is an unique equivalence class
$\Xi_n(Y,\eta)$ of $\eta$-thickenings of $Y$ in dimension~$n$. \hfill \ensuremath{\Box}
\end{Proposition}

A representative $(N,g)$ of $\Xi_n(Y,\eta)$ is called a (or sometimes {\it the}) \dfn{stable $\eta$-thickening 
of $Y$ of dimension $n$}.

\begin{proof}
This follows from Wall's classification of stable thickenings by their stable tangent bundle
\cite[Proposition~5.1]{WallThick}, whence by their stable normal bundle. More precisely for the uniqueness, 
Wall's theorem provides a diffeomomorphism between any two stable thickenings, which may be not of degree one.
But a stable thickeninbg $N$ is of the form $N_-\times I$ (see \remref{RBdThick} below) and thus $N$ admits a 
self-diffeomorphism of degree $-1$. 
\end{proof}

\begin{Remark}\label{RBdThick}\rm 
When $n>2k-1\geq 6$, the boundary ${\rm Bd} N$ of a stable $\eta$-thickening $(N,g)$ is a \as{k} manifold 
(See Definition~B in the introduction). 
Indeed, let $\gamma_0\:Y\to N$ be a homotopy inverse of $g$. As $(N,{\rm Bd} N)$ is ($n-k$)-connected and 
$n-k>k-1$, the map $\gamma_0$ is homotopic to a map $\gamma\:Y\to {\rm Bd} N$. 
Up to homotopy equivalence, one may assume that $Y$ is a finite simplicial complex and that $\gamma$ is an embedding
into a ${\mathcal C}^\infty$-triangulation of ${\rm Bd} N$. Then, $\gamma(Y)$ has a smooth regular neighborhood $N_-$ \cite{HirschNhb}
and $(N_-,g_{|N_-})$ is a stable $\eta$-thickening of $Y$ of dimension $n-1$.  
By the s-cobordism theorem, the manifold $N$ is diffeomorphic to $N_-\times [0,1]$ and one thus gets
a manifold triad $(N,N_-\cup {\rm Bd} N_-\times [0,1],N_-\times\{1\})$.
But the pair $(N_-,{\rm Bd} N_-)$ is $(n-k-1)$-connected. 
By the procedure of eliminating handles (see \cite{KeBMS}), $N_-\times \{0\}$ and $N_-\times\{1\}$ both admit a handle decomposition with handles of index $\leq k-1$. Therefore, ${\rm Bd} N$ is \as{k}. 
\hfill \ensuremath{\Box}
\end{Remark}


The uniqueness of the Spivak class admits a different form for stable thickenings. 

\begin{Lemma}\label{LSpiThick}
Let $(N_j,g_j)$ ($j=1,2$) be two stable $\eta$-thickenings of $Y$ of dimension $n>2(k-1)\geq 6$, 
with Spivak class $(N_j)\in {\mathcal H}_n(N_j,{\rm Bd} N_j)$.
Then there exists a simple homotopy equivalence $f\:(N_1,{\rm Bd} N_1)\to (N_2,{\rm Bd} N_2)$ satisfying $g_1\simeq f\kern .7pt {\scriptstyle \circ} \kern 1pt g_2$
and ${\mathcal H}_*f((N_1))=(N_2)$.
\end{Lemma}

\begin{proof}
The data $(N_j,g_j,(N_j))$ determine surgery problems $\alpha_j\:({\mathcal N}_j,{\rm Bd}{\mathcal N}_j)\to (N_j,{\rm Bd} N_j)$ such that
${\mathcal H}_*\alpha_j(({\mathcal N}_j)_{PT})=(N_j)$, where $({\mathcal N}_j)_{PT}\in{\mathcal H}_n({\mathcal N}_j,{\rm Bd}{\mathcal N}_j)$ is 
the Thom-Pontryagin class of ${\mathcal N}_j$. By Wall's $(\pi-\pi)$-theorem, one may suppose that $f_j$ are 
simple homotopy equivalences. Hence, $({\mathcal N}_j,g_j\kern .7pt {\scriptstyle \circ} \kern 1pt f_j)$ are stable $\eta$-thickenings of $Y$. 
By \proref{PWallThick}, there is a diffeomorphism $h\:{\mathcal N}_1\to{\mathcal N}_2$ such that $g_2\kern .7pt {\scriptstyle \circ} \kern 1pt\alpha_2\kern .7pt {\scriptstyle \circ} \kern 1pt h$ is homotopic to $g_1\kern .7pt {\scriptstyle \circ} \kern 1pt\alpha_1$. Being a diffeomorphism of degree one, $h$ satisfies ${\mathcal H}_*h(({\mathcal N}_1)_{TP})=({\mathcal N}_2)_{TP}$.
Therefore the map $f=\alpha_2\kern .7pt {\scriptstyle \circ} \kern 1pt h\kern .7pt {\scriptstyle \circ} \kern 1pt\alpha_1'$, where $\alpha_1'$ is a homotopy inverse of $\alpha_1$, satisfies the conclusion of \lemref{LSpiThick}. 
\end{proof}
\end{ccote}

\begin{ccote}\label{endsMan} Ends of manifolds. \ \rm
A general reference for ends of spaces is \cite{RanEnds}.
We restrict to the case of a $\sigma$-compact finitely dominated open manifolds $U$ with one end $\epsilon$. 
Such an end is called \dfn{tame} \cite[Definition~10, p.~xiii]{RanEnds} if it admits a
sequence $U\supset Z_1 \supset Z_2 \supset \cdots $ of finitely dominated closed neighborhoods with
$$
\bigcap_{j=1}^\infty Z_j = \emptyset \ , \ \pi_1(Z_1)\approx \pi_1(Z_2) \approx  \cdots \approx \pi_1(\epsilon)  \, .
$$

Recall that, according to our definition in~\ref{Pesp}, a Poincar\'e\  pair is finitely dominated.
\begin{Proposition}\label{Pcompletion}
Let $U$ be a connected $\sigma$-compact finitely dominated $n$-dimensional manifold with compact boundary ${\rm Bd} U$ and with one end $\epsilon$, which is tame and satisfies $\pi_1(\epsilon)\approx \pi_1(U)$. 
Set ${\mathcal P}={\rm Wall}(U)\in \tilde K_0(\pi_1(U),\omega_U)$.
Then there exists a Poincar\'e\  cobordism $({\mathbb U},{\rm Bd} U,{\mathbb P})$, 
with $\pi_1({\mathbb P})\approx\pi_1({\mathbb U})$, together with a homotopy equivalence $\alpha\:U\to{\mathbb U}$.
Moreover, with the identification
$\pi_1({\mathbb P})\approx\pi_1({\mathbb U})\approx\pi_1(U)$ obtained using $\alpha$, the formulae
$$
{\rm Wall}({\mathbb U})={\mathcal P} \ \hbox{ and } \ {\rm Wall}({\mathbb P}) = {\mathcal P} + (-1)^{n}{\mathcal P}^*
$$
hold true in $\tilde K_0(\pi_1(U),\omega_U)$.
\end{Proposition}

\begin{proof}
We use the \dfn{space of ends} $e(U)$ of $U$, which is the space of proper maps $c\:[0,\infty)\to U$ \cite[Definition~1.2]{RanEnds}. 
It is equipped with the \dfn{origin map} $\vartheta\:e(U)\to U$, given by $\vartheta(c)=c(0)$.
Define ${\mathbb U}$ to be the mapping cylinder of $\vartheta$:
$$ 
{\mathbb U} = e(U)\times [0,1] \amalg U  \big/ \{(c,1)\sim c(0)\} \, ,
$$ 
which retracts by deformation onto $U$, so the inclusion $\alpha\:U\to{\mathbb U}$ is a homotopy equivalence.
Set ${\mathbb P}=e(U)$. 

The conclusions of the proposition come from \cite[Proposition~10.5]{RanEnds}. We just need to check that  the hypotheses of the latter, i.e. forward and reverse tameness of $\epsilon$, are implied by our tameness hypothesis. This is guaranteed by \cite[Proposition~8.9 and~10.13]{RanEnds}.
\end{proof}
\end{ccote}


\subsection{Cohomology anti-simple manifolds}\label{Scasm}

Let $M$ be a connected (smooth or PL-manifold) of dimension $n$ and let $k$ be an integer
with $n\geq 2k$. We shall compare the two Definitions~B of the introduction, i.e. 
$M$ is \dfn{\as{k}} if it admits a handle decomposition without handles of index $j$ 
for $k\leq j \leq n-k$ and $M$ is \dfn{\has{k}} if it is \ccs{k}{n-k}.

Obviously, a \as{k}-manifold is \has{k}. Only the empty manifold is \has{0}. A closed manifold is \as{1} \ 
if and only if it is a homotopy sphere. 

The above definition may also be used for Poincar\'e\  spaces (with or without boundary) of formal dimension $n$. It will be mainly used for closed manifold. In this case, ``cohomology antisimple'' could be called ``homology antisimple'' thanks to the following
 
\begin{Proposition}\label{has/chas}
Let $M$ be a closed connected manifold of dimension $n$ (or a Poincar\'e\  space of formal dimension $n$ without boundary). 
The following condition are equivalent.
\begin{itemize}
 \item[(i)] $M$ is \has{k}.
 \item[(ii)] $H_j(M;R)=0$ for any ${\mathbb Z}\pi_1(M)$-module $R$ when $k\leq j \leq n-k$.
\end{itemize}
\end{Proposition}

\begin{proof}
Since, by Poincar\'e\  duality, $H^j(M;R) \approx H_{n-j}(M;R)$.
\end{proof}

By \secref{SCSC}, a \has{k} manifold $M$ determines a cell-dispensability obstruction $w_k(M)\in\tilde K_0(\pi_1(M))$. 
We call it the \dfn{antisimple obstruction} fo $M$, thanks to the following

\begin{Proposition}\label{Pcasas}
Let $M$ be a closed connected manifold of dimension $n\geq 8$ which is \has{k} for $k\geq 4$. 
The following condition are equivalent.
\begin{itemize}
 \item[(a)] $M$ is \as{k}.
 \item[(b)] $w_k(M)=0$.
\end{itemize}
\end{Proposition}

\begin{proof}

If is $M$ is \as{k}, then $M$ has the homotopy type of a finite CW-complex $X$ having no cells in dimension 
$j$ for $k\leq j \leq n-k$. By \lemref{WL2bb}, one has $0=w_k(X)=w_k(M)$, which proves $(a)\Rightarrow (b)$.

Conversely, suppose that $w_k(M)=0$. As $M$ is \has{k}, there is by \proref{WP1}
a homotopy equivalence $f\:Y\to M$, where 
$Y$ is a finite CW-complex without $j$-cells for $k\leq j\leq n-k$. Let $K$ be a finite simplicial complex
homotopy equivalent to $Y^{k-1}$. As $n\geq 2k$ one may suppose that $f$ is an PL-embedding for a $C^\infty$-triangulation
of $M$. The complex $f(K)$ admits a smooth regular neighborhood $T$, which admits a handle decomposition 
with handles of index $\leq k-1$. Let $V=M-{\rm int\,}T$, which we see as a cobordism between $N={\rm Bd}T$
and the empty set.
As $k\geq 4$, one has $\pi_1(N)\approx\pi_1(V)$ and 
$H_j(\tilde V,\tilde N)\approx H_j(\tilde M,\tilde T)=0$ for $j\leq n-k$. The procedure of eliminating handles 
in a cobordism (see \cite{KeBMS}) then produces a handle decomposition of $(V,N)$ with only handles of index $> n-k$,
thus proving (a).
\end{proof}

We now turn our attention to realizing elements of $\tilde K_0({\mathbb Z}\pi)$ as antisimple obstructions $w_k(M)$ for a
\has{k} manifold $M$ with $\pi_1(M)=\pi$. We first establish necessary conditions to be fulfilled by $w_k(M)$. 
If $P$ is a finitely generated projective ${\mathbb Z}\pi$-module, so is its \dfn{dual} $P^*=\hom_{{\mathbb Z}\pi}(P,{\mathbb Z}\pi)$,
endowed with the left ${\mathbb Z}\pi$-action  $(a\beta)(u)=\beta(u)\bar a$ (recall that the involution $a\mapsto\bar a$
on ${\mathbb Z}\pi$ involves the orientation character $\omega$ of $M$: see \ref{holoc}).
This induces an involution ${\mathcal P}\mapsto{\mathcal P}^*$ on $\tilde K_0({\mathbb Z}\pi)$. An element ${\mathcal P}\in\tilde K_0({\mathbb Z}\pi)$
is called \dfn{$n$-self-dual}, i.e ${\mathcal P}=(-1)^{n+1}{\mathcal P}^*$ 


\begin{Proposition}\label{Pdua}
Let $M$ be a closed connected manifold of dimension $n\geq 6$ which is \has{k} for $k\geq 3$. 
Then $w_k(M)$ is $n$-self-dual.
\end{Proposition}

\begin{proof}
Consider a handle decomposition ${\mathcal H}$ for $M$
$$
{\mathcal H} : \ D^n = {\mathcal H}_0 \subset {\mathcal H}_1 \subset \cdots \subset {\mathcal H}_n = M \ ,
$$
where ${\mathcal H}_j$ is the union of handles of index $\leq j$ in ${\mathcal H}$. This handle decomposition 
makes $M$ homotopy equivalent to a cell complex and thus gives rise to 
a chain complex of free ${\mathbb Z}\pi$-modules $C_j({\mathcal H})=H_j(\tilde {\mathcal H}_j,\tilde {\mathcal H}_{j-1})$ 
whose homology is that of $\tilde M$. As seen in \lemref{LW1b} and its proof, one has
\beq{Pdua-e1}
w_k(M) = (-1)^k[B_{k-1}({\mathcal H})]\in \tilde K_0({\mathbb Z}\pi) \, .
\end{equation}
We now use the ``dual'' handle decomposition ${\mathcal H}^*$ of ${\mathcal H}$ (see e.g. \cite[p.~394]{MilnorWT}), producing a 
chain complex $C_j({\mathcal H}^*)$, whose homology is also that of $\tilde M$. By \remref{Rj>k}, one has 
\beq{Pdua-e2}
w_k(M) = w_{n-k}(M) = (-1)^{n-k}[B_{n-k-1}({\mathcal H}^*)] \in \tilde K_0({\mathbb Z}\pi) \, .
\end{equation}
The correspondence $\Theta$ associating to a $j$-handle $e$ its dual ($n-j$)-handle $\Theta(e)$ 
produces a chain isomorphism
$$
\dia{
C_{n-k+1}({\mathcal H}^*) \ar[r]^{\partial} \ar[d]^{\Theta}_\approx & C_{n-k}({\mathcal H}^*) \ar[d]^{\Theta}_\approx 
\ar[r]^{\partial} &  C_{n-k-1}({\mathcal H}^*) \ar[d]^{\Theta}_\approx 
\\
C_{k-1}({\mathcal H})^* \ar[r]^{\partial^*}   & C_{k}({\mathcal H})^* \ar[r]^{\partial^*} 
&  C_{k+1}({\mathcal H})^* 
}
$$
Therefore,
\beq{Pdua-e3}
B_{n-k-1}({\mathcal H}^*) \approx {\rm Image\,}\big(\partial^*\: C_{k}({\mathcal H})^* \to C_{k+1}({\mathcal H})^*\big) \, .
\end{equation}
Since $B_{k-1}({\mathcal H})$ is projective, the exact sequence
\beq{Pdua-e4}
0 \to Z_k({\mathcal H}) \to C_k({\mathcal H}) \fl{\partial} B_{k-1}({\mathcal H}) \to 0
\end{equation}
splits. Passing to the dual modules, we get the split exact sequence
\beq{Pdua-e5}
0 \to B_{k-1}({\mathcal H})^* \fl{\partial^*} C_k({\mathcal H})^* \to Z_k({\mathcal H})^* \to 0  \ . 
\end{equation}
Therefore,
$$
\begin{array}{rcll}
 (-1)^{n-k} w_k(M) &=& [B_{n-k-1}({\mathcal H}^*)] & \comeq{by \eqref{Pdua-e2}} \\[2mm] &=&
 [Z_k({\mathcal H})^*] & \comeq{by \eqref{Pdua-e3} and \eqref{Pdua-e5} } \\[2mm] &=&
 [Z_k({\mathcal H})]^* \\[2mm] &=&
 -[B_{k-1}({\mathcal H})]^*  & \comeq{by \eqref{Pdua-e4}} \\[2mm] &=&
 (-1)^{k+1} w_k(M)^*  & \comeq{by \eqref{Pdua-e1}  ,}
\end{array}
$$
proving that $w_k(M)=(-1)^{n+1}w_k(M)^*$. 
\end{proof}

\subsection{Realization of antisimple obstructions}\label{Srea}

Let $\pi$ be a finitely presented group and let $\omega\:\pi\to\{\pm 1\}$ be a homomorphism.
In this section, we shall realize classes in $\tilde K_0({\mathbb Z}\pi,\omega)$ as antisimple obstruction of a 
\has{k} closed manifold $M$. Making sense of this question requires some identification of $(\pi_1(M),\omega_M)$
with $(\pi,\omega)$. More strongly, we will fix the stable normal ($k-1$)-type of $M$, using the notion
of an $(Y,\eta)$-referred manifold (see~\ref{MPfnormal}). We will thus consider the following

\begin{Problem}\label{ProbRea}
Let  $Y$ be a connected finite cell complex of dimension $k-1$ and let $\eta$ be a stable vector bundle over $Y$.
Let ${\mathcal P}\in\tilde K_0({\mathbb Z}\pi_1(Y),\omega_\eta)$. Does there exist an
$(Y,\eta)$-referred closed manifold $(M,g)$ such that $M$ is \has{k} and satisfies $g_*(w_k(M))={\mathcal P}$?
\end{Problem}

Problem~\ref{ProbRea} will partly answered in \thref{Trea} below (see also \proref{PreaPoin}).
Before going to this, we show that a $(Y,\eta$)-referred compact manifold enjoys a decomposition involving
a stable $\eta$-thickening of $Y$ in the sense of~\ref{coThick}.

\begin{Proposition}\label{Pdecomp}
Let $Y$ be a connected finite cell complex of dimension $k-1$ and
let $\eta$ be a stable vector bundle over $Y$.
Let $(M,g)$ be an $(Y,\eta$)-referred compact manifold of dimension $r\geq 2k\geq 6$. 
Then, there are codimension-$0$ compact submanifolds $N$ and $T$ of $M$ giving a decomposition
\beq{Pdecomp-dec}
M= N\cup T \ , \ N\cap T = {\rm Bd} N \ , \ {\rm Bd} T = {\rm Bd}  M \amalg {\rm Bd} N  \ ,
\end{equation}
such that $(N,g_{|N})$ is a stable $\eta$-thickening of $Y$ and $g_{g|T}$ is an $(Y,\eta$)-referred map.
Moreover, for any ${\mathbb Z}\pi_1(Y)$-module $R$, the restriction homomorphism \\
$\rho^j\:H^j(M,T;R)\to H^j(M;R)$
satisfies the following properties:
\begin{itemize}
 \item[(i)] $\rho^j$ is an isomorphism for $k\leq j\leq r-k-1$ and $\rho^{r-k}$ is injective;
 \item[(ii)] if $M$ is a closed manifold, then $\rho^j$ is an isomorphism for $k\leq j\leq r-k$.
 Consequently, $M$ is \has{k} if and only if
$T$ is \has{k}, in which case $g_*(w_k(M))=(g_{|T})_*(w_k(T))$ in $\tilde K_0({\mathbb Z}\pi_1(Y),\omega_\eta)$.
\end{itemize}
\end{Proposition}

\begin{proof}
The $(Y,\eta)$-reference map $g\:M\to Y$ admits a homotopy section $\gamma\:Y\to M$ 
(see \lemref{Lrefretr} and its proof).
Up to homotopy equivalence, one may assume that $Y$ is a finite simplicial complex and that $\gamma$ is an embedding
into a ${\mathcal C}^\infty$-triangulation of $M$ with $\gamma(Y)\cap \partial M=\emptyset$. Then, $\gamma(Y)$ has a smooth regular neighborhood $N$ in $M\setminus \partial M$ \cite{HirschNhb}.
Hence, the embedding $\gamma$ factors through an embedding $\gamma^1\:Y\to N$ which is a simple homotopy equivalence,
and $g_{|N}\:N\to Y$ is a homotopy inverse of $\gamma^1$. It follows that $g_{|N}$ is a homotopy equivalence,
which is simple by the composition rule for the Whitehead torsion \cite[Lemma~7.8]{MilnorWT}.
Since $g$ is $\eta$-normal, so is $g_{|N}$.
Since $r\geq 2k\geq 6$, one has $\pi_1({\rm Bd} N)\approx\pi_1(N)$ and, by Poincar\'e\  duality,
$H_j(\tilde N,\widetilde{{\rm Bd} N})\approx H^{r-j}(N;{\mathbb Z}\pi_1(N))\approx H^{r-j}(Y;{\mathbb Z}\pi_1(Y))=0$ for $j\leq n-k$.
Therefore, the pair $(N,{\rm Bd} N)$ is ($r-k$)-connected and $(N,g_{|N})$ is a stable $\eta$-thickening of $Y$. 
We thus get the decomposition \eqref{Pdecomp-dec} by setting $T=M{\scriptstyle\setminus} {\rm int}N$.

By van Kampen theorem, $\pi_1g_{g|T}\:\pi_1(T)\to\pi_1(Y)$ is an isomorphism. Since
$H_j(\tilde M,\tilde T)\approx H_j(\tilde N,\widetilde{{\rm Bd} N})$ the pair $(M,T)$ is ($r-k$)-connected.
Therefore, $g_{|T}$ is an $Y$ reference map, which is $\eta$-normal since $g$ is so.

By excision and Poincar\'e\  duality, one has isomorphisms
\beq{Pdecomp-eq0}
H^j(M,T;R) \fl{\approx} H^j(N,{\rm Bd} N;R) \fl{\approx} H_{r-j}(N;R) = 0 \ \hbox{ for } \ j\leq r-k \, ,
\end{equation}
which proves (i). 

When ${\rm Bd} M=\emptyset$, we claim that the restriction homomorphism $\beta\:H^*(M,T;R)\to H^*(M;R)$ is injective. 
Indeed, \eqref{Pdecomp-eq0} sits in a commutative diagram
$$
\dia{
H^{j}(M,T;R) \ar[d]^\beta \ar[r]^(0.46){\rm\tiny excis.}_(0.45)\approx &  H^{j}(N,{\rm Bd} N;R) 
\ar[r]^(0.54){\rm PD}_(0.54)\approx & H_{r-j}(N;R)  \ar[d]^{\sigma_*}
\\ 
H^{j}(M;R) \ar[rr]^{\rm PD}_\approx && H_{r-j}(M;R) 
}
$$
where $\sigma_*$ is induced by the inclusion $\sigma\:N\to M$. As the reference map $g\:M\to Y$ produces
a homotopy retraction of $\sigma$, the homomorphism  $\sigma_*$ is injective. Therefore,
the cohomology sequence of the pair $(M,T)$ splits into short exact sequences
$$
0 \to H^j(M,T;R) \to H^j(M;R) \to H^j(T;R) \to 0 \, .
$$
This, together with \eqref{Pdecomp-eq0} proves the first assertion of (ii). 
Finally, one has $H_*(\tilde M,\tilde T)\approx H_*(\tilde N,\widetilde{{\rm Bd} N})$, so the pair $(M,T)$ is ($r-k$)-connected.
As $r-k\geq k$, one has $g_*(w_k(M))=(g_{|T})_*(w_k(T))$ by \lemref{WL2bb}.
\end{proof}

Our first result concerning Problem~\ref{ProbRea} is an affirmative answer in the category of Poincar\'e\  spaces. Recall that, according to our definition in~\ref{Pesp}, a Poincar\'e\  space or pair is finitely dominated.


\begin{Proposition}\label{PreaPoin}
Let $Y$ be a connected finite cell complex of dimension $k-1\geq 2$ and let $\eta$ be a stable vector bundle over $Y$.
Let ${\mathcal P}\in \tilde K_0({\mathbb Z}\pi_(Y),\omega_\eta)$ and let $r\geq 2k$ be an integer.
Then, there exists an $(Y,\eta$)-referred Poincar\'e\  space 
$({\mathbb P},q)$ of formal dimension $r$, 
such that ${\mathbb P}$ is \has{k} satisfying $w_k({\mathbb P})={\mathcal P}$ and ${\rm Wall}({\mathbb P}) = {\mathcal P} + (-1)^{r}{\mathcal P}^*$ 
(see Convention~\ref{Conv} below). 
Moreover, ${\mathbb P}$ admits up to homotopy an $\eta$-normal Poincar\'e\  decomposition (see~\ref{PdecNor})
\beq{PreaPoin-dec}
{\mathbb P} \simeq N \cup {\mathbb T} \ , \ N\cap {\mathbb T} = {\rm Bd} N  = {\rm Bd}{\mathbb T} \ ,
\end{equation}
where $(N,q_{|N})$ is a stable $\eta$-thickening of dimension $r$ of $Y$.
The Poincar\'e\  space ${\mathbb T}$ 
is \has{k} and satisfies $w_k({\mathbb T})=w_k({\mathbb P})$ and ${\rm Wall}({\mathbb T}) = {\rm Wall}({\mathbb P})$. 
\end{Proposition}

\begin{Convention}\label{Conv}\rm
Let $Y$ be a connected finite cell complex. 
Let $(A,\alpha)$ be an $(Y,\eta)$-referred space. Then, $\tilde K_0$-equalities are understood to hold in 
$\tilde K_0(\pi_1(Y),\omega_\eta)$, using the map $\alpha$ and its restrictions to subspaces of $A$. For instance, 
the equation $w_k({\mathbb T})=w_k({\mathbb P})$ in \proref{PreaPoin} should be understood as $(q_{|{\mathbb T}})_*(w_k({\mathbb T}))=q_*(w_k({\mathbb P}))$.
\hfill \ensuremath{\Box}
\end{Convention}


\begin{proof}
We set $(\pi,\omega)=(\pi_1(Y),\omega_\eta)$ and we use Convention~\ref{Conv} throughout the proof without notice.
Consider the infinite $k$-dimensional CW-complex $L$ of \proref{PW4}. By \remref{WR10}, 
$L$ is the direct limit $L_\infty$ of a direct system 
\beq{Trea-e0}
Y\hookrightarrow L_1\hookrightarrow L_2 \hookrightarrow \cdots \hookrightarrow L_\infty=L \fl{g} Y
\end{equation}
of finite cell complexes of dimension $k$. The $Y$-referred map $g\:L\to Y$ is the direct limit of 
$Y$-referred maps $g_j\:L_j\to Y$. The composed map in \eqref{Trea-e0} is ${\rm id}_Y$.

We now see that Direct system \eqref{Trea-e0} may be realized up to homotopy equivalence by a system of 
($r+1$)-dimensional manifolds (compact with boundary when $j<\infty$), getting a commutative diagram
\beq{Trea-e1}
\dia{ 
Y \ar@{^{(}->}[r] \ar@{<-}[d]^(.5){\phi_0}
& 
L_1 \ar@{^{(}->}[r] \ar@{<-}[d]^(.5){\simeq}_(.5){\phi_1} & L_2 \ar@{^{(}->}[r]
\ar@{<-}[d]^(.5){\simeq}_(.5){\phi_2}
& \cdots \ar@{^{(}->}[r] & L_\infty=L     \ar@{<-}[]!<3.3ex,-2ex>;[d]!<3.3ex,1ex>^(.5){\simeq}_(.5){\phi} 
\ar[r]^(0.60)g & Y
\\
U_0 \ar@{^{(}->}[r] & U_1 \ar@{^{(}->}[r]^{\alpha_0} & U_2 \ar@{^{(}->}[r]^{\alpha_1} & \cdots \ar@{^{(}->}[r] & U_\infty=U
}
\end{equation}
where the vertical arrows are homotopy equivalences (simple for $j<\infty$).
We start with $(U_0,\phi_0)$ being a stable $(Y,g_0^*\eta)$-thickening of dimension $r+1$. We can thus choose a stable
isomorphism from the stable normal bundle on $U_0$ and $\phi_0^*(g_0^*\eta)$, or, equivalently, 
a stable trivialization ${\mathcal F}_0$ of $TU_0\oplus \phi_0^*(g^*\eta)$ 
(i.e. $(\phi_0,{\mathcal F}_0)$ is a {\it normal map} in the language of \cite{WallBookSurgery}). 
Recall that the inclusions in \eqref{Trea-e0} are obtained by attaching cells of dimension $k-1$ and $k$.
By surgery below the middle dimension
on ${\rm Bd} U_0$, \cite[Chapter~1]{WallBookSurgery}, the data $(U_0,\phi_0,{\mathcal F}_0)$ thus determines $(U_1,\phi_1,{\mathcal F}_1)$, then
$(U_2,\phi_2,{\mathcal F}_2)$, etc. Note that $(U_j,\phi_j)$ is the stable 
$(L_j,g_j^*\eta)$-thickening of dimension $r+1$ and ${\mathcal F}_j$ (obtained from ${\mathcal F}_0$ without further choice) 
determines the embedding $\alpha_j\:U_j\to U_{j+1}$.

At the limit when $j\to\infty$, one gets an open manifold $U$ together with an 
$(Y,\eta)$-reference map  $g\kern .7pt {\scriptstyle \circ} \kern 1pt\phi\: U\to Y$. 
The manifold $U$ has one end $\epsilon$, determined by the cofinal system of neighborhoods 
$E_j=U{\scriptstyle\setminus} {\rm int} U_j$. The inclusions $E_{j+1}\hookrightarrow E_j\hookrightarrow U$ induce an isomorphism
on the fundamental groups. As $U\simeq L$ is finitely dominated by \proref{PW4} and ${\rm Bd} E_j$ is a closed manifold,
the open manifold $E_j$ is finitely dominated (see \cite[Complement~6.6]{SieTh}). Therefore, 
the end $\epsilon$ is {\it tame} (see~\ref{endsMan}).

As mentioned in \remref{RWallObs}, $w_k(L)=w_k(U)$ is Wall's finiteness obstruction ${\rm Wall}(U)$. 
One has ${\rm Wall}(U)={\rm Wall}(Z_i)$ by the sum theorem \cite[Theorem~6.5]{SieTh} and thus, by \cite[Proposition~6.11]{SieTh},
${\rm Wall}(U)$ coincides with $\sigma(\epsilon)$, the Siebenmann obstruction  to complete $U$ into a  
compact manifold with boundary. 

By \proref{Pcompletion}, there exists a Poincar\'e\  pair $({\mathbb U},{\mathbb P})$, 
with $\pi_1({\mathbb P})\approx\pi_1({\mathbb U})$, together with a homotopy equivalence $\alpha\:U\to{\mathbb U}$.
Recall from \proref{PW4} that ${\mathbb U}\simeq L$ is is cohomology silent  in degrees $\geq k$. By  Poincar\'e\  duality, one has
$$
H_j(\widetilde{{\mathbb U}},\tilde{\mathbb P}) \approx H_j({\mathbb U},{\mathbb P};{\mathbb Z}\pi) \approx H^{r+1-j}({\mathbb U};{\mathbb Z}\pi) = 0  \quad {\rm for } \
j\leq n+1-k \, .
$$
Hence, the pair $({\mathbb U},{\mathbb P})$ is ($r+1-k$)-connected. Since $U\simeq {\mathbb U}$, the map $g\kern .7pt {\scriptstyle \circ} \kern 1pt\phi\: U\to Y$ extends 
to an $Y$-reference map $q\:{\mathbb U}\to Y$. By the connectivity of $({\mathbb U},{\mathbb P})$, the restriction of $q$ to ${\mathbb P}$ 
(also called $q$) is an $Y$-reference map. Also, this connectivity implies that ${\mathbb P}$ is \has{k} and that 
$w_k({\mathbb P})=w_k(L)={\mathcal P}$.
As $\alpha\:U\to {\mathbb U}$ is a homotopy equivalence, one has ${\rm Wall}({\mathbb U})={\rm Wall}(U)={\rm Wall}(L)={\mathcal P}$. The formula 
\beq{wobd}
{\rm Wall}({\mathbb P}) = {\mathcal P} + (-1)^{r}{\mathcal P}^*
\end{equation}
is asserted in \proref{Pcompletion}.

We now prove that the reference map $q\:{\mathbb U}\to Y$ is $\eta$-normal (and so is $q\:{\mathbb P}\to Y$ by Part (3) of 
\lemref{Lnormal}). We use the ideas of Pedersen-Ranicki \cite[proof of Lemma~3.1]{PeRa} (with different notations).

By Siebenmann's famous theorem \cite[Theorem~7.5]{SieTh}, 
the open manifold $U\times S^1$ is the interior of a compact manifold $A$ with boundary
$V={\rm Bd} A$. The Poincar\'e\  duality argument used above to prove that the pair $({\mathbb U},{\mathbb P})$ is ($r+1-k$)-connected proves that 
$(A,V)$ is ($r+1-k$)-connected.
The projections of $U\times S^1$ onto $S^1$ together with the map $g\kern .7pt {\scriptstyle \circ} \kern 1pt\phi\:U\to Y$ gives rise to a map 
$g_A\:A\to Y$ which is $\eta$-normal and a map $p_A\:A\to S^1$. Let $(\hat A,\hat V)\to (A,V)$ be the infinite cyclic cover associated to the map $p_A$.
It is proven in \cite[proof of Lemma~3.1]{PeRa} that there are homotopy equivalences 
\begin{itemize}
 \item[(1)] $h_1\: (A,V)\fl{\simeq} ({\mathbb U},{\mathbb P})\times S^1$
 \item[(2)] $h_2\: \hat{V} \fl{\simeq} {\mathbb P}$.
\end{itemize}
The composite map
\beq{Prea-eq7}
A \fl{h_1} {\mathbb U}\times S^1 \fl{h_1'} A \fl{g_A}  Y \, , 
\end{equation}
where $h_1'$ is a homotopy inverse of $h_1$, is homotopic to $g_A$ and is thus $\eta$-normal. By \lemref{Lnormal},
we deduce that $g_A\kern .7pt {\scriptstyle \circ} \kern 1pt h_1'$ is $\eta$-normal, and so is its restriction to ${\mathbb U}\times pt$. 
But this restriction is homotopic to the $Y$-reference map $q\:{\mathbb U}\to Y$, which is thus $\eta$-normal.

It remains to construct the decomposition of \eqref{PreaPoin-dec}. We check that the map 
$\bar g_A= (g_A,p_A)\: A\to Y\times S^1$ is an $(Y\times S^1,\bar\eta)$-reference map for $A$,
where $\bar\eta$ is induced from $\eta$ by the projection to $Y$. As the pair $(A,V)$ is 
is ($r+1-k$)-connected, the restriction of $\bar g_A$ to $V$ is a an $(Y\times S^1,\bar\eta)$-reference map for $V$.
By \proref{Pdecomp}, there is a manifold decomposition $V= N_0 \cup T_0$, where $(N_0,\bar g_A)$ is a stable 
$\bar\eta$-thickening of $Y\times S^1$. As $\bar g_A|U\times S^1 \simeq q\times {\rm id}_{S^1}$,
\proref{PWallThick} implies that $N_0$ is diffeomorphic to $N\times S^1$, 
where $(N,q)$ is a stable $\eta$-thickening of $Y$.

Using the homotopy equivalence $h_2\:\hat V \to {\mathbb P}$, one thus gets a decomposition 
${\mathbb P}\simeq (N\times {\mathbb R}) \cup \hat T_0$.
The manifold $N$ is diffeomorphic to $N\cup_{{\rm Bd} N} C$ where $C={\rm Bd} N \times [0,1]$. Form the space
$$
{\mathbb T} =  \big(C\times \{0\}\big) \cup \hat T_0  \subset (N\times{\mathbb R}) \cup \hat T_0   \, .
$$
containing a subspace ${\rm Bd}{\mathbb T}={\rm Bd} (N\times \{0\})$. One has the decomposition 
$$
{\mathbb P} \simeq (N\times\{0\})\cup {\mathbb T}\simeq N\cup{\mathbb T} \, .
$$

As the inclusions ${\rm Bd} N\hookrightarrow N\hookrightarrow {\mathbb P}$ induce isomorphisms 
$\pi_1({\rm Bd} N)\approx\pi_1(N)\approx\pi_1({\mathbb P})$, it follows from van Kampen's theorem and the property of
amalgamated products (see \cite[Theorem~2.6, p.~187]{LySchu}) that $\pi_1({\rm Bd} N)\approx\pi_1({\mathbb T})\approx\pi_1({\mathbb P})$.
By excision, $H_*(\tilde N,\widetilde{{\rm Bd} N})\approx H_*(\tilde {\mathbb P},\tilde {\mathbb T})$. Therefore, the pair $({\mathbb P},{\mathbb T})$ is ($r-k$)-connected and thus the map of $q_{|{\mathbb T}}\:{\mathbb T}\to Y$ is an $Y$-reference map.

As the map $q\:N\to Y$ is $\eta$-normal, that \eqref{PreaPoin-dec} is an $\eta$-normal Poincar\'e\  decomposition
follows from \corref{CPdecNor2}.

Finally, the equation ${\rm Wall}({\mathbb T}) = {\rm Wall}({\mathbb P})$
comes from the sum theorem \cite[Theorem~6.5]{SieTh}. That ${\mathbb T}$ is \has{k} and the 
equation $w_k({\mathbb T})=w_k({\mathbb P})$ are proven by the corresponding arguments given in the proof of 
Part (ii) of \proref{Pdecomp}.
\end{proof}


We now turn our attention to Problem~\ref{ProbRea} as it is stated, that is in the category of closed manifolds.
Let $Y$ be a connected finite cell complex of dimension $k-1$ and
let $\eta$ be a stable vector bundle over $Y$. Set $(\pi,\omega)=(\pi_1(Y),\omega_\eta)$.
Let $(M,g)$ be an $(Y,\eta$)-referred closed manifold of dimension $r$ such that $M$ is \has{k}.
By \proref{Pdua}, its antisimple obstruction is $r$-self-dual, i.e. satisfies $g_*(w_k(M))=(-1)^{r+1}g_*(w_k(M))^*$.
It thus defines a class
\beq{tate}
[g_*(w_k(M))] \in H^{r+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi,\omega)) 
\end{equation}
in the Tate cohomology group 
\small
$$
H^{r+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi,\omega)) = 
\{{\mathcal P}\in \tilde K_0({\mathbb Z}\pi,\omega)\mid {\mathcal P}=(-1)^{r+1}{\mathcal P}^*\}\big/ \{{\mathcal P}+(-1)^{r+1}{\mathcal P}^*\} \, .
$$
\rm
This Tate cohomology group is an abelian group of exponent two, which occurs 
in the Ranicki exact sequence
\beq{ranicki}
L_{r+1}^h(\pi,\omega) \to L_{r+1}^p(\pi,\omega) \to
H^{r+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi,\omega)) \fl{\delta_{R}} L_{r}^h(\pi,\omega) \to L_{r}^p(\pi,\omega)
\end{equation}
where $L_j^h$ (respectively: $L_j^p$) are the surgery obstruction groups for surgery data with target
a finite (respectively: finitely dominated) Poincar\'e\  complex. This sequence was first obtained by Ranicki
using his algebraic setting of surgery \cite[Theorem~4.3]{RanFound}. A geometrical version  
of Sequence~\eqref{ranicki} will be used, which was provided by Pedersen-Ranicki \cite[p.~243]{PeRa}.
Our partial solution of Problem~\ref{ProbRea} is the following

\begin{Theorem}\label{Trea}
Let $Y$ be a connected finite cell complex of dimension $k-1\geq 2$ and let $\eta$ be a stable vector bundle over $Y$.
Set $(\pi,\omega)=(\pi_1(Y),\omega_\eta)$ and let $r\geq 2k$ be an integer.
Let ${\mathcal P}\in \tilde K_0({\mathbb Z}\pi,\omega)$ be an $r$-self-dual element.
Suppose that 
$[{\mathcal P}]\in H^{r+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi,\omega))$ belongs to the kernel of the homomorphism $\delta_R$
of \eqref{ranicki}.
Then, there exists an $(Y,\eta$)-referred closed manifold $(M,q)$ of dimension $r$ 
such that $M$ is \has{k} and satisfies $q_*(w_k(M))={\mathcal P}$.
\end{Theorem}

\begin{proof}
We consider the Poincar\'e\  pair $({\mathbb U},{\mathbb P})$ of the proof of \proref{PreaPoin}, equipped with its 
$(Y,\eta)$-reference map $q\:{\mathbb U}\to Y$ and its restriction $q\:{\mathbb P}\to Y$.
One has ${\rm Wall}({\mathbb U})={\mathcal P}$ and ${\rm Wall}({\mathbb P}) = {\mathcal P} + (-1)^{r}{\mathcal P}^*$ (using Convention~\ref{Conv}).
Since ${\mathcal P}=(-1)^{r+1}{\mathcal P}^*$ by hypothesis, 
one has ${\rm Wall}({\mathbb P})=0$ and, by \cite[Theorem~F]{WallFin}, ${\mathbb P}$ is homotopy equivalent to a finite complex.
Thus, by changing the pair $({\mathbb U},{\mathbb P})$ by a homotopy equivalence, one may suppose 
that  ${\mathbb P}$ is a {\it finite} Poincar\'e\  space. 

Being $\eta$-normal, the map $q$ determines a surgery problem with target ${\mathbb U}$ (see~\ref{MPfnormal}).
In this language and using the point of view of \cite[pp.~242-44]{PeRa}, 
the data $({\mathbb U},q)$ represents a class
in the geometric L-group $L_{r+1}^{1,p,h}(Y)$. The definition of the latter is akin to that of 
$L_{r+1}^{1}(Y)=L_{r+1}^{1,s}(Y)$ (also working for $L_{r+1}^{1,h}(Y)$) given in \cite[Chapter~9]{WallBookSurgery}.
With these geometric L-groups, Pedersen and Ranicki obtain, for $r\geq 5$, an isomorphism $\sigma_*$ of exact sequences
\small
\beq{pedranicki}
\dia{
L_{r+1}^{1,h}(Y) \ar[r] \ar[d]_\approx^(0.45){\sigma^h}    & 
L_{r+1}^{1,p}(Y) \ar[r] \ar[d]_\approx^(0.45){\sigma^p} &
L_{r+1}^{1,p,h}(Y) \ar[r]^(0.5){\delta_{PR}} \ar[d]_\approx^(0.45){\sigma^{p,h}}&  
L_{r}^{1,h}(Y) \ar[r] \ar[d]_\approx^(0.45){\sigma^h} & 
L_{r}^{1,h}(Y) \ar[d]_\approx^(0.45){\sigma^p}
\\ 
L_{r+1}^h(\pi) \ar[r] & L_{r+1}^p(\pi) \ar[r] &
H^{r+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi)) \ar[r]^(0.65){\delta_{R}} &  L_{r}^h(\pi) \ar[r] & L_{r}^p(\pi)
}
\end{equation}  
\rm
where the bottom line is Ranicki's exact sequence~\eqref{ranicki}, writing $L^*_*(\pi)$ for $L^*_*(\pi,\omega)$
and $\tilde K_0({\mathbb Z}\pi)$ for $(\tilde K_0({\mathbb Z}\pi),\omega)$.
The arrows in \eqref{pedranicki} satisfy the following properties
\begin{itemize}
 \item $\sigma^{p,h}({\mathbb U},q) = [{\rm Wall}({\mathbb U})]$;
 \item $\delta_{PR}$ sends the class of $({\mathbb U},q)$ to that of $({\mathbb P},q)$.
  \item $\sigma^h$ is the surgery obstruction measured in $L_r^h(\pi)$.
\end{itemize}
Hence, in the language of~\ref{MPfnormal}, one has 
$$
\sigma({\mathbb P},q)=\sigma^h\kern .7pt {\scriptstyle \circ} \kern 1pt\delta_{PR}({\mathbb U},q) = \delta_R\kern .7pt {\scriptstyle \circ} \kern 1pt\sigma^{p,h}({\mathbb U},q) = \delta_R({\mathcal P}) \, . 
$$
By hypothesis, $[{\rm Wall}({\mathbb U})]=[{\mathcal P}]\in\ker\delta_R$, hence, $\sigma({\mathbb P},q)=0$.
This means that there exists a $q^*\eta$-normal homotopy equivalence $\theta\:M\to{\mathbb P}$ 
where $M$ is a closed $r$-manifold. The latter thus admits the $(Y,\eta)$-reference map $q\kern .7pt {\scriptstyle \circ} \kern 1pt\theta$.
Being homotopy equivalent to ${\mathbb P}$, the closed manifold $M$ is \has{k} and $w_k(M)=w_k({\mathbb P})={\mathcal P}$ by \lemref{WL2bb}.
\end{proof}

\subsection{Cohomology antisimple cobordisms}\label{SScob}

Let $n\geq 2k\geq 6$ be integers. A cobordism $(W,M,M')$ is called \dfn{\has{k}} if the following three conditions hold.
\begin{itemize}
\item $M$ and $M'$ are closed connected manifold of dimension $n$ which are \has{k}; 
\item The pairs $(W,M)$ and $(W,M')$ are weakly ($k-1$)-connected (see \ref{connectivity});
\item $W$ is \ccs{k}{n-k}. 
\end{itemize}
For example, an $h$-cobordism between closed connected \has{k} mani\-folds is \has{k}.

Recall from \eqref{tate} that a closed connected manifold $M$ of dimension $n$ which is \has{k} 
defines a class 
$[w_k(M)] \in H^{n+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi_1(M),\omega_M))$.

\begin{Proposition}\label{PcobL2}
Let $(W,M,M')$ be a \has{k} cobordism between closed connected \has{k} manifolds of dimension $n\geq 2k\geq 6$. 
Then $[w_k(M)]=[w_k(M')]$ in $H^{n+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi_1(W),\omega_W)$. 
\end{Proposition}

\begin{proof}
The proof involves two steps.

\sk{1}\noindent{\it Step 1: Reduction to the case where $(W,M)$ and $(W,M')$ are ($k-1$)-connected.} \  
The cobordism $(W,M,M')$ admits a handle decomposition of the form
$$
M\times I = {\mathcal H}_{k-1} \subset  {\mathcal H}_k \subset\cdots\subset {\mathcal H}_{n+2-k} = W \ ,
$$
where ${\mathcal H}_j$ is the union of handles of index $\leq j$. Indeed, since the pair $(W,M)$ is weakly ($k-1$)-connected, it is 
($k-2$)-connected and the procedure of eliminating handles in a cobordism (see \cite{KeBMS}), 
applied to a given handle decomposition ${\mathcal H}$, permits us to 
to get rid of the handles of index $<k-1$ and $>n+2-k$. The same procedure applied to the dual handle decomposition ${\mathcal H}'$
makes it of the form
$$
M'\times I = {\mathcal H}_{k-1}' \subset  {\mathcal H}_k' \subset\cdots\subset {\mathcal H}_{n+2-k}' = W \ .
$$
The manifold $V={\mathcal H}_{k-1}$ and $V'={\mathcal H}_{k-1}'$ give cobordisms $(V,M,M_1)$ and $(V',M',M_1')$ having 
handle decomposition involving only handles of index $k-1$. One has $W=V\cup W_1\cup V'$, 
defining a cobordism $(W_1,M_1,M_1')$. The inclusions of the above manifolds into $W$ all induce isomorphisms on
fundamental groups with orientation characters, so all these will be identified with $(\pi,\omega)=(\pi_1(W),\omega_W)$.
One has the following facts.
\begin{itemize}
\item[(1)] {\it $V$ is \ccs{k}{n-k}.} \ Indeed, for any ${\mathbb Z}\pi$-module $R$, 
the exact sequence $H^{j}(V,M;R) \to H^j(V;R) \to H^j(M;R)$ implies the claim, since
$M$ is \has{k} and $H^{j}(V,M;R)=0$ if $j\neq k-1$.
\item[(2)] {\it $w_k(V)=w_k(M)$.} \ Let $N$ be the union of handles of index $\leq k-1$ in a handle decomposition of $M$
and let $T=M{\scriptstyle\setminus} {\rm int N}$. Since the pair $(M,N)$ is ($k-1$)-connected, the 
the ($k-1$)-handles of $V$ might be attached on $N$, producing a cobordism $N_+$ from $N$, and 
$V=N_+ \cup_{{\rm Bd} N \times I} (T\times I)$. The pair $(V,N_+)$ is ($k-1$)-connected and 
$$
H_k(\tilde V,\tilde N_+) \approx H_k(\tilde T,\widetilde{{\rm Bd} N}) \approx H_k(M,N) \, .  
$$
As $w_k(V)=[H_k(\tilde V,\tilde N_+)]$ and $w_k(M) =[H_k(\tilde M,\tilde N)]$, the claim follows.
\item[(3)] {\it $M_1$ is \has{k} and $w_k(M_1)=w_k(M)$.} \ This follows from (1) and (2), since $(V,M_1)$
is ($n+2-k$)-connected and $n+2-k>n-k>k$. 
\item[(4)] {\it $W_1$ is \ccs{k}{n-k}.} \ As $(W,W_1)$ is ($n+2-k$)-connected,
this follows from the corresponding property of $W$. 
\end{itemize}
Points (1)--(3) also hold true for $(V',M')$. Therefore, it is equivalent to prove \proref{PcobL2} for $(W,M,M')$
or for  $(W_1,M_1,M_1')$, so we may assume that $(W,M)$ and $(W,M')$ are ($k-1$)-connected.


\sk{1}\noindent{\it Step 2: Proof of \proref{PcobL2} when $(W,M)$ and $(W,M')$ are ($k-1$)-connected.} \ 
Applying \lemref{LXT} to both ends of $W$ gives the equalities
\beq{PcobL2-e1}
w_k(M) + [H_k(W,M)] = w_k(W) = w_k(M') + [H_k(W,M')]  \, .
\end{equation}
We will compute  $[H_k(W,M')]$. As in Step 1, one checks that the 
cobordism $(W,M,M')$ admits a handle decomposition of the form
$$
M\times I = {\mathcal H}_k \subset  {\mathcal H}_{k+1} \subset\cdots\subset {\mathcal H}_{n+1-k} = W \ ,
$$
where ${\mathcal H}_j$ is the union of handles of index $\leq j$. 
Since $n\geq 2k$, one has $n+2-k>k$. The chain complex $C_*(\tilde W,\tilde M)$ is thus the equivalent to 
$$
0 \to C_{n+1-k}(\tilde {\mathcal H}) \to\cdots\to C_k(\tilde {\mathcal H}) \to 0 \, .
$$
Write $C_j=C_j(\tilde{\mathcal H})$, $Z_j=Z_j(\tilde{\mathcal H})$ and $B_j=B_j(\tilde{\mathcal H})$.
For $k<j<n+1-k$, one has $H_j(\tilde W)=0=H_j(\tilde M)$ by \lemref{LW1}.
Therefore, for $k<j<n+1-k$, $H_j(\tilde W,\tilde M)=0$, and thus $Z_j=B_j$. 
One then has exact sequences
$$
\begin{array}{lll}
 0\to B_{k}\to C_k \to H_k(\tilde W,\tilde M)\to 0 \\
 0\to B_{k+1}\to C_{k+1} \to B_{k}\to 0\\
 \hskip 9mm \cdots \ etc \, \cdots \\
 0\to H_{n+1-k}(\tilde W,\tilde M) \to C_{n+1-k} \to B_{n-k}\to 0
\end{array}
$$
As $H_k(\tilde W,\tilde M)$ is finitely generated and projective, so are all the above modules
and one has 
$$
[H_{k}(\tilde W,\tilde M)]= -[B_k]=[B_{k+1}]=-[B_{k+2}]\cdots = (-1)^{r+1}[B_{k+r}]  \, .
$$
As, $[H_{n+1-k}(\tilde W,\tilde M)]=-B_{n-k}$, one has  
\beq{PcobL2-e2}
[H_{n+1-k}(\tilde W,\tilde M)] = (-1)^{n}[H_{k}(\tilde W,\tilde M)] \, . 
\end{equation}
Now, by Poincar\'e\  duality, 
$$
H_{n+1-k}(\tilde W,\tilde M)=H_{n+1-k}(W,M;{\mathbb Z}\pi)
\approx H^{k}(W,M';{\mathbb Z}\pi) \, .
$$
To compute the latter, one uses the handle decomposition ${\mathcal H}'$ of $(W,M')$ which 
is the dual handle decomposition to ${\mathcal H}$. 
Then, ${\mathcal H}'$ has no handle of index $<k$. Therefore
\begin{eqnarray*}
 H^{k}(W,M';{\mathbb Z}\pi) &\approx & \ker\big(C_k(\tilde{\mathcal H}')^*\fl{\delta} C_{k+1}(\tilde{\mathcal H}')^*\big) \nonumber
 \\ &=&
\{\alpha\in C_k(\tilde{\mathcal H}')^*\mid\alpha\kern .7pt {\scriptstyle \circ} \kern 1pt\partial=0\} \nonumber \\ &\approx & H_k(\tilde W,\tilde M')^* \, .
\end{eqnarray*}
Thus, $ H_k(\tilde W,\tilde M')\approx H^{k}(W,M';{\mathbb Z}\pi)^*$ and, by \eqref{PcobL2-e1} and \eqref{PcobL2-e2}, one has
\begin{eqnarray*}
w_k(M') &=& w_k(M) + [H_k(\tilde W,\tilde M)] - [H_k(\tilde W,\tilde M')] \\  &=&
w_k(M) + [H_k(\tilde W,\tilde M)] - [H^{k}(W,M';{\mathbb Z}\pi)]^* \\  &=&
w_k(M) + [H_k(W,M)] - [H_{n+1-k}(\tilde W,\tilde M)]^* \\  &=&
w_k(M) + [H_k(W,M)] + (-1)^{n+1} [H_k(W,M)]^*   \ ,
\end{eqnarray*}
which proves the proposition. 
\end{proof}

The rest of this section is devoted to the proof of the following result.

\begin{Theorem}\label{TcobL2}
Let $M$ be a closed connected \has{k} manifold of dimension $n\geq 2k\geq 8$. 
The following conditions are equivalent.
\begin{itemize}
 \item[(a)] $M$ is cohomology $k$-antisimply cobordant to an \as{k} closed manifold.
 \item[(b)] $[w_k(M)]=0$ in $H^{n+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi_1(M),\omega_M)$. 
\end{itemize}
\end{Theorem}


We need some preliminary results before starting the proof.
For two integer $s\leq t$, we denote by $\ecs{t}{s}$ the class of CW-spaces which are \ccs{s}{t}.

\begin{Lemma}\label{Ldececs}
Let $X=X_-\cup X_+$, $X_-\cap X_+=X_0$ be a CW-decomposition. Suppose that $X_\pm$ and $X_0$ are in $\ecs{s}{t}$
and that $H^{s-1}(X_+;R)\to H^{s-1}(X_0;R)$ is onto for any ${\mathbb Z}\pi_1(X)$-module $R$. Then $X\in\ecs{s}{t}$. 
\end{Lemma}

\begin{proof}
Let $R$ be ${\mathbb Z}\pi_1(X)$-module and consider the Mayer-Vietoris sequence
$$
\dia{ 
H^{j-1}(X_0;R) \ar[r]^(.55){\delta^{j-1}} & H^j(X;R) \ar[r] & H^j(X_-;R)\oplus H^j(X_+;R)
}
$$
By our hypotheses, $X_0\in\ecs{s}{t}$ and $\delta^{s-1}=0$, which implies the lemma.
\end{proof}


\begin{Lemma}\label{LcobL2}
Let $Y$ be a connected finite cell complex of dimension $k-1\geq 2$ and let 
$\eta$ be a stable vector bundle over $Y$.
Let $n\geq 2k$ be an integer.
Let ${\mathcal P}\in \tilde K_0({\mathbb Z}\pi_(Y),\omega_\eta)$ being $n$-self-dual.
Suppose that ${\mathcal P}$ represents $0$ in $H^{n+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi(Y),\omega_\eta))$.
Then, there exists an $(Y,\eta$)-referred compact ($n+1$)-dimensional manifold $(A,q)$ (see below) such that
\begin{itemize}
 \item[(a)] $M={\rm Bd} A$ is \has{k} and satisfies $q_*(w_k(M))={\mathcal P}$.
 \item[(b)] $A\in\ecs{k}{n-k}$. 
 \end{itemize}
\end{Lemma}

Here, an $(Y,\eta$)-referred manifold $(A,q)$ means that $q\:A\to Y$ and as well as $q_{|{\rm Bd} A}$ are both
$(Y,\eta)$-reference maps.

\begin{proof}
We set $(\pi,\omega)=(\pi_1(Y),\omega_\eta)$. All spaces are equipped with a $(Y,\eta)$-reference map, so we use Convention~\ref{Conv} throughout the proof. Reference maps are often dropped from the notation. 
 
We refer to the proof of \proref{PreaPoin} for $r=n$ and use its notations.
There, a Poincar\'e\  pair $({\mathbb U},{\mathbb P})$ of formal dimension $n+1$
is constructed, together with an $(Y,\eta)$-reference map $q\:{\mathbb U}\to Y$, such that
${\rm Wall}({\mathbb U})={\mathcal P}$ and ${\rm Wall}({\mathbb P}) = {\mathcal P} + (-1)^{n}{\mathcal P}^*$.
Moreover, ${\mathbb P}$ is \has{k} with $w_k({\mathbb P})={\mathcal P}$. The space ${\mathbb U}$ is a completion of the one-end open manifold $U$
of \eqref{Trea-e1}, satisfying ${\rm Wall}(U)={\mathcal P}$. We mentioned that $U=U_0\cup E_0$ where $(U_0,q_{|U_0})$ is a 
stable $\eta$-thickening of $Y$. Thus, ${\mathbb U}=U_0\cup {\mathbb E}$, where $({\mathbb E},{\rm Bd} U_0,{\mathbb P})$ is a Poincar\'e\  cobordism with ${\rm Wall}({\mathbb E})={\mathcal P}$. As the pair $({\mathbb U},{\mathbb E})$ is highly connected, the restriction $q_{\scriptscriptstyle{\mathbb E}}\:{\mathbb E}\to Y$  of $q$ to ${\mathbb E}$ is an $Y$-reference map. By \corref{CPdecNor2}, $q_{\scriptscriptstyle{\mathbb E}}$
is $\eta$-normal.

The hypothesis that ${\mathcal P}$ represents $0$ in $H^{n+1}({\mathbb Z}_2;\tilde K_0({\mathbb Z}\pi,\omega))$ means that \\
${\mathcal P}=-{\mathcal Q}-(-1)^{n+1}{\mathcal Q}^*$ for some ${\mathcal Q}\in \tilde K_0({\mathbb Z}\pi,\omega)$. 
We now use \proref{PreaPoin} for $r=n+1$, replacing ${\mathcal P}$ by ${\mathcal Q}$. 
One gets a an $(Y,\eta$)-referred Poincar\'e\  complex (without boundary) 
$({\mathbb P}_1,q_1)$ of formal dimension $n+1$, having an $\eta$-normal Poincar\'e\  decomposition
$$
{\mathbb P}_1 \simeq N_1 \cup {\mathbb T}_1 \ , \ N_1\cap {\mathbb T}_1 = {\rm Bd} N_1  = {\rm Bd}{\mathbb T}_1 \ ,
$$
where $(N_1,{q_1}_{|N_1})$ is a stable $\eta$-thickening of $Y$ in dimension $n+1$ and ${\mathbb T}_1$ 
is an $(Y,\eta)$-referred Poincar\'e\  space with boundary ${\rm Bd}({\mathbb T}_1)={\rm Bd} N_1$
such that ${\rm Wall}({\mathbb T}_1) = {\mathcal Q} + (-1)^{n+1}{\mathcal Q}^*$.

As $q$ and $q_1$ are $\eta$-normal, one can choose Spivak classes $({\mathbb U})\in{\mathcal H}_{n+1}({\mathbb U},{\rm Bd}{\mathbb U})$ and 
$({\mathbb P}_1)\in{\mathcal H}_{n+1}({\mathbb P}_1)$. By \proref{PPdecNor} (ii), they induce Spivak classes 
$(U_0)\in{\mathcal H}_{n+1}(U_0,{\rm Bd} U_0)$ and $(N_1)\in{\mathcal H}_{n+1}(N_1,{\rm Bd} N_1)$. By \proref{LSpiThick},
there exists a simple homotopy equivalence $f\:(U_0,{\rm Bd} U_0)\to (N_1,{\rm Bd} N_1)$ commuting with the $Y$-reference
maps and satisfying ${\mathcal H}_*f((U_0))=(N_1)$. Form the space 
$$
{\mathbb A} = {\mathbb E} \cup_{{\rm Bd}(U_0)} {\mathcal C}(f) \cup_{{\rm Bd} N_1={\rm Bd} {\mathbb T}_1} {\mathbb T}_1 \, ,
$$
where ${\mathcal C}(f)$ is the mapping cylinder of $f_{|{\rm Bd} U_0}$:
$$
{\mathcal C}(f) = {\rm Bd} U_0 \times [0,1] \amalg {\rm Bd} N_1 \big/ \{(x,1)\sim f(x)\} \, .
$$
Set ${\mathbb E}^+ = {\mathbb E} \cup {\mathcal C}(f)$ and ${\rm Bd}{\mathbb E}^+ = {\mathbb P} \amalg {\rm Bd}{\mathbb T}_1$. 


That $f$ commutes with the reference maps on ${\mathbb E}^+\simeq {\mathbb E}$ and ${\mathbb T}_1$
implies that these references maps extend to a map 
$q_{\scriptscriptstyle{\mathbb A}}\:{\mathbb A}\to Y$. This map $q_{\scriptscriptstyle{\mathbb A}}$ is an $Y$-reference map.
Indeed, all fundamental groups under consideration are isomorphic and identified with~$\pi$. One has a morphism of 
of Mayer-Vietoris sequences
$$
\dia{ 
H_j(\widetilde{\partial N_1}) \ar[r] \ar[d]^{(q_N)_*}   & 
H_j(\tilde{\mathbb E}^+)\oplus H_j(\tilde{\mathbb T}_1) \ar[r]  
\ar[d]^{(q_{\scriptscriptstyle{\mathbb E}})_*\oplus (q_{\scriptscriptstyle{{\mathbb T}_1}})_*}    & 
H_j(\tilde{\mathbb A}) \ar[d]^{(q_{\scriptscriptstyle{\mathbb A}})_*}
\ar[r]^(.45
){\partial_{MV}} & H_{j-1}(\widetilde{\partial N_1})  \ar[d]^{(q_N)_*}
\\
H_j(\tilde Y) \ar[r] & H_j(\tilde Y)\oplus H_j(\tilde Y) \ar[r] & H_j(\tilde Y) \ar[r]^(.45)0 & H_{j-1}(\tilde Y)
}
$$
where the bottom line is the Mayer-Vietoris sequence of $Y=Y\cup Y$, $Y\cap Y=Y$.
The homomorphism $(q_N)_*\: H_j(\widetilde{\partial N_1})\to H_j(\tilde Y)$ is an isomorphism for $j\leq n+2-k > k$,
whence $\partial_{MV}=0$ on this range. As $q_{\scriptscriptstyle{\mathbb E}}$ and $q_{\scriptscriptstyle{{\mathbb T}_1}}$
are $Y$-reference maps, one deduces that $q_{\scriptscriptstyle{\mathbb A}}$ is an $Y$-reference map.

We now refer to Diagram~\eqref{PdecNor-dia} and \proref{PPdecNor}, 
with $X_-={\mathbb E}^+$, $X_+={\mathbb T}_1$ and $Z_0={\rm Bd}{\mathbb T}_1$. 
Since ${\mathcal H}_*f((U_0))=(N_1)$, one has $\partial_{\mathcal H}(({\mathbb E}^+),-({\mathbb T}_1))=0$.
By \corref{CPdecNor},  $({\mathbb A},{\mathbb P})$ is a Poincar\'e\  pair of formal dimension~$n+1$ and $q_{\scriptscriptstyle{\mathbb A}}$ 
is an $(Y,\eta)$-reference map.

We claim that ${\mathbb A}\in \ecs{k}{n-k}$. Indeed, one has the following
\begin{itemize}
 \item ${\mathbb E}^+\in \ecs{k}{n-k}$. For, $E_0\in\ecs{k}{n-k}$ by \proref{Pdecomp} (for $r=n+1$), so
 ${\mathbb E}^+\simeq{\mathbb E}\simeq E_0\in \ecs{k}{n-k}$.
 \item ${\mathbb T}_1\in \ecs{k}{n+1-k}$ by \proref{PreaPoin} (for $r=n+1$).
 \item ${\rm Bd} U_0 \in \ecs{k}{n-k}$ by \remref{RBdThick}.
\end{itemize}
As the inclusion ${\rm Bd} U_0\hookrightarrow{\mathbb T}_1$ homotopy commutes with the $Y$-reference maps, it induces
a homomorphism $H^{k-1}({\mathbb T}_1;R)\to H^{k-1}({\rm Bd} U_0;R)$ which is onto for for any ${\mathbb Z}\pi_1(X)$-module $R$.
By \lemref{Ldececs}, we deduce that ${\mathbb A} \in \ecs{k}{n-k}$.

By Siebenmann's sum theorem \cite[Theorem~6.5]{SieTh}, one has
$$
{\rm Wall}({\mathbb A}) = {\rm Wall}({\mathbb E}) + {\rm Wall}({\mathbb T}_1) = {\mathcal P} + {\mathcal Q} + (-1)^{n+1}{\mathcal Q}^* =0 \, .
$$
On the other hand, one has ${\rm Wall}({\mathbb P})={\mathcal P}+ (-1)^n{\mathcal P}^*$ by \proref{PreaPoin}. 
Since ${\mathcal P}=(-1)^{n+1}{\mathcal P}^*$ by hypothesis, one has ${\rm Wall}({\mathbb P})=0$.
By \cite[Theorem~F]{WallFin}, $({\mathbb A},{\mathbb P})$ has the homotopy type of a {\it finite} Poincar\'e\  pair. 
The surgery obstruction $\sigma({\mathbb A},q_{\scriptscriptstyle{\mathbb A}})$ (see~\ref{MPfnormal}) is thus defined in 
$L^h_{n+1}(\pi,\pi,\omega)=0$ by Wall's ($\pi-\pi$)-theorem.
Thus, up to homotopy equivalence,
one may replace $({\mathbb A},{\mathbb P})$ is an $(Y,\eta)$-referred {\it compact manifold pair} $(A,M)$ of dimension $n+1$.
The manifold pair $(A,M)$ has the required properties for \lemref{LcobL2}.
\end{proof}


\begin{proof}[Proof of \thref{TcobL2}]
That $(b)\!\Rightarrow\! (a)$ is guaranteed by \proref{PcobL2}, only requiring $n\geq 6$.
For the converse, let us start with some preliminary constructions. Choose a handle decomposition of $M$ 
and let $N$ be the union of handles of index $\leq k-1$. Then, $N$ is a stable $n$-thickening
of a connected finite subcomplex $Y$ of $M$ of dimension $k-1$. By the classification of stable thickening
(see the proof of \thref{PWallThick}), $N$ is determined up to diffeomorphism
by the the stable vector bundle $\nu$ over $Y$, i.e. the restriction of the stable normal bundle of $M$ over $Y$.
Let $(\pi,\omega)=(\pi_1(Y),\omega_\nu)$. The inclusion $Y\subset M$, which is ($k-1$)-connected, induces an identification
$(\pi_1(M),\omega_M)\approx (\pi,\omega)$. Using this identification, let ${\mathcal P}=w_k(M)\in\tilde K_0({\mathbb Z}\pi,\omega)$.
Set $T=M{\scriptstyle\setminus} {\rm int} N$. 

Let $(A,q)$ be an $(Y,\nu)$-referred compact manifold of dimension $n+1$, as produced by \lemref{LcobL2}, 
so that ${\rm Bd} A$ is a closed connected \has{k} manifold with $w_k({\rm Bd} A)=-{\mathcal P} \in\tilde K_0({\mathbb Z}\pi,\omega)$
(identification via $q$). Let $M_0={\rm Bd} A$. 
[Note the asymmetry of the pairs $(M,Y)$ and $(M_0,Y)$: the pair $(M,Y)$ is ($k-1$)-connected but there is no retraction
of $M$ onto $Y$, while $M_0$ retracts onto $Y$ but the pair $(M_0,Y)$ is only weakly ($k-1$)-connected 
(see \lemref{Lrefretr})]. 
By \proref{Pdecomp}, there is a manifold decomposition
$$
M_0 = N_0 \cup T_0 \quad , \quad N_0\cap T_0 = {\rm Bd} N_0 = {\rm Bd} T_0
$$
where $(N_0,q_{|N_0})$ is a stable $\nu$-thickening of $Y$ and $T_0$ is \has{k} with $w_k(T_0)=w_k({\rm Bd} A)$.
Using a collar neighborhood of $M_0$ in $A$, the inclusion $N_0\hookrightarrow M_0$ may be extended
to an embedding $\alpha\: N_0\times [0,2]\to A$ with $\alpha^{-1}({\rm Bd} A)=N_0\times \{0\}$. 
Let $A_0=A{\scriptstyle\setminus} (\rm{int} N_0 \times [0,2))$. Note that $A_0$ is homeomorphic to $A$ (diffeomorphic modulo
rounding corners). 

By \proref{PWallThick}, there is a degree-one diffeomorphism $h\:N\to N_0$. Let $h_{\rm Bd} \:{\rm Bd} N \to {\rm Bd} N_0$ 
be restriction of $h$ to ${\rm Bd} N$. Form the space
$$
W = T\times [0,1] \cup_{\beta} A_0 \, ,
$$
where $\beta\:{\rm Bd} T\times [0,1] \to {\rm Bd} T_0\times [0,1]$ is the diffeomorphism given by 
$\beta(x,t)=(\alpha\kern .7pt {\scriptstyle \circ} \kern 1pt h_{\rm Bd} (x),t)$. 
After rounding the corners (see appendix of \cite{BSD}), $W$ is a smooth cobordism between the manifolds
$$
{\mathcal M} = T \cup_{h_{\rm Bd}} T_0 \ \hbox{ and } \  {\mathcal M}_1 \approx T \cup_{h_{\rm Bd}} N_0  
$$
(since ${\rm Bd} N_0\times [1,2] \cup N_0\times\{2\}\approx N_0$).
\thref{TcobL2} will follow from the three assertions
\begin{itemize}
 \item[(A)] ${\mathcal M}$ is \as{k}.
 \item[(B)] ${\mathcal M}_1$ is diffeomorphic to $M$.
 \item[(C)] $W$ is a \has{k} cobordism.
\end{itemize}

We now start proving the above assertions. There is an embedding $Y\hookrightarrow {\rm Bd} N$.
We leave to the reader to prove, using Van Kampen's theorem,
that the various fundamental groups $\pi_1(W)$, $\pi_1({\mathcal M})$, are thus all identified with $\pi_1(Y)$.

\sk{1}\noindent 
{\em Proof of (A).} \ 
We first prove that ${\mathcal M}$ is \has{k} (i.e. ${\mathcal M}\in\ecs{k}{n-k}$).
Let $R$ be a ${\mathbb Z}\pi$-module. Given that $T_0\in\ecs{k}{n-k}$ and the exact sequence
$$
H^{j-1}(T_0;R) \fl{\delta}   H^j({\mathcal M},T_0;R) \to H^j({\mathcal M};R) \to H^j(T_0;R) \ ,
$$
it is enough to prove that $\delta$ is surjective when $k\leq j\leq n-k$. By excision, one has
$$
H^j({\mathcal M},T_0;R) \approx H^j(T,{\rm Bd} N;R) \approx  H^j(M,N;R) \, .
$$
As $M\in\ecs{k}{n-k}$ and $\dim Y = k-1$, one deduces from the exact sequence
$$
H^{j-1}(N;R) \to H^j(M,N;R) \to H^j(M;R)
$$
that $H^j({\mathcal M},T_0;R)\approx H^{j-1}(N;R)=0$ for $j>k$. For $j=k$, one uses the following commutative diagram
\beq{TcobL2-L1-dia}
\dia{ 
H^{k-1}(T_0;R) \ar[d]^(0.43)\delta \ar[r]^(0.47){a} & H^{k-1}({\rm Bd} T;R) \ar[d]^(0.43){\delta_1} \ar@{<-}[r] & 
H^{k-1}(N;R) \ar[d]^(0.43){\delta_2}
\\
H^k({\mathcal M},T_0;R) \ar[r]^(0.47)\approx & H^k(T,{\rm Bd} T;R) \ar@{<-}[r]^(0.54)\approx & H^k(M,N;R)
}
\end{equation}
where the horizontal arrows are induced by inclusions, via the identifications \\ 
${\rm Bd} T={\rm Bd} N \approx {\rm Bd} N_0 = {\rm Bd} T_0$. 
The homomorphism $\delta_2$ is onto since $H^k(M;R)=0$. This implies that $\delta_1$ is onto. Now, the homomorphism $a$ may be identified with
the homomorphism $a_0$ in the commutative diagram
$$
\dia{ 
 H^{k-1}(M_0;R)  \ar[rr]^{(\sigma_0)^*} \ar[d] &&  H^{k-1}(N_0;R) \ar[d]^(0.45)b  
\\
H^{k-1}(T_0;R) \ar[r]^(0.47){a_0}  &  H^{k-1}({\rm Bd} T_0;R) \ar[r]^\approx & H^{k-1}({\rm Bd} N_0;R)
} .
$$
The homomorphism $(\sigma_0)^*$ is onto, since the inclusion $\sigma_0\:N_0\to M_0$ admits a homotopy retraction 
(see \lemref{Lrefretr}). The homormorphism $b$ induced by the inclusion is also onto for, by Poincar\'e\  duality
$$
H^k(N_0,{\rm Bd} N_0;R) \fl{\approx} H_{n-k}(N_0;R) = 0  \, ,
$$
since $n-k>k-1$. Therefore, $a$ is onto in Diagram \eqref{TcobL2-L1-dia}, implying that $\delta$ is onto, 
which proves that ${\mathcal M}$ is \has{k}.

We now compute the antisimple obstruction $w_k({\mathcal M})$. 
By excision, one has isomorphisms
\beq{DexcisMM0}
\dia{
H_*(\tilde {\mathcal M},\widetilde{{\rm Bd} N}) \ar@{<-}[r]^(.39)\approx & H_*(\tilde T,\widetilde{{\rm Bd} N}) \oplus H_*(\tilde T_0,\widetilde{{\rm Bd} N}) \ar[r]^\approx & H_*(\tilde M,\tilde N) \oplus H_*(\tilde M_0,\tilde N_0)
}
\end{equation}
By construction, the pair $(M,N)$ is ($k-1$)-connected and so is the pair $(M_0,N_0)$ by \lemref{Lrefretr}.
Therefore, \eqref{DexcisMM0} implies that $({\mathcal M},{\rm Bd} N)$ is ($k-1$)-connected. The first isomorphism 
of~\eqref{DexcisMM0} for $*=k$ then implies that $\pi_k({\mathcal M},N) \approx \pi_k(M,N)\oplus \pi_k(M_0,N_0)$, which,
by Definition~\eqref{defwk}, implies that 
$$
w_k({\mathcal M})=w_k(M)+w_k(M_0) = {\mathcal P} - {\mathcal P} =0 \, .
$$
As $n\geq 8$, Assertion~(A) follows from \proref{Pcasas}.

\sk{1}\noindent 
{\em Proof of (B).} \
Let $c\:{\rm Bd} N\times [0,1)\to N$ and $c_0\:{\rm Bd} N_0\times [0,1)\to N_0$ be embeddings of open collar neighborhoods
of ${\rm Bd} N$ and ${\rm Bd} N_0$ in $N$ and $N_0$. Let $T^+=T\cup_{{\rm Bd} T={\rm Bd} N} {\rm Bd} N\times [0,1)$. As smooth manifolds,
$M$ and ${\mathcal M}_1$ may be presented as follows
$$
M = T^+ \amalg N\big/ \{(x,t)\sim c(x,t) \hbox{ for }  (x,t)\in {\rm Bd} N\times (0,1)\}
$$
and
$$
{\mathcal M}_1 = T^+ \amalg N_0\big/ \{(x,t)\sim c_0(h_{\rm Bd}(x),t) \hbox{ for }  (x,t)\in {\rm Bd} N_0\times (0,1)\}  \, .
$$
Thus, a diffeomorphism $s\: M\fl{\approx} {\mathcal M}_1$ may be defined as
$$
s(x) = 
\left\{
\begin{array}{ll}
x & \hbox{if $x\in T^+$} \\
h(x) & \hbox{if $x\in {\rm int\,}N$.}
\end{array} 
\right.
$$


\sk{1}\noindent 
{\em Proof of (C).} \
We consider the morphism of Mayer-Vietoris sequences for the inclusion of $4$-ads
$$
(\tilde {\mathcal M},\tilde T,\tilde T_0,\widetilde{{\rm Bd} N}) \hookrightarrow
(\tilde W,\tilde T\times [0,1],\tilde A_0,\widetilde{{\rm Bd} N}\times [0,1]) \, .
$$
For $j\leq k-1$, all the homomorphisms, except possibly $H_j({\mathcal M})\to H_j(W)$, are isomorphisms .
But then the latter is an isomorphism by the five lemma. The same happens for the inclusion
$$
(\tilde {\mathcal M}_1,\tilde T,N,\widetilde{{\rm Bd} N}) \hookrightarrow
(\tilde W,\tilde T\times [0,1],\tilde A_0,\widetilde{{\rm Bd} N}\times [0,1]) \, .
$$
Therefore, both pairs $(W,{\mathcal M})$ and $(W,{\mathcal M}_1)$ are weakly ($k-1$)-connected. 
The proof that $W\in\ecs{k}{n-k}$ is the same as that of ${\mathcal M}\in\ecs{k}{n-k}$ (see the proof of (A)): just replace 
${\mathcal M}$ by $W$ and $T_0$ by $A_0$. 
\end{proof}

\subsection{Examples}\label{Sexpl}

Let  $Y$ be a connected finite cell complex of dimension $k-1\geq 2$ and let $\eta$ be a stable vector bundle over $Y$.
Let $(\pi,\omega)=(\pi_1(Y),\omega_\eta)$. 
Set $H^+=H^{2j}=H^{2j}({\mathbb Z}_2,\tilde K_0(\pi,\omega))$ and 
$H^-=H^{2j+1}=H^{2j+1}({\mathbb Z}_2,\tilde K_0(\pi,\omega))$.

Let ${\mathcal P}$ be a  $\pm$-symmetric element of $\tilde K_0({\mathbb Z}\pi,\omega)$,
i.e. ${\mathcal P}$ satisfies ${\mathcal P}=\pm{\mathcal P}^*$. Suppose that 
${\mathcal P}$ represents zero in $H^\pm$. By \thref{Trea}, 
there exist $(Y,\eta)$-referred closed connected manifolds $(M,g)$ of dimension $r\geq 2k$
such that $M$ is \has{k} with $g_*(w_k(M))={\mathcal P}$. 
By \thref{TcobL2}, when $r\geq 8$,
such manifolds are cohomology $k$-antisimply cobordant to an \as{k} closed manifolds.

A source of such examples is given by the odd torsion $T_{\rm odd}$ of $\tilde K_0({\mathbb Z}\pi)$.
Indeed, $T_{\rm odd}$ splits under the involution $*$ into a direct sum of
eigenspaces $T_{\rm odd}^{\pm}$ for the eigenvalues $\pm 1$. This produces $\pm$-symmetric element of $\tilde K_0({\mathbb Z}\pi,\omega)$ which represent zero in $H^\pm$,
since the latter is a $2$-group. The group  $T_{\rm odd}$ is not zero for many finite groups,
such as $p$-groups for $p$ an odd prime, or the symmetric groups (see \cite{OliverProj} for a survey and references).

More essential examples may arise when $\pi$ is a finite $2$-group. For example, when $\pi=({\mathbb Z}_2)^3$, then
$\tilde K_0({\mathbb Z}\pi)\approx{\mathbb Z}_2$ \cite[Theorem~12.9]{WallNorms}. Thus, $H^\pm\approx{\mathbb Z}_2$. On the other hand,
the homomorphism
\beq{deltan}
\delta^{(n)} = \delta_R^{(n)} : H^{n+1} \to L_n^h(\pi,\omega)
\end{equation}
of \eqref{ranicki} vanishes for $n\equiv 2\, ({\rm mod} 4)$ \cite[Theorem~C]{HamMil} (warning: 
the conventions for $H^\pm$ in \cite{HamMil} are the opposite of ours, since the authors use there the involution
${\mathcal P}\mapsto -{\mathcal P}^*$). By \thref{Trea} and \proref{PcobL2}, for $n=4j+2\geq 2k$, there exists
an $(Y,\eta)$-referred closed connected $n$-manifolds $(M,g)$ which is \has{k} but not
cohomology $k$-antisimply cobordant to an \as{k} manifold. As \cite[Theorem~C]{HamMil} applies to a large class of
finite $2$-groups, one might be able to get other examples.

When $\pi$ is the generalized quaternion group $Q2^s$ ($s\geq 3$), one sees in \cite[\S~6 and~(1.5)]{HamMil}
that $H^\pm\approx{\mathbb Z}_2$. On the other hand, 
the homomorphism $\delta^{(n)}$ of~\eqref{deltan} vanishes for 
$\not\equiv 1\, ({\rm mod} 4)$ \cite[Theorem~D]{HamMil}. As above, one deduces from \thref{Trea} and \proref{PcobL2} that, for $2k\leq n\neq 4j+1$, there an $(Y,\eta)$-referred closed connected $n$-manifolds $(M,g)$ which is \has{k} but  not cohomology $k$-antisimply cobordant to an \as{k} manifold.

Finally, \proref{Pcasas}, \thref{TcobL2} together with \proref{PproductFor} implies the following

\begin{Proposition}\label{CCproductFor}
Let $M$ be a closed connected $n$-dimensional manifold ($n\geq 8$) which is \has{k} for $k\geq 4$. 
Let $A$ be a closed connected manifold of dimension $a$ with $a\leq n-2k$. Then the manifold $M\times A$ is \has{k+a} and
\begin{itemize}
\item[(1)] if $\chi(A)=0$, then $M\times A$ is \as{(k+a)}
\item[(2)] if $\chi(A)$ is even, then $M\times A$ is cohomology $(k+a)$-antisimply cobordant to an \as{(k+a)} closed manifold. \hfill \ensuremath{\Box}
\end{itemize}
\end{Proposition}




\section{Referred antisimple manifolds}


Through this section $Y$ will be a fixed connected finite cell complex of dimension $k-1\geq 2$ equipped
with a stable vector bundle $\eta$ over it. We set $(\pi,\omega)=(\pi_1(Y),\omega_\eta)$.
We denote by ${\rm Wh}(\pi,\omega)$ the Whitehead group ${\rm Wh}(\pi)$ endowed with the involution $\alpha\mapsto\alpha^*$
using the orientation character $\omega$. 
When we consider an $(Y,\eta$)-referred manifold
$(X,g)$, its fundamental group and orientation character $(\pi_1(X),\omega_X)$ is thus is identified via $g_*$ with 
$(\pi,\omega)$. This gives an identification ${\rm Wh}(\pi_1(X),\omega_X)\approx{\rm Wh}(\pi,\omega)$.
Equalities between elements of ${\rm Wh}(\pi_1(X'),\omega_{X'})$
and ${\rm Wh}(\pi_1(X''),\omega_{X''})$ for ($Y,\eta$)-manifolds $(X',g')$ and $(X'',g'')$
should be understood to hold in ${\rm Wh}(\pi,\omega)$ via the identifications (like in Convention~\ref{Conv}).
The intervalle $[0,1]$ is denoted by $I$.

As done in the proof of \proref{Pdecomp}, if $(M,g)$ be an $(Y,\eta$)-referred closed manifold of dimension $r\geq 2k\geq 6$, a homotopy section $\gamma\:Y\to M$ of $g$ gives rise to 
a manifold decomposition $N\cup T\fl{\approx} M$ such that $(N,g_{|N})$ is a stable $\eta$-thickening of $Y$
When $M$ is \as{k}, we get a richer decomposition.


\begin{Proposition}\label{PdecAsm}
Let $(M,g)$ be an $(Y,\eta$)-referred \as{k} closed manifold of dimension $r\geq 2k\geq 6$. 
Then,

\begin{itemize}
 \item[(1)] a homotopy section $\gamma\:Y\to M$ of $g$ gives rise to 
a manifold decomposition
$$
M=N \cup V \cup N^* \ , \  N \cap N^* = \emptyset \ ,
\ V\cap N = {\rm Bd} N \ , \ V\cap N^* = {\rm Bd} N^*
$$
such that
\begin{itemize}
 \item[(i)] $(N,g_{|N})$ and $(N^*,g_{|N^*})$  are stable $\eta$-thickenings of $Y$ of dimension~$r$.
 \item[(ii)] the compact $r$-manifold $V$ is an h-cobordism between ${\rm Bd} N$ and ${\rm Bd} N^*$. 
\end{itemize}
\item[(2)] The Whitehead torsion $\tau(V,{\rm Bd} N)\in Wh(\pi,\omega)$ is $r$-self-dual, i.e. satisfies \\
$\tau(V,{\rm Bd} N)=(-1)^{r+1}\tau(V,{\rm Bd} N^*)^*$.
\item[(3)]
The class $[\tau(V,{\rm Bd} N)]\in H^{r+1}({\mathbb Z}_2;{\rm Wh}(\pi_1(Y))$ does not depend on the section 
$\gamma$. 
\end{itemize}  
\end{Proposition}

Thanks to (3) above, the class of $[\tau(V,{\rm Bd} N)] \in H^{r+1}({\mathbb Z}_2;{\rm Wh}(\pi_1(Y))$ will be denoted by 
$\tau(M,g)$.


\begin{proof}
As in the proof of \proref{Pdecomp}, 
one may suppose that $\gamma\:Y\to M$ is an embedding. Let 
$N$ be a smooth regular neighborhood of $\gamma(Y)$. We thus get the manifold decomposition $M= N\cup T$
of \proref{Pdecomp}.
By general position, $\gamma$ is isotopic to an embedding
$\gamma^*\:Y\to T$. Let $N^*$ be a smooth regular neighborhood of $\gamma^*(Y)$ in ${\rm int}(T)$.
Let $V=M {\scriptstyle\setminus} {\rm int}(N\cup N^*)$. 

To prove that the cobordism $(V,{\rm Bd} N,{\rm Bd} N^*)$ is an h-cobordism, let us 
consider a handle decomposition ${\mathcal H}$ for $M$
$$
{\mathcal H} : \ D^n = {\mathcal H}_0 \subset {\mathcal H}_1 \subset \cdots \subset {\mathcal H}_n = M \ ,
$$
where ${\mathcal H}_j$ is the union of handles of index $\leq j$ in ${\mathcal H}$. We suppose that ${\mathcal H}$ is \as{k},
i.e. ${\mathcal H}_{k-1}={\mathcal H}_{n-k+1}$ and that ${\mathcal H}$ has only one $r$-dimensional handle. 
Consider the dual handle decomposition ${\mathcal H}^*$ of ${\mathcal H}$ (see e.g. \cite[p.~394]{MilnorWT}).

By general position, one may assume that $\gamma(Y)\subset N\subset  {\rm int\,}{\mathcal H}_{k-1}$. 
Let $X={\mathcal H}_{k-1}{\scriptstyle\setminus} {\rm int\,}N$.
The pairs $(M,N)$ and $(M,{\mathcal H}_{k-1})$ are both weakly ($k-1$)-connected. Therefore, the pair $({\mathcal H}_{k-1},N)$
is weakly ($k-1$)-connected. Since the homotopy dimensions of $Y$ and of ${\mathcal H}_{k-1}$ are both $\leq k-1$,
one deduces that $\gamma\:Y\to {\mathcal H}_{k-1}$ is a homotopy equivalence. It follows easily that $(X,{\rm Bd} N,{\rm Bd}{\mathcal H}_{k-1})$
is an h-cobordism (see e.g. \cite[proof of Proposition~4.6]{HauJah}).

Similarely, one may suppose that $N^*\subset {\rm int\,}{\mathcal H}_{k-1}^*$ and get an h-cobordism \\
$(X^*,{\rm Bd} N^*,{\rm Bd}{\mathcal H}_{k-1}^*)$. One has that $V\cap V^*={\rm Bd}{\mathcal H}_{k-1}={\rm Bd}{\mathcal H}_{k-1}^*$, 
so the cobordism $V= X\cup X^*$ is an h-cobordism.

To prove (2), recall from \remref{RBdThick} that $N\approx N_-\times I$
and thus ${\rm Bd} N \approx \bar N_-\cup N_-\times\{1\}$, where $\bar N_-=N_-\times\{0\} \cup {\rm Bd} N_-\times I$.
Analogously,  $N^*\approx N_-^*\times I$ 
and ${\rm Bd} N^* \approx \bar N_-^*\cup N_-^*\times\{1\}$, where $\bar N_-^*=N_-^*\times\{0\} \cup {\rm Bd} N_-^*\times I$.
For $t=0,1$, consider the embedding $\gamma_t\:Y\to N_-\times\{t\}$ homotopic in $N$ to $\gamma$. 
We do the same for $\gamma_t^*\:Y\to N_-^*\times\{t\}$ homotopic in $N^*$ to $\gamma^*$. 

The two embeddings $\gamma_0$ and  $\gamma_0^*$ are isotopic in $M$. By general position, they are isotopic in $V$. Therefore, $V\approx (\bar N_-\times I)\cup Z$, where $(Z,N_-\times 1,N_-^*\times 1)$ is an h-cobordism.
Set $\bar\gamma_1\:Y\to Z$ to be $\gamma_1$ post-composed with the inclusion $N_-\times\{1\}\hookrightarrow Z$
and $\bar\gamma_1^*\:Y\to Z$ to be $\gamma_1^*$ post-composed with the inclusion $N_-^*\times\{1\}\hookrightarrow Z$
One has
$$
\tau(V,{\rm Bd} N) = \tau(Z,N_-\times\{1\}) = \tau(\bar\gamma_1) 
$$
and 
$$
(-1)^{r+1}\tau(V,{\rm Bd} N) = \tau(V,{\rm Bd} N^*) = \tau(Z,N_-^*\times\{1\}) = \tau(\bar\gamma_1^*) \, . 
$$
Since $\gamma_1$ and $\gamma_1^*$ are homotopic in ${\rm Bd} Z$, this proves (2). 

To prove (3), let $\delta\:Y\to M$ be another homotopy section of $g$, giving rise to a manifold decomposition
$M\approx N(\delta) \cup V(\delta)\cup N^*(\delta)$ which we have to compare with the decomposition
$M\approx N(\gamma) \cup V(\gamma)\cup N^*(\gamma)$. As above, by general position and given the connectivity of 
$(M,N(\gamma))$ and $(M,N(\delta))$, there is a self-homotopy equivalence $\phi$ of $Y$ such that 
$\gamma=\delta\kern .7pt {\scriptstyle \circ} \kern 1pt\phi$. Therefore, we shall have $N(\delta)\subset {\rm int} N(\gamma)$
and $N^*(\delta)\subset {\rm int} N^*(\gamma)$, 
whence h-cobordisms $(S,{\rm Bd} N(\delta),{\rm Bd} N(\gamma))$ and $(S^*,{\rm Bd} N^*(\delta),{\rm Bd} N^*(\gamma))$. One has
$V(\delta) = S \cup V(\gamma) \cup S^*$. Therefore
\begin{eqnarray*}
 \tau(V(\delta),{\rm Bd} N(\delta)) &=& \tau(V(\gamma),{\rm Bd} N(\gamma)) + \tau(S,{\rm Bd} N(\delta)) + \tau (S^*,{\rm Bd} N^*(\gamma))
 \\ &=&
 \tau(V(\gamma),{\rm Bd} N(\gamma)) + \tau(S,{\rm Bd} N(\delta)) + (-1)^{r+1}\tau(S,{\rm Bd} N(\delta))^* \, ,
\end{eqnarray*}
which proves (3).
\end{proof}

\begin{Remark}\label{RTdoub}\rm
In case $\tau(M,g)=0$, then, by the s-cobordism theorem, $V\cup N^*$ is diffeomorphic to $N$ and $M$ is diffeomorphic to
the gluing of two copies of $N$ by a self-diffeomorphism of ${\rm Bd} N$. We call such an antisimple manifold a 
\dfn{twisted double}.\hfill \ensuremath{\Box}
\end{Remark}


Two $(Y,\eta)$-referred manifold $(M_i,g_i)$ ($i=0,1$) are called \dfn{h-cobordant} if there is an h-cobordism
$(W,M_0,M_1)$ admitting a $k$-connected $\eta$-normal map $G\:W\to Y$ extending $g_0$ and $g_1$. 

\begin{Proposition} 
Let $(M,g)$ be an $(Y,\eta$)-referred \as{k} closed manifold of dimension $r\geq 2k\geq 6$. 
Then,
\begin{itemize}
 \item[(1)] The class $\tau(M,g)\in H^{r+1}({\mathbb Z}_2;Wh(\pi_1(Y))$ is an invariant of the h-cobordism class of $(M,g)$.
\item[(2)] $\tau(M,g)=0$ if and only if $(M,g)$ is h-cobordant to a twisted double.
\end{itemize} 
\end{Proposition}

\begin{proof}
Let $(W,G)$ be an ($Y,\eta$)-referred h-cobordism between $(M_0,g_0)$ and $(M_1,g_1)$. 
Let $\Gamma\:Y\to W$ be a homotopy section of $G$. Composed with the projections on $M_i$, it gives rise to homotopy sections $\gamma_i\:Y\to M_i$ of $g_i$. One can also construct a homotopy $\hat\Gamma\:Y\times I \to W$  between 
$\gamma_0$ and $\gamma_1$ and a homotopy $\hat\Gamma^*\:Y\times I \to W$  between 
$\gamma_0^*$ and $\gamma_1^*$. By general position, this can be done by embeddings and we thus get  
a decomposition
$$
W \approx (N\times I) \cup Z \cup (N^*\times I) 
$$
where $(Z,V(\gamma_0),V(\gamma_1))$ is an h-cobordism.  One has

\noindent\hskip -6mm
\begin{minipage}{12.7cm}
{\small
\begin{eqnarray}
 \tau(V(\gamma_0),{\rm Bd} N(\gamma_0)) + \tau(Z,V(\gamma_0))  &=& 
\tau(V(\gamma_1),{\rm Bd} N(\gamma_1)) + \tau(Z,V(\gamma_1)) \nonumber  \\ &=&
\tau(V(\gamma_1),{\rm Bd} N(\gamma_1)) + (-1)^{r}\tau(Z,V(\gamma_0))^* \label{eqhcob}
\end{eqnarray}
}
\end{minipage}\\
which proves (1).

If $(M,g)$ is h-cobordant to a twisted double, then $\tau(M,g)=0$ by (1) already proven.
Conversely, suppose that $\tau(M,g)=0$. This means that there is a manifold decomposition
$M\approx N \cup V \cup N^*$ with $(V,{\rm Bd} N,{\rm Bd} N^*)$ being an h-cobordism with 
$\tau(V,{\rm Bd} N)= \alpha+ (-1)^{r+1}\alpha^*$ for some $\alpha\in{\rm Wh}(\pi,\omega)$. Let $(Z,M,M')$ be an h-cobordism
with $\tau(Z,M)=-\alpha$. Using \eqref{eqhcob}, this proves that $\tau(V',N')=0$ and thus $M'$ is a twisted double.
\end{proof}

We now turn our attention to the construction of antisimple manifolds which are not h-cobordant to
twisted doubles. Analogously to Ranicki's exact sequence \eqref{ranicki}, the main ingredient is the {\it Rothenberg exact sequence} 
\cite[Proposition~4.1]{Sha}
\beq{Rothenberg}
L_{r+1}^s(\pi,\omega) \to L_{r+1}^h(\pi,\omega) \fl{T}  H^{r+1}({\mathbb Z}_2;Wh(\pi,\omega)) \fl{\delta_{\rm Rot}} L_{r}^s(\pi,\omega)
\end{equation}

\begin{Proposition}\label{PAsreali}
Let $a\in H^{r+1}({\mathbb Z}_2;Wh(\pi,\omega))$ such that $\delta_{\rm Rot}(a)=0$. Then, for $r\geq 2k\geq 6$ there exists an $(Y,\eta)$-referred \as{k} closed manifold $(M,g)$ with $\tau(M,g)=a$.
\end{Proposition}

\begin{proof}
The map $T$ in \eqref{Rothenberg} may be geometrically described as follows (see \cite[p.~313]{Sha}). Let $(N,g)$ be a stable $\eta$-thickening
of $Y$ of dimension $r$. Set ${\rm Bd} (N\times I)\approx N_- \cup N_+$, where  $N_-=N\times\{0\}\cup {\rm Bd} N\times [0,1/2]$
and $N_+=N\times\{1\}\cup {\rm Bd} N\times [1/2,1]$.
An element $\sigma\in  L_{r+1}^h(\pi,\omega)$ may be represented as the surgery obstruction of a degree one normal map $f\: (X,X_-,X_+)\to (N\times I,N_-,N_+)$, where $X$ is a compact ($r+1$)-dimensional manifold with ${\rm Bd} X = X_-\cup X_+$,
such that $f_-=f_{|X_-}\:X_-\to N_-$ is a diffeomorphism and $f_+=f_{|X_+}\:X_+\to N_+$ is a homotopy equivalence.
One has $T(\sigma)=\tau(f_+)$ (via the identification by $g_*$).

As mentioned in \remref{RBdThick}, $N=N^{(n)}$ is itself an s-cobordism between an $\eta$-thickening 
$N^{(n-1)}$ of $Y$ and itself. It follows that $X_+$ is an h-cobordism between two copies 
$N^{(n-1)}_\pm$ of $N^{(n-1)}$, with $\tau(N_+,N^{(n-1)})=-\tau(f)$. One deduces that $M={\rm Bd} X$
is an \as{k} closed manifold, ($Y,\eta)$-referred via $g$, with $\tau(M,g)=a$ (signs are irrelevant
since $H^{r+1}({\mathbb Z}_2;Wh(\pi,\omega))$ is a group of exponent $2$).
\end{proof}

\begin{Examples}\rm
(1)
$L_{\rm odd}^h(\pi)=0$ if $\pi$ is a finite group of odd order \cite{BakOdd} ($\omega$ is then trivial). Hence, by \proref{PAsreali}, any class $a\in H^{2j}({\mathbb Z}_2;Wh(\pi))$ may be realized as $\tau(M,g)$ for an $(Y,\eta)$-referred
closed \as{k} manifold of dimension $n=2j\geq 6$. 
It follows from \cite[\S~4]{Bass1} that  $H^{2j}({\mathbb Z}_2;Wh(\pi))\neq 0$ if $\pi$ is finite abelian, 
of exponent $\neq 1,2,3,4$ or $6$.

(2) Let $Y_1,\dots,Y_s$ be finite CW-complexes of dimension $\leq k-1$, such that $\pi_1(Y_i)=G_i$
is a finite group of odd order. Let $Y=Y_1\vee\cdots Y_s$ and $G=\pi_1(Y)\approx G_1 * \cdots * G_s$.
One has $L_{\rm odd}^h(G)=0$ by \cite[Theorem~5]{CappellMV}. Hence, as in (1) above, 
any class $a\in H^{2j}({\mathbb Z}_2;Wh(G))$ may be realized as $\tau(M,g)$ for an $(Y,\eta)$-referred
closed \as{k} manifold of dimension $n=2j\geq 6$.
Recall that $H^{2j}({\mathbb Z}_2;Wh(G))\approx\bigoplus_{i=1}^s H^{2j}({\mathbb Z}_2;Wh(G_i))$ by \cite{StallingsWhFP}.

(3)  When $Y$ is acyclic, an $(Y,\eta)$-referred closed \as{k} manifold is a homology sphere. As an example,
consider the Poincar\'e\  $3$-sphere minus a disk, which collapses to an acyclic $2$-dimensional finite complex $Y$ with
$$
\pi_1(Y) = \Delta = \langle a,b\, |\, a^5=b^3=(ab)^2 \rangle  ,
$$
the binary icosahedral group. Let $C_5= \langle t\, |\, t^5=1 \rangle $ 
be the cyclic group of order $5$ with a given generator $t$. There is a homomorphism $\nu\: C_5\to\Delta$
with $\nu(t)=a^2$. One can prove that the induced homomorphism
$$
{\mathbb Z}_2 \approx H^{2j}({\mathbb Z}_2;{\rm Wh}(C_5)) \fl{\nu_{2j}} H^{2j}({\mathbb Z}_2;{\rm Wh}(\Delta))
$$
is injective \cite[Chapter~5]{HauTh}. By naturality of the Rothenberg exact sequence and Example~(1) above,
this produces \as{3} homology spheres in dimension $r=2j\geq 6$ which are not h-cobordant to a twisted double.
Using (1) and (2) above, we can get such homology spheres with fundamental group a free product of finitely many copies of $\Delta$.

(4) Suppose that $Y$ is the lens space $L(p,q)$ with $p\equiv 3\ {\rm mod\,}4$, or $(p,q)=(5,2)$
and take for $\eta$ the trivial bundle.
Then, any ($Y,\eta$)-referred \as{k} closed manifold $M$
of dimension $r=2j+1\geq 2k\geq 6$ is a twisted double. Indeed, in the decomposition of \proref{PdecAsm},
the h-cobordism $(V,{\rm Bd} N,{\rm Bd} N^*)$ is inertial, i.e. ${\rm Bd} N^*$ is diffeomorphic to ${\rm Bd} N$. But, there is 
no non-trivial inertial h-cobordism starting from ${\rm Bd} N = Y\times S^{2j}$  under our hypotheses
\cite[Theorem~6.1 and Lemma~6.5]{Hau2}. Hence $M$ is a twisted double by \remref{RTdoub}.
\end{Examples}




\footnotesize
