\section{Introduction}

The structural theory for regular Riemannian foliations, due mainly to P.~Molino \cite{molino}, establishes that the leaf closures of such a foliation are described by the action of a locally constant sheaf of lie algebras of transverse Killing vector fields (see Section \ref{section: (Singular) Riemannian foliations} for more details). Killing foliations are regular Riemannian foliations for which this sheaf is constant \cite{mozgawa}. This class include those foliations given by orbits of isometric actions, as well as Riemannian foliations of simply connected manifolds. The transverse action of the Molino sheaf on a Killing foliation can be seen as a transverse version of classical isometric actions of tori, and many results of this area thus have a transverse analog \cite{caramello}, \cite{caramello2}, \cite{goertsches}, \cite{goertsches2}, \cite{toben}.

One of the most elementary algebraic invariants of a foliation is its basic cohomology, which can be seen as the De Rham cohomology of its (often quite singular) leaf space. For the case of a regular Killing foliation there is also the equivariant basic cohomology with respect to the transverse action of the Molino sheaf. This invariant was introduced in \cite{goertsches} and further studied in \cite{goertsches2}. In those papers the authors proved transverse analogs of the Borel--Hsiang and Atiyah--Bott/Berline--Vergne localization results, which were later generalized in \cite{lin} to the case of infinitesimal transverse actions on regular Riemannian foliations. Moreover, in \cite{caramello2} the authors showed that part of the structure of the equivariant basic cohomology remains constant under regular deformations (see also \cite{caramello}), which also linked this object to the theory of torus actions on orbifolds.

In this paper we are interested in the generalization of the notion of Killing foliations to the realm of singular foliations, in the sense of letting the dimension of the leaves vary. The theory of singular Riemannian foliations is still in plain development, having seen relatively recent answers to fundamental questions \cite{alex4}, \cite{mendes}, and new interesting examples \cite{lytchak}. They also appear naturally in the theories of isometric actions and submanifolds (for instance, as the set of all parallel submanifolds of an isoparametric submanifold \cite{thorbergsson}), and play interesting roles in many other areas of geometry, for instance in the study of the regularity of the Sharafutdinov projection onto the soul \cite{wilking}.

Singular Killing foliations include foliations given by the orbits of isometric actions (see Example \ref{example: homogeneous Riemannian foliations are Killing}) and orbit-like foliations on simply connected manifolds. This last fact follows since orbit-like foliations verify the strong Molino conjecture, as we show in Proposition \ref{proposition: strong molino conjecture for orbit-like}. In fact, the class of singular Killing foliations will include all Riemannian foliations on simply connected manifolds, if the strong Molino conjecture holds true in general (see Section \ref{section: (Singular) Riemannian foliations}).

Our results concerning equivariant basic cohomology generalize some of those in \cite{goertsches} to the singular setting. Moreover, when specified to the regular case, they do not involve hypothesis of $\f$ being transversely oriented. The main result is Theorem \ref{theorem: borel localization}, which is a singular transverse version of Borel's localization theorem \cite{borel} stating that the inclusion of the set of closed leaves $M^{\mathfrak{a}}\to M$ induces an isomorphism in equivariant cohomology, modulo torsion. Our technical tools to obtain this result involve the study of the equivariancy of the Molino sheaf under the so-called homothetic transformations of $\f$ (see Section \ref{section: homothetic invariance of Molino sheaf}). This enables us to construct a suitable covering of $M$ by saturated open sets which equivariantly retract by deformations onto leaf closures (see Proposition \ref{proposition: good cover}), and the proof then follows the usual cohomological techniques. This result also covers a gap in the proof of \cite{wolak} that the basic cohomology of a singular Riemannian foliation is finite dimensional when $M$ is compact (see Theorem \ref{theorem: basic cohomology is finite dimensional}).

In particular, the basic Euler characteristic $\chi(\f)$ is well defined for any singular Riemannian foliation, when $M$ is compact. In the case of singular Killing foliations, a consequence of Borel's localization is that this invariant localizes to the set of closed leaves. This is Theorem \ref{theorem: localization of bec}, which generalizes \cite[Theorem D]{caramello} to singular Killing foliations. A direct corollary is that for any singular Killing foliation with finitely many closed leaves, of a compact manifold, $\chi(\f)$ is precisely the number of closed leaves (see Corollary \ref{corollary: localization of bec finite case}). Another application is that $\dim H(\f^\mathfrak{a})\leq \dim H(\f)$, with equality holding if, and only if, $\f$ is equivariantly formal. This is Theorem \ref{theorem: comparison dimensions}, which generalizes \cite[Theorem 1]{goertsches}.

In analogy to the classical case of Lie group actions, and more generally regular Killing foliations \cite{goertsches2}, a ``concrete'' version of Borel's localization theorem, in terms of a localization formula for the integral of basic equivariant forms to the set of closed leaves, is expected to hold. We intend to investigate this in a future article.


\section{Equivariant basic cohomology of singular foliations}

When a Lie algebra acts infinitesimally and transversely on a (regular) foliation, one can define its equivariant basic cohomology, which is an algebraic invariant that captures information from the action and the topology of the leaf space. This object was first studied in \cite{goertsches}, generalizing the ideas from classical equivariant cohomology of Lie group actions. In this section we will see that this notion also generalizes for the case of \emph{singular} foliations. Let us begin by fixing our notation and recalling the main notions and tools concerning smooth foliations. We work in the smooth category, so all objects are considered of differentiability class $\mathcal{C}^\infty$, unless otherwise explicitly stated.

\subsection{Preliminaries}

Let $M$ be an $n$-dimensional manifold. A \textit{singular foliation} of $M$ is a partition $\f$ of $M$ into connected, immersed submanifolds, called \textit{leaves}, such that the module $\mathfrak{X}(\f)$ of smooth vector fields that are tangent to the leaves is transitive on each $L\in\f$. That is, for each $L\in\f$ and each $x\in L$, any given $v\in T_xL$ can be extended to a smooth vector field $V\in\mathfrak{X}(\f)$. We denote the leaf containing $x\in M$ by $\f(x)$. The tangent spaces of the leaves form a distribution of varying rank, that we denote by $T\f$. The \textit{dimension of $\f$} is defined as
$$\dim(\f)=\max_{L\in\f}\dim(L),$$
and its \textit{codimension} by $\codim(\f)=\dim(M)-\dim(\f)$. We say that $\f$ is \textit{closed} when all leaves of $\f$ are closed submanifolds of $M$. When all leaves have the same dimension we say that $\f$ is a \textit{regular foliation}, or simply a \textit{foliation}, of $M$. The space of leaves $M/\f$ of $\f$ is the quotient space of $M$ by the equivalence relation that identifies points in the same leaf. A subset $J\subset M$ is \textit{saturated} when $J=\pi^{-1}\pi(J)$, for $\pi:M\to M/\f$ the canonical projection. In other words, $J$ is saturated if it contains all the leaves that it intersects.

\begin{example}
An important class singular foliations consists of those given by Lie group actions, which are called \textit{homogeneous}: if a Lie group $H$ acts smoothly on $M$, then the partition of $M$ by the connected components of the orbits of $H$ is a singular foliation $\f_H$. If all stabilizers of this action have the same dimension, in particular if the action is almost free, then $\f_H$ is a regular foliation.
\end{example}

If $(N,\g)$ is another singular foliation, we say that a smooth map $f:M\to N$ is \textit{foliate} when it maps leaves of $\f$ into leaves of $\g$. A smooth map $f:M\times[0,1]\to N$ is a \textit{foliate homotopy} (between $f_0$ and $f_1$) when it is foliate with respect to the product foliation of $\f$ and the trivial foliation of $[0,1]$ by points. Alternatively, $f$ is a foliate homotopy if $f_t:M\ni x\mapsto F(x,t)\in N$ is foliate for each $t\in[0,1]$. If $J\subset M$ is a saturated submanifold, then a \textit{foliate deformation retraction} $f:M\times[0,1]\to M$ of $M$ onto $J$ is a foliate homotopy which is also a deformation retraction of $M$ onto $J$.

A vector field $X\in\mathfrak{X}(M)$ is \textit{foliate} when $[X,Y]\in \mathfrak{X}(\f)$ for any $Y\in \mathfrak{X}(\f)$. These are the vector fields whose flows $\Phi^X_t$ are foliate maps. We denote the Lie algebra of foliate vector fields by $\mathfrak{L}(\f)$. The \textit{transverse vector fields} are the elements $\mathfrak{l}(\f):=\mathfrak{L}(\f)/\mathfrak{X}(\f)$. Notice that $\mathfrak{l}(\f)$ inherits the Lie bracket from $\mathfrak{L}(\f)$ and is, thus, also a Lie algebra. A transverse vector field $X$ is \textit{tangent} to a saturated submanifold $J\subset M$ when it admits a representative $\tilde{X}\in\mathfrak{L}(\f)$ which is tangent to $J$. Notice that in this case any representative of $X$ is tangent to $J$. Let $f:(M,\f)\to (N,\g)$ be foliate, $X\in\mathfrak{l}(\f)$ and $Y\in\mathfrak{l}(\g)$. We say that $X$ and $Y$ are \textit{$f$-related} when they admit $f$-related foliate representatives. In the case of a foliate map $f:M\to M$, we say that $X\in\mathfrak{l}(\f)$ is $f$-invariant when it is $f$-related to itself.

A tensor field $\xi\in\mathcal{T}^k(M)$ is called \textit{$\f$-basic} when it satisfies $\iota_X\xi=0$ and $\mathcal{L}_X\xi=0$ whenever $X\in\mathfrak{X}(\f)$. We can consider, in particular basic differential forms. By Cartan's formula $\mathcal{L}_X=\iota_Xd+d\iota_X$, a form $\omega\in\Omega^k(M)$ is basic if and only if, $i_X\omega=0$ and $i_X(d\omega)=0$, for all $X\in\mathfrak{X}(\f)$. The space of $\f$-basic $k$-forms is denoted by $\Omega^k(\f)$. Then
$$\Omega(\f):=\bigoplus_{k=0}^q\Omega^k(\f)$$
is the \textit{algebra of basic forms} of $\f$. It is clear that $\Omega(\f)$ is closed under the usual exterior derivative $d$, so it is a $\mathbb{Z}$-graded differential algebra (defining $\Omega^k(\f)=\{0\}$ if $k\notin\{0,\dots,q\}$).

The cohomology groups of the complex
$$\cdots \stackrel{d}{\longrightarrow} \Omega^{k-1}(\f) \stackrel{d}{\longrightarrow} \Omega^k(\f) \stackrel{d}{\longrightarrow} \Omega^{k+1}(\f) \stackrel{d}{\longrightarrow} \cdots .$$
are the \textit{basic cohomology groups} of $\f$, denoted $H^i(\f)$. We denote $H(\f)=\bigoplus H^i(\f)$, which is a graded algebra with the usual exterior product. Notice that when $\f$ is the trivial foliation by points, $H(\f)$ reduces to the De Rham cohomology of $M$. If $f:(M,\f)\to(N,\g)$ is foliate, then it pulls $\g$-basic forms back to $\f$-basic forms, thus induces a linear map $f^*:H(\g)\to H(\f)$. For $\f$-saturated open sets $U,V\subset M$, the short exact sequence 
$$0\longrightarrow \Omega(\f|_{U\cup V})\stackrel{i_U^*\oplus i_V^*}{\longrightarrow}\Omega(\f|_U)\oplus \Omega(\f|_V)\stackrel{j_U^*-j_V^*}{\longrightarrow} \Omega(\f|_{U\cap V})\longrightarrow 0.$$
induces a Mayer--Vietoris sequence in basic cohomology, where $i_U:U\to U\cup V$, $i_V:V\to U\cup V$, $j_U:U\cap V\to U$ and $j_V:U\cap V\to V$ are the natural inclusions.

When $\dim H(\f))<\infty$, we define the \textit{basic Euler characteristic} of $\f$ as
$$\chi(\f)=\sum_i(-1)^i\dim(H^i(\f)).$$
Basic cohomology $H(\f)$ can in fact be infinite-dimensional, even for a regular foliation $\f$ (see \cite{ghys2}). We will see in Theorem \ref{theorem: basic cohomology is finite dimensional} that it is finite-dimensional for the case of singular Riemannian foliations of compact manifolds.

\subsection{Equivariant basic cohomology}\label{section: equivariant basic cohomology}

Our definition of the equivariant basic cohomology of a singular foliation will be a direct generalization of the regular case, which was introduced in \cite{goertsches} via the concept of $\mathfrak{g}^\star$-algebras. We also refer to \cite{guillemin} for a thorough introduction to the topic of equivariant cohomology. Let us begin by recalling that a \textit{$\mathfrak{g}^\star$-algebra} consists of a differential $\mathbb{Z}$-graded commutative algebra $(A, d)$ acted upon by a finite-dimensional Lie algebra $\mathfrak{g}$, that is, so that for each $X\in\mathfrak{g}$ there are derivations $\mathcal{L}_X:A\to A$ and $\iota_X:A\to A$, depending linearly on $X$ and of degree $0$ and $-1$ respectively, satisfying
$$\iota_X^2=0,\ \ \ \mathcal{L}_X\mathcal{L}_Y-\mathcal{L}_Y\mathcal{L}_X=\mathcal{L}_{[X,Y]},\ \ \ \mathcal{L}_X\iota_Y+\iota_Y\mathcal{L}_X=\iota_{[X,Y]}\ \ \mbox{and}\ \ \mathcal{L}_X=d \iota_X+\iota_X d.$$
If $A$ and $B$ are $\mathfrak{g}^\star$-algebras, an algebra morphism $f:A\to B$ is a morphism of $\mathfrak{g}^\star$-algebras if it commutes, in the $\mathbb{Z}$-graded sense, with $d$, $\mathcal{L}_X$ and $\iota_X$.

Now let $(M,\f)$ be a singular foliation and $\mathfrak{g}$ a finite-dimensional Lie algebra. An \textit{infinitesimal transverse action} of $\mathfrak{g}$ on $\f$ is a Lie algebra homomorphism $\mu:\mathfrak{g}\to \mathfrak{l}(\f)$. The isotropy $\mathfrak{g}_x=\{X\in \mathfrak{g}\ |\ \mu(X)_x=0\}$ is constant for points in the same leaf, so we define $\mathfrak{g}_L=\mathfrak{g}_x$, for $x\in L\in \f$, which is the infinitesimal transverse analog of the stabilizer of a point. More generally, for a saturated submanifold $J\subset M$, consider the \textit{isotropy subalgebra}
$$\mathfrak{g}_J=\{X\in \mathfrak{g}\ |\ \mu(X)|_J=0\}=\bigcap_{x\in J} \mathfrak{g}_x.$$
Notice that $\mathfrak{g}_L=\mathfrak{g}_{\overline{L}}$, for $L\in \f$. Somewhat conversely, for $\mathfrak{h}<\mathfrak{g}$, the \textit{$\mathfrak{h}$-fixed locus}
$$M^{\mathfrak{h}}=\{x\in M\ |\ \mu(X)_x=0\ \mbox{for all}\ X\in\mathfrak{h}\}$$
is saturated (but not a submanifold in general). A saturated submanifold $J\subset M$ is \textit{$\mathfrak{h}$-invariant} when $\mu(X)$ is tangent to $J$, for each $X\in \mathfrak{h}$.

Let us prove that an infinitesimal transverse action induces a $\mathfrak{g}^\star$-algebra structure on $\Omega(\f)$. More precisely, for $X\in \mathfrak{g}$, consider the derivations $\mathcal{L}_X$ and $\iota_X$ on $\Omega(\f)$ given by $\mathcal{L}_X\omega=\mathcal{L}_{\tilde{X}}\omega$ and $\iota_{X}\omega=\iota_{\tilde{X}}\omega$, where $\tilde{X}\in\mathfrak{L}(\f)$ is foliate representative of $\mu(X)$. One readily checks that they are well defined: if $\tilde{X}_1,\tilde{X}_2\in\mathfrak{L}(\f)$ are different representatives for $\mu(X)$, then $\tilde{X}_1-\tilde{X}_2\in\mathfrak{X}(\f)$ and thus $\mathcal{L}_{\tilde{X}_1-\tilde{X}_2}\omega=0=\iota_{\tilde{X}_1-\tilde{X}_2}\omega$ for any $\omega\in\Omega(\f)$. Notice that $\mathcal{L}_X$ and $\iota_X$ have degree $0$ and $-1$, respectively.

\begin{proposition}\label{proposition: g star algebra stricture on basic complex}
Let $(M,\f)$ be a singular foliation and $\mu:\mathfrak{g}\to \mathfrak{l}(\f)$ an infinitesimal transverse action. Then, with the derivations $\mathcal{L}_X$ and $\iota_{X}$ defined above, $(\Omega(\f),d)$ becomes a $\mathfrak{g}^\star$-algebra.
\end{proposition}

\begin{proof}
The algebraic properties that $\mathcal{L}_X$, $\iota_X$ and $d$ must satisfy all follow directly from the same properties for the usual Lie derivative, interior multiplication and exterior derivative. We thus only have to show that $\mathcal{L}_X\omega$ and $\iota_X\omega$ indeed remain in $\Omega(\f)$ when $\omega\in\Omega(\f)$. In fact, for $Y\in\mathfrak{X}(\f)$, we have $\iota_Y\iota_{\tilde{X}}\omega=-\iota_{\tilde{X}}\iota_Y\omega=0$ (because $\iota_Y\omega=0$) and $\mathcal{L}_Y\iota_{\tilde{X}}\omega=\iota_{\tilde{X}}\mathcal{L}_Y\omega+\iota_{[Y,\tilde{X}]}\omega=0$ (because $\mathcal{L}_Y\omega=0$ and $[Y,{\tilde{X}}]\in\mathfrak{X}(\f)$), so $\iota_X\omega\in\Omega(\f)$. Similarly, $\iota_Y\mathcal{L}_{\tilde{X}}\omega=\mathcal{L}_{\tilde{X}}\iota_Y\omega-\iota_{[\tilde{X},Y]}\omega=0$ and $\mathcal{L}_Y\mathcal{L}_{\tilde{X}}\omega=\mathcal{L}_{\tilde{X}}\mathcal{L}_Y\omega-\mathcal{L}_{[\tilde{X},Y]}\omega=0$, hence $\mathcal{L}_X\omega\in\Omega(\f)$.
\end{proof}

Given Proposition \ref{proposition: g star algebra stricture on basic complex}, we can now define the equivariant basic cohomology of $\f$ in the usual Cartan's model way which we recall here for the reader's convenience (see also \cite{guillemin} or \cite{goertsches}). Let $\sym(\mathfrak{g}^*)$ denote the symmetric algebra over the dual $\mathfrak{g}^*$, which we interpret as the polynomial algebra on $\mathfrak{g}$. Consider the coadjoint action of $\mathfrak{g}$ on $\mathfrak{g}^*$ given by $(\mathrm{ad}^*_{X}\phi)(Y)=\phi(-[X,Y])$, for $X,Y\in\mathfrak{g}$ and $\phi\in\mathfrak{g}^*$. It extends naturally to $\sym(\mathfrak{g}^*)$ by $(\mathrm{ad}^*_{X}f)(Y)=f(-[X,Y])$. For $X\in\mathfrak{g}$ and $\omega=\sum f_i\otimes \omega_i \in \sym(\mathfrak{g}^*)\otimes \Omega(\f)$, we define
$$X\omega=\sum \left(\mathrm{ad}^*_{X}f_i\otimes \omega_i + f_i\otimes \mathcal{L}_X\omega_i\right).$$
The \textit{basic Cartan complex of $(\f,\mathfrak{g})$} is the subspace
$$C_{\mathfrak{g}}(\f)=(\sym(\mathfrak{g}^*)\otimes \Omega(\f))^{\mathfrak{g}}$$
of $\mathfrak{g}$-invariant elements $\omega$ that is, those satisfying $X\omega=0$. Notice that an element $\omega\in C_{\mathfrak{g}}(\f)$ can be seen as a $\mathfrak{g}$-equivariant polynomial map $\mathfrak{g}\to \Omega(\f)$. Moreover, if $\mathfrak{g}$ is Abelian then its coadjoint action is trivial, so $\omega\in C_{\mathfrak{g}}(\f)$ is just a polynomial map $\mathfrak{g}\to \Omega(\f)^\mathfrak{g}$.

The basic Cartan complex $C_{\mathfrak{g}}(\f)$ is naturally a $\sym(\mathfrak{g}^*)^{\mathfrak{g}}$-algebra, with ring multiplication $(\omega\eta)(X)=\omega(X)\wedge\eta(X)$ and $\sym(\mathfrak{g}^*)^{\mathfrak{g}}$-module multiplication induced by $f\mapsto f\otimes 1$.  Moreover, it is indeed a cochain complex with morphisms $d_\mathfrak{g}$, the \textit{equivariant derivative}, given by
$$(d_\mathfrak{g}\omega)(X)=d(\omega(X))-\iota_X(\omega(X)),$$
which is a degree $1$ derivation with respect to the usual grading
\begin{equation}\label{grading of the Cartan complex}
C_{\mathfrak{g}}^n(\Omega(\f))=\bigoplus_{2k+l=n}(\sym^k(\mathfrak{g}^*)\otimes \Omega^l(\f))^{\mathfrak{g}}.
\end{equation}
We define the \textit{$\mathfrak{g}$-equivariant basic cohomology of $\f$} as the cohomology of its Cartan complex:
$$H_\mathfrak{g}(\f)=H(C_\mathfrak{g}(\Omega(\f),d_\mathfrak{g})).$$
It inherits an $\sym(\mathfrak{g}^*)^{\mathfrak{g}}$-algebra structure from $C_{\mathfrak{g}}(\f)$. When $\mathfrak{g}=0$ is the trivial Lie algebra, $H_{\mathfrak{g}}(\f)$ is just the usual basic cohomology of $\f$. Another simple, but important particular case, is the following.

\begin{example}\label{example: equivariant cohomology of trivial action}
Suppose the transverse action of $\mathfrak{g}$ on $\f$ is trivial. Then one easily sees that $C_{\mathfrak{g}}(\f)=\sym(\mathfrak{g}^*)^{\mathfrak{g}}\otimes \Omega(\f)$ and $(d_\mathfrak{g}\omega)(X)=d(\omega(X))$, so one obtains
$$H_{\mathfrak{g}}(\f)\cong \sym(\mathfrak{g}^*)^{\mathfrak{g}}\otimes H(\f),$$
as $\sym(\mathfrak{g}^*)^{\mathfrak{g}}$-algebras.
\end{example}

\subsection{Homotopy invariance}

Let us now study the invariance of equivariant basic cohomology under homotopies, so let $(M,\f)$ and $(N,\g)$ be singular foliations with transverse $\mathfrak{g}$-actions $\mu:\mathfrak{g}\to\mathfrak{l}(\f)$ and $\nu:\mathfrak{g}\to\mathfrak{l}(\g)$, respectively. A foliate map $f:M\to N$ is called \textit{$\mathfrak{g}$-equivariant} if $\mu(X)$ and $\nu(X)$ are $f$-related, for all $X\in\mathfrak{g}$. In this case it is easy to check that the pullback $f^*:\Omega(\g)\to\Omega(\f)$ is a degree $0$ morphism of $\mathfrak{g}^\star$-algebras. In fact, it is clear that it commutes with $d$, and a straightforward calculation shows that it also commutes with each $\iota_X$. The commutation with $\mathcal{L}_X$ then follows easily using Cartan's formula or also by a direct computation. Hence, we obtain an induced morphism in equivariant basic cohomology $(f^*)_*:H_\mathfrak{g}(\g)\to H_\mathfrak{g}(\f)$, which we will often denote simply by $f^*$ when there is no risk of confusion. A foliate homotopy $f:M\times[0,1]\to N$ is \textit{$\mathfrak{g}$-equivariant} if $f_t$ is $\mathfrak{g}$-equivariant for each $t\in[0,1]$.

We recall that, for $\mathfrak{g}^\star$-algebras $A$ and $B$, two morphisms $f_0,f_1:A\to B$ are \textit{chain homotopic} when there is a linear map $Q:A\to B$ of degree $-1$ that satisfies
$$\iota_XQ+Q\iota_X=0,\ \ \ \mathcal{L}_XQ-Q\mathcal{L}_X=0\ \ \mbox{and}\ \ dQ+Qd=f_1-f_0$$
for all $X\in\mathfrak{g}$. This implies that the induced maps on equivariant cohomology are equal, that is $(f_0)_*=(f_1)_*:H_{\mathfrak{g}}(A)\to H_{\mathfrak{g}}(B)$ \cite[Proposition 2.4.1]{guillemin}. The following proposition is a generalization of \cite[Lemma 2.5.1]{lin} to singular foliations, and the proof is an adaptation of a classical proof of the homotopy invariance of equivariant cohomology \cite[p. 22]{guillemin}.

\begin{proposition}[Homotopy invariance of equivariant basic cohomology]\label{proposition: Homotopy invariance of equivariant basic cohomology}
Let $(M,\f)$ and $(N,\g)$ be singular foliations acted upon transversely by a Lie algebra $\mathfrak{g}$, and let $f:M\times[0,1]\to N$ be a foliate $\mathfrak{g}$-equivariant homotopy. Then
$$f_0^*=f_1^*:H_{\mathfrak{g}}(\g)\longrightarrow H_{\mathfrak{g}}(\f).$$
\end{proposition}

\begin{proof}
As we saw above, the conclusion will follow if we show that $f_0^*:\Omega(\g)\to\Omega(\f)$ and $f_1^*:\Omega(\g)\to\Omega(\f)$ are chain homotopic. For each $t\in [0,1]$, consider $V_t:M\to TN$ given by
$$V_t(x)=\left.\frac{d}{dt}f(x,s)\right|_{s=t},$$
and for $\omega\in\Omega^{k+1}(\g)$, let $f_t^*(\iota_{V_t}\omega)\in\Omega^k(M)$ denote the form
$$f_t^*(\iota_{V_t}\omega)_x(v_1,\dots,v_k)=\omega_{f_t(x)}(V_t(x),\dif f_t(v_1),\dots,\dif f_t(v_k)).$$
One has the identity (see \cite[p. 158]{guillemin2})
\begin{equation}\label{eq: homotopy invariance}\frac{d}{dt}f_t^*\omega=f_t^*(\iota_{V_t}d\omega)+df_t^*(\iota_{V_t}\omega).\end{equation}
We claim that $f_t^*(\iota_{V_t}\omega)\in\Omega^k(\f)$. In fact, since each $f_t$ is foliate, $\dif f_t$ maps $T\f$ into $T\g$, and then it follows easily that $\iota_Xf_t^*(\iota_{V_t}\omega)=0$ when $X\in\mathfrak{X}(\f)$. Hence, as $d\omega \in\Omega^{k+2}(\g)$, we also have $\iota_Xf_t^*(\iota_{V_t}d\omega)=0$, and from equation \eqref{eq: homotopy invariance}
$$\iota_Xdf_t^*(\iota_{V_t}\omega)=\iota_X\frac{d}{dt}f_t^*\omega-\iota_Xf_t^*(\iota_{V_t}d\omega)=0.$$

Now notice that integrating equation \eqref{eq: homotopy invariance} we obtain
$$f_1^*\omega-f_0^*\omega=\int_0^1\frac{d}{dt}f_t^*\omega\;dt=\int_0^1f_t^*(\iota_{V_t}d\omega)\;dt+\int_0^1 df_t^*(\iota_{V_t}\omega)\;dt=Qd\omega+dQ\omega,$$
so if we define $Q:\Omega(\g)\to\Omega(\f)$ by
$$Q\omega=\int_0^1f_t^*(\iota_{V_t}\omega)\;dt,$$
then it is a degree $-1$ linear map satisfying $f_1^*-f_0^*=dQ+Qd$.

It remains to show that $Q$ commutes with each $\iota_X$ and $\mathcal{L}_X$, for $X\in\mathfrak{g}$. Choose $\tilde{X}^M\in\mathfrak{L}(\f)$ and $\tilde{X}^N\in\mathfrak{L}(\g)$ to be $f_t$-related foliate representatives of the transverse fields corresponding to the action of $X$ on $\f$ and $\g$, respectively. Then
\begin{align*}
f_t^*(\iota_{V_t}\iota_X\omega)_x(v_1,...,v_{k-2}) & =  (\iota_X\omega)_{f_t(x)}(V_t(x),\dif (f_t)_xv_1,\dots, \dif (f_t)_xv_{k-2})\\
 & =  \omega_{f_t(x)}(\tilde{X}^N_{f_t(x)},V_t(x),\dif (f_t)_xv_1,\dots, \dif (f_t)_xv_{k-2})\\
 & =  -\omega_{f_t(x)}(V_t(x),\dif (f_t)_x\tilde{X}^M_x,\dif (f_t)_xv_1,\dots, \dif (f_t)_xv_{k-2})\\
 & =  -f_t^*(\iota_{V_t}\omega)_x(\tilde{X}^M_x,v_1,\dots, v_{k-2})\\
 & =  -\iota_Xf_t^*(\iota_{V_t}\omega)_x(v_1,...,v_{k-2}).
\end{align*}
Hence
\begin{equation}\label{eq: homotopy invariance 2} f_t^*(\iota_{V_t}\iota_X\omega)= -\iota_Xf_t^*(\iota_{V_t}\omega),\end{equation}
which integrated yields $\iota_XQ+Q\iota_X=0$. Now from Cartan's formula and equations \eqref{eq: homotopy invariance} and \eqref{eq: homotopy invariance 2},
\begin{align*}
\mathcal{L}_Xf_t^*(\iota_{V_t}\omega) & = \iota_X df_t^*(\iota_{V_t}\omega)+d\iota_Xf_t^*(\iota_{V_t}\omega)\\
 & = \iota_X\frac{d}{dt}f_t^*\omega-\iota_Xf_t^*(\iota_{V_t}d\omega)+d\iota_Xf_t^*(\iota_{V_t}\omega)\\
 & = \frac{d}{dt}f_t^*\iota_X\omega+f_t^*(\iota_{V_t}\iota_Xd\omega)-df_t^*(\iota_{V_t}\iota_X\omega)\\
 & = f^*_t(\iota_{V_t}d\iota_X\omega)+f_t^*(\iota_{V_t}\iota_Xd\omega)\\
 & = f^*_t(\iota_{V_t}\mathcal{L}_X\omega).
\end{align*}
Integrating this equation we get $\mathcal{L}_XQ-Q\mathcal{L}_X=0$.
\end{proof}

Two singular foliations $(M,\f)$ and $(N,\g)$ with transverse $\mathfrak{g}$-actions are \textit{$\mathfrak{g}$-homotopy equivalent} when there exist foliate $\mathfrak{g}$-equivariant maps $f:M\to N$ and $g:N\to M$ such that $g\circ f\simeq \mathrm{id}_M$ and $f\circ g\simeq \mathrm{id}_N$ by foliate $\mathfrak{g}$-equivariant homotopies.

\begin{corollary}\label{corollary: invariance of ebc under homotopies and deformation retracts}
If $\f$ and $\g$ are $\mathfrak{g}$-homotopy equivalent singular foliations then $H_{\mathfrak{g}}(\f)\cong H_{\mathfrak{g}}(\g)$ as $\sym(\mathfrak{g}^*)^{\mathfrak{g}}$-algebras. In particular, if $h:(M,\f)\to (M,\f)$ is a foliate $\mathfrak{g}$-equivariant deformation retraction onto a $\mathfrak{g}$-invariant submanifold $J\subset M$, then $H_{\mathfrak{g}}(\f)\cong H_{\mathfrak{g}}(\f|_J)$.
\end{corollary}


\section{Molino's structural theory of Riemannian foliations}\label{section: (Singular) Riemannian foliations}

A singular foliation $\f$ is a \textit{singular Riemannian foliation} when there exists a Riemannian metric $\mathrm{g}$ on $(M,\f)$ such that every geodesic which is perpendicular to a leaf of $\f$ remains perpendicular to all leaves it intersects. In this case $\mathrm{g}$ is an \textit{adapted metric} for $\f$. If furthermore $\mathrm{g}$ can be chosen to be complete, then we say that $\f$ is \textit{complete}. A main class of examples is that of isometric homogeneous foliations.

\begin{example}[{\cite[Section 6.1]{molino}}]\label{example: homogeneous srf}
Let $(M,\mathrm{g})$ be a Riemannian manifold on which a Lie group $H$ acts by isometries. Then $\mathrm{g}$ is an adapted metric for $\f_H$, which is thus a singular Riemannian foliation.
\end{example}

When $\f$ is regular, the adapted metric $\mathrm{g}$ is also called \textit{bundle-like}. It induces a \textit{transverse metric} for $\f$, that is, a basic $(2,0)$-tensor field $\mathrm{g}^T$ on $M$ satisfying $\mathrm{g}^T(X,Y)=\mathrm{g}^T(Y,X)$, for any $X,Y\in TM$, and $\mathrm{g}^T(X,X)>0$ whenever $X$ is not tangent to $\f$. This means $\mathrm{g}^T$ projects to a Riemannian metric on the local quotient $S$ of a submersion $\pi_i:U\to S$ locally defining $\f$ by its fibers. A transverse field $X\in\mathfrak{l}(\f)$ is a \textit{transverse Killing vector field} when $\mathcal{L}_{\tilde{X}}\mathrm{g}^T=0$, for a representative $\tilde{X}\in \mathfrak{L}(\f)$. It is clear that such a field projects to a Killing vector field on $S$.

There is a rich structural theory which describes the leaf closures of a complete (regular) Riemannian foliation $\f$, known as Molino theory. We refer to \cite{molino} and \cite{alex2} for details on this topic. One of its results states that the collection $\overline{\f}=\{\overline{L}\subset M\ |\ L\in\f\}$ is a singular Riemannian foliation, and its leaves, the closures of leaves in $\f$, are determined by the orbits of a locally constant sheaf of Lie algebras of germs of transverse Killing vector fields, the Molino sheaf $\mathscr{C}_\f$. This means that, for each $x\in M$, there is a small neighborhood $U\ni x$ where 
$$T_y\overline{\f}(y)=T_y\f(y)\oplus\{X_y\ |\ X\in \mathscr{C}_{\f}(U)\}$$
for each $y\in U$. When $M$ is connected, the typical stalk $\mathfrak{g}_\f$ of the Molino sheaf is the \textit{structural algebra of $\f$}, an important invariant of the foliation.

A complete, regular Riemannian foliation with globally constant Molino sheaf is called a \textit{Killing foliation}. The terminology, from \cite{mozgawa}, is motivated by the case of a homogeneous foliation $\f_H$, given by the locally free, isometric action of a Lie group $H$ on a complete, connected Riemannian manifold $M$, for in this case $\mathscr{C}_{\f}$ is the sheaf of germs of transverse Killing fields induced by the fundamental Killing vector fields of the action of $\overline{H}<\mathrm{Iso}(M)$. The structural algebra of a Killing foliation is Abelian \cite[Section 5.5]{molino}, and to emphasize this we will denote it by $\mathfrak{a}$ (also omitting $\f$ if it is clear from the context). Since for a Killing foliation $\mathscr{C}_\f$ is globally constant, we have $\mathfrak{a}\cong\mathscr{C}_\f(M)$. That is, $\mathfrak{a}$ acts transversely on $\f$, and its action describes the leaf closures. We denote this succinctly by $\mathfrak{a}\f=\overline{\f}$. 

Now suppose $\f$ is a complete singular Riemannian foliation, and let $\Sigma^k$ be the union of all $k$-dimensional leaves of $\f$. We call $\Sigma^k$ the \textit{$k$-locus} of $\f$.  One proves that each connected component $\Sigma^k_\alpha$ of $\Sigma^k$ is an embedded submanifold of $M$, called a \textit{stratum} (see \cite[Section 6.2]{molino}). This gives a stratification
$$M=\bigsqcup_{k,\alpha} \Sigma^k_\alpha$$
of $M$. We will often omit $k$ in $\Sigma^k_\alpha$ when it is not needed. The restriction $\f_\alpha:=\f|_{\Sigma_\alpha}$, endowed with $\mathrm{g}_\alpha:=\mathrm{g}|_{\Sigma_\alpha}$, is a regular Riemannian foliation, for each $\alpha$. Moreover, if $L\subset \Sigma_\alpha$ then $\overline{L}\subset\Sigma_\alpha$ (\cite[Lemma 6.4]{molino}), and each $\Sigma_\alpha$ is transversally totally geodesic, meaning that a geodesic which is normal to the leaves and tangent to $\Sigma_\alpha$ remains within it at least for some time (\cite[Proposition 6.3]{molino}). We will say that a transverse field $X\in\mathfrak{l}(\f)$ is a \textit{transverse Killing vector field} if its restriction to each stratum $\Sigma_\alpha$ is a transverse Killing vector field for $(\f_\alpha,\mathrm{g}_\alpha^T)$.

It will be useful to define some more terminology. The $\dim(\f)$-locus of $\f$ is an open and dense submanifold of $M$, also called the \textit{regular locus} of $\f$ and denoted by $\Sigma_{\mathrm{reg}}$. Each connected component of $\Sigma_{\mathrm{reg}}$ is a \textit{regular stratum} of $\f$. All other strata are called \textit{singular} and have codimension at least $2$ (in particular, when $M$ is connected $\Sigma_{\mathrm{reg}}$ is connected, and hence a stratum). Their union $\Sigma_{\mathrm{sing}}$ is the \textit{singular locus} of $\f$. The most singular strata, that is, those containing the leaves of least dimension, are called \textit{minimal}. Their union is the \textit{minimal locus} of $\f$, also denoted by $\Sigma_{\mathrm{min}}$. The \textit{closed locus} of $\f$ is the union of all closed leaves, denoted by $\Sigma_{\mathrm{cl}}$.

There is an analog of Molino's structural theorem for singular Riemannian foliations, but with a caveat. Its main part is the fact that the collection $\overline{\f}$ of the closures of leaves of a complete singular Riemannian foliation $\f$ is again a singular Riemannian foliation, known as \emph{Molino's conjecture}. It remained open for more than three decades, and was proven to hold recently in \cite[Theorem]{alex4}.

The caveat resides in the generalization of the Molino sheaf to the singular setting. Although a restriction $\f_\alpha$ is not necessarily complete, one can still apply Molino's theory to it. This can be done by the approach to Molino theory via pseudogroups \cite[Appendix D]{molino} (since the holonomy pseudogroup of $\f_\alpha$ is complete), or, alternatively, the metric on $\Sigma_\alpha$ can be changed so that $\f_\alpha$ becomes complete. In particular, for the case of $\f_{\mathrm{reg}}$, it is possible to prove that the corresponding Molino sheaf $\mathscr{C}_{\mathrm{reg}}$ extends continuously to a locally constant sheaf $\mathscr{C}_{\f}$ on $M$, called the \textit{Molino sheaf of $\f$} \cite[Lemma 6.5]{molino}. That is, a section of $\mathscr{C}_{\mathrm{reg}}$ admits a continuous extension through the singular locus, but we do not know whether in general this extension is smooth. In other words, a section of $\mathscr{C}_{\f}$ is a ``continuous transverse Killing vector field'' $X$: it restricts to a transverse Killing vector field on each stratum, but it is represented by a local vector field $\tilde{X}$ on $M$ which is only continuous.

The Molino sheaf $\mathscr{C}_\alpha$ of each $\f_\alpha$ is the quotient of the restriction of $\mathscr{C}_{\f}$ to $\Sigma_\alpha$ by the kernel of the restriction map on sections, that is, by the subsheaf consisting of those sections whose restriction to $\Sigma_\alpha$ vanish \cite[Proposition 6.8]{molino}. It is clear, hence, that the orbits of $\mathscr{C}_{\f}$ describe the closures of the leaves of $\f$. In analogy with the regular case, when $M$ is connected we say that the typical stalk $\mathfrak{g}_\f$ of $\mathscr{C}_{\f}$ is the \textit{structural Lie algebra} of $\f$. It coincides with the structural algebra of $\f_{\mathrm{reg}}$. From the relation between $\mathscr{C}_\alpha$ and $\mathscr{C}_{\f}$, it follows that the structural algebra $\mathfrak{g}_\alpha$ of $\f_\alpha$ is a quotient of $\mathfrak{g}_\f$. Also as in the regular case, we have the following.

\begin{example}[Molino sheaf of a homogeneous singular Riemannian foliation]\label{exe: molino sheaf of homogeneous singular foliation}
Let $(M,\mathrm{g})$ be a complete, connected Riemannian manifold with an isometric homogeneous foliation $\f$, given by the connected components of the orbits of $H<\mathrm{Iso}(M)$. The fundamental vector fields of the action of the closure $\overline{H}<\mathrm{Iso}(M)$ are $\mathrm{g}$-Killing and foliate. The Molino sheaf $\mathscr{C}_{\f}$ is the sheaf of Lie algebras of germs of transverse Killing vector fields induced by them. Hence $\overline{\f}$ is also homogeneous, given by the connected components of the orbits of $\overline{H}$.
\end{example}

In the above example we see that $\mathscr{C}_{\f}$ is a sheaf of Lie algebras of germs of genuine transverse Killing vector fields. The question of whether in general $\mathscr{C}_{\mathrm{reg}}$ extends smoothly to such a sheaf is known as the ``strong'' Molino conjecture (since it implies the usual one). Molino states this conjecture in \cite[p. 215]{molino}. We will see that it holds for an important class of foliations in Proposition \ref{proposition: strong molino conjecture for orbit-like}. In general, we will say that $\f$ is a singular Riemannian foliation \textit{with smooth $\mathscr{C}_{\f}$} when it satisfies the strong Molino conjecture.

\subsection{Homotheties}

Let $(M,\f)$ be a complete singular Riemannian foliation and let $J\subset M$ be a saturated submanifold contained in some stratum $\Sigma^k$. An open subset $P\subset J$ is a \textit{homothety center} when it is connected and admits a tubular neighborhood $\mathrm{Tub}_r(P)$, for some $r>0$. That is, $\mathrm{Tub}_r(P)$ is the diffeomorphic image of $\{V\in\nu P\ |\ \|V\|<r\}$ under the normal exponential map $\exp^\perp:\nu J\to M$. This is the case, for instance, when $P$ is relatively compact in $J$. If $J$ is compact (or more generally if $J$ itself is a homothety center), then any connected open subset $P\subset J$ is a homothety center. Any closed leaf $L\in \f$ is a homothety center, even if it is not compact, by \cite[Proposition 16]{mendes}. The same holds for any leaf closure $\overline{L}\in\overline{\f}$, since $\overline{\f}$ is again a singular Riemannian foliation. We denote by $\rho_P:\mathrm{Tub}_r(P)\to P$ the orthogonal projection. By shrinking $r$ if necessary, we can assume further that for any $y\in \mathrm{Tub}_r(P)$ the leaf $\f(y)$ is transverse to $\rho_P^{-1}(\rho_P(y))$. Each connected component of $\f(y)\cap \mathrm{Tub}_r(P)$ is a \textit{plaque} of $\f$.

The neighborhood $\mathrm{Tub}_r(P)$ is called a \textit{distinguished tubular neighborhood} for $P$. For all $\lambda\in(0,\infty)$ such that $\mathrm{Tub}_{\lambda r}(P)$ is a distinguished tubular neighborhood of $P$, we define the \textit{homothetic transformation}
$$h_\lambda:\mathrm{Tub}_{r}(P)\ni\exp^\perp(V)\longmapsto \exp^\perp(\lambda V)\in \mathrm{Tub}_{\lambda r}(P)$$
around $P$. Notice that each $h_\lambda$ is a diffeomorphism, and $h_\lambda\circ h_\mu=h_{\lambda\mu}$, when both sides of the equation make sense. We extend the definition to $\lambda=0$, getting $h_0=\rho_P$. One checks that in fact $h_\lambda$ converges to $\rho_P$ in the compact-open topology, as $\lambda\to 0$ (see \cite[Section 2.2]{inagaki}). These maps are of particular interest in the theory of singular Riemannian foliations because of Molino's homothetic lemma, which asserts that they are foliate with respect to the restrictions of $\f$ to $\mathrm{Tub}_{r}(P)$ and $\mathrm{Tub}_{\lambda r}(P)$ (see \cite[Lemma 6.2]{molino} for the case $\lambda>0$; for $\lambda=0$ this follows easily from the properties of $\f$). Special cases of interest are $P=\overline{L}$ and, when $M$ is compact, $P$ taken as a saturated submanifold of the minimal locus, in particular $P=\Sigma_\alpha\subset\Sigma_{\mathrm{min}}$.

When $J=L\in\f$, a fiber $S_x=\rho_P^{-1}(x)$ is a \textit{slice} for $\f$ at $x\in L$, and $\f|_{S_x}$ is called the \textit{slice foliation} at $x$. Its pullback by $\exp^\perp$ is a singular Riemannian foliation of $B_r(0)\subset T_xS_x=\nu_xL$ which, by the homothetic lemma, can be extended via homotheties to a singular Riemannian foliation $\f_x$ on the whole of $\nu_xL$, called the \textit{infinitesimal foliation at $x$}.

\subsection{Orbit-like foliations}

Relevant subclasses of singular Riemannian foliations are defined in terms of infinitesimal foliations, for instance, $\f$ is called \textit{infinitesimally closed} when $\f_x$ is closed for each $x\in M$, and \textit{orbit-like} when $\f_x$ is closed and homogeneous for each $x\in M$. By Molino's theory, the closure $\overline{\f}$ of a complete regular Riemannian foliation $\f$ is orbit-like, which led him to study this class of foliations \cite{molino2}. Other examples are given by the so-called \textit{holonomy foliations}, whose leaves are the holonomy tubes with respect to a metric connection on an Euclidean vector bundle over a Riemannian manifold $L$ (see, e.g., \cite[Example 2.7]{alex4}). Orbit-like and infinitesimally closed foliations were also prominent in the proof of Molino's conjecture in \cite{alex4} (also \cite{alex9}). In fact, let us now see that the arguments in \cite{alex9} adapt directly to conclude that the strong Molino conjecture holds for orbit-like foliations.

\begin{proposition}\label{proposition: strong molino conjecture for orbit-like}
If $\f$ is orbit-like, then $\mathscr{C}_\f$ is smooth. In other words, the strong Molino conjecture holds for orbit-like foliations.
\end{proposition}

\begin{proof}
Let $X\in \mathscr{C}_\f(U)$ be a local section and $x\in U$. We must prove that $X$ admits a smooth representative $\tilde{X}$ in some neighborhood $V\ni x$. Let $\Sigma$ be the stratum containing $x$ and let $V=\mathrm{Tub}_r(P)\subset U$ be a distinguished tubular neighborhood around an open homothety domain $P\subset\Sigma$ containing $x$. Take a slice $S\subset P$ through $x$ for $\f|_P$. We are interested in the so called \textit{local reduction} (\cite[Section 2.8]{alex9})
$$N=\exp(\{\xi\in (\nu P)|_S\ |\ \|\xi\|<r\}),$$
which generalize the notion of slice. As in the case of slices, by shrinking $V$ if necessary we can assume that there is a submersion ${p}:V\to N$, whose fibers are contained in the leaves of $\f$.

The connected components of the intersections of the leaves of $\f$ with $N$ define a singular foliation $\f_{N}$. By \cite[Proposition 2.20]{alex9}, there exists an adapted metric $\mathrm{g}_{N}$ on $N$ making $\f_{N}$ into a singular Riemannian foliation. The main point of this construction is that, since $\f$ is orbit-like, $\f_{N}$ is homogeneous, given by the orbits of a compact group. One can then lift smooth flows on the orbit space $N/\f_{N}$ to smooth flows on $N$, by a fundamental result of G.~Schwarz \cite{schwarz}.

The metric $\mathrm{g}_{N}$ preserves the transverse metric of $\f$, meaning that the distance between leaves of $\f_{N}$ with respect to $\mathrm{g}_{N}$ coincides with the distance between the corresponding plaques of $\f$ defining them. It follows that the restriction $X|_N$ defines a continuous isometric flow $\varphi$ on $N/\f_{N}$, since it restricts to a transverse Killing vector field on the dense open subset $\Sigma_{\mathrm{reg}}\cap N$. By \cite[Theorem 3.1]{alex9} $\varphi$ is smooth, and so by Schwarz's theorem it lifts to a smooth flow $\tilde{\varphi}$ on $N$. The infinitesimal generator $\tilde{X}_{N}$ is hence a smooth representative for $X|_N$. Choose any smooth field $\tilde{X}$ on $V$ $p$-related to $\tilde{X}_{N}$. Since $X$ is $\f$-invariant and the fibers of ${p}$ are contained in the leaves of $\f$, it follows that $\tilde{X}$ is a local representative of $X$.
\end{proof}

\subsection{Good covers and the dimension of basic cohomology}

Since homothetic transformations with respect to a distinguished tubular neighborhood $\mathrm{Tub}_{r}(P)$ restrict to the identity on $P$ and are defined for $\lambda\in[0,1]$, they provide a \textit{homothetic retraction}
$$h_P:\mathrm{Tub}_{r}(P)\times[0,1]\longrightarrow \mathrm{Tub}_{r}(P)$$
of $\mathrm{Tub}_{r}(P)$ onto $P$ by $h_P(x,t)=h_{1-t}(x)$. These maps, which are foliate strong deformation retractions, will be very useful to us. Let us begin by applying them in the construction of suitable open covers for $(M,\f)$. We say that a locally finite cover $\mathcal{U}=\{U_i\}$ of $M$ by $\f$-saturated open sets is a \textit{good cover} for $(M,\f)$ when any non-empty intersection $U$ of elements of the cover admits a foliate deformation retraction onto the closure of some leaf $L_U\subset U$. In the next proposition we will prove that singular Riemannian foliations of compact manifolds admit finite good covers. This result were previously stated in \cite{wolak} but we will provide an alternative proof here, because the original one relies on \cite[Lemma 2]{wolak}, which does not hold. In fact, it claims that there exists an $r>0$ which works as the radius of a tubular neighborhood $\mathrm{Tub}_{r}(\overline{L})$ for any $L\in \f$. The closed homogeneous foliation of the sphere $\mathbb{S}^2\subset \mathbb{R}^2\times\mathbb{R}$ by the usual action of $\mathrm{SO}(2)$ on the first factor $\mathbb{R}^2$ shows that this $r>0$ may not exist: the radius of a tubular neighborhood of a regular leaf must be smaller than the distance of that leaf to the singular leaves on the poles, which can be taken arbitrarily small. The claim even fails for a closed regular foliation, for a similar reason: if there is a leaf with non-trivial holonomy $L$, then the radius of a tubular neighborhood of a leaf with trivial holonomy $L'$ must be smaller than the distance $d(L,L')$.

\begin{proposition}\label{proposition: compact implies finite type}\label{proposition: good cover}
A singular Riemannian foliation $\f$ of a compact manifold $M$ admits a finite good cover $\mathcal{U}$ with the additional property that each non-empty intersection of its elements retracts to a leaf closure in it by a concatenation of homothetic retractions.
\end{proposition}

\begin{proof}
We begin by noticing that, since $\overline{\f}$ is again a Riemannian foliation, we can assume that $\f$ is closed: a cover for $\overline{\f}$ with the desired properties will work for $\f$ as well, since $\overline{\f}$-saturated sets are $\f$-saturated.

To establish the result for closed foliations, let us first consider the case of a closed regular Riemannian foliation $\f$ of a compact manifold $M$. In this case $M/\f$ is naturally a compact Riemannian orbifold (see \cite[Proposition 3.7]{molino}), and a finite good cover for $\f$ is given by the collection $\mathcal{U}=\{U_j\}$ of preimages under $\pi:M\to M/\f$ of a finite good cover $\mathcal{W}=\{W_j\}$ of $M/\f$, which in turn can be obtained via convex neighborhoods, as in the manifold case (see \cite[Proposition 4.4.2]{choi}). Moreover, by choosing the convex neighborhoods appropriately we can assume that, if $M/\f$ previously had an arbitrary finite open cover $\mathcal{A}$, then $\mathcal{W}$ could have been chosen to be a refinement of it (that is, with each $W_j$ contained in some $A\in\mathcal{A}$). By construction, a non-empty intersection $W$ of elements of $\{W_j\}$ retracts by deformation to a point $x_W$ by the geodesic convexity, so the deformation retraction of $U=\pi^{-1}(W)$ onto the corresponding leaf ${L_U=\pi^{-1}(x_W)}$ is (a restriction of) a homothetic retraction.

Now assume that $\f$ is a closed singular Riemannian foliation of $M$, and let $k_0<\dots<k_\ell=\dim(\f)$ be the dimensions of the leaves in $\f$. In this step we want to build a finite cover $\mathcal{U}_0$ of the minimal locus $\Sigma_{\mathrm{min}}=\Sigma^{k_0}$ by saturated open subsets of $M$, with the property that each intersection of elements in this cover retracts to some leaf closure in $\Sigma^{k_0}$ by a concatenation of homothetic retractions. Since $\Sigma^{k_0}$ is a disjoint union of a finite collection $\{\Sigma_i\}$ of compact strata, we can choose $r>0$ such that $\mathrm{Tub}_{2r}(\Sigma_i)$ are pairwise disjoint tubular neighborhoods. By the previous case, for each $i$ there exists a finite good cover $\{V_{ij}\}$ for $\f|_{\Sigma_i}$ by saturated open sets in $\Sigma_i$. Setting $U_{ij}:=\rho_{\Sigma_i}^{-1}(V_{ij})$ we see that the collection $\mathcal{U}_0=\{U_{ij}\}$, for $i$ and $j$ varying over all possible values, covers $\mathrm{Tub}_{2r}(\Sigma^{k_0}):=\bigsqcup_i \mathrm{Tub}_{2r}(\Sigma_i)$ by saturated open sets. Moreover, a non-empty intersection of elements in $\mathcal{U}_0$ is of the form $U=U_{i_0j_0}\cap\dots\cap U_{i_0 j_l}$, (all terms with the same $i$-index, by the choice of $r$). Hence $U$ retracts, by a homothetic retraction with respect to $\Sigma_{i_0}$, to the intersection $V=V_{i_0j_0}\cap\dots\cap V_{i_0 j_l}\subset\Sigma_{i_0}$. By construction, $V\subset \Sigma_{i_0}$ in turn homothetically retracts onto some leaf $L_U\subset\Sigma_{i_0}$. Observe once again that if $\f|_{\Sigma^{k_0}}$ previously had an arbitrary finite open cover $\mathcal{A}$ by saturated open sets in $\Sigma^{k_0}$, then we could have chosen $\{V_{ij}\}$ to be a refinement of it; this will be important for the next step.

Let us now show that it is possible to extend $\mathcal{U}_0$ to a finite cover $\mathcal{U}_1$ of $\Sigma^{k_0}\sqcup\Sigma^{k_1}$ by saturated open subsets in $M$, maintaining the property that each intersection of elements in $\mathcal{U}_1$ retracts, by a concatenation of homothetic retractions, to some leaf closure in $\Sigma^{k_0}\sqcup\Sigma^{k_1}$. By \cite[Theorem 1.2]{alex1}, if we blow $M$ up along each connected component of $\Sigma^{k_0}$ using the neighborhood $\mathrm{Tub}_{r}(\Sigma^{k_0})$, we obtain:
\begin{enumerate}[(1)]
\item a compact Riemannian manifold $\hat{M}$,
\item a singular Riemannian foliation $\hat{\f}$ of $\hat{M}$, whose leaves have dimensions $k_1<\dots<k_\ell$, and
\item a foliate smooth map $\hat{\pi}:\hat{M}\to M$ which restricts to an isometry $\hat{M}\setminus \mathrm{Tub}_{r}(\hat{\pi}^{-1}((\Sigma^{k_0})))\to M\setminus \mathrm{Tub}_{r}(\Sigma^{k_0})$.
\end{enumerate}
Consider the minimal stratum $\hat{\Sigma}_{\mathrm{min}}$ of $\hat{\f}$, and cover it with
$$\mathcal{A}=\left\{\mathrm{Tub}_{r}(\hat{\pi}^{-1}((\Sigma^{k_0})))\cap \hat{\Sigma}_{\mathrm{min}}, \mathrm{Tub}_{2r}(\hat{\pi}^{-1}((\Sigma^{k_0})))\cap \hat{\Sigma}_{\mathrm{min}}, \hat{\Sigma}_{\mathrm{min}}\mathbin{\big\backslash}\overline{\mathrm{Tub}_{r}(\hat{\pi}^{-1}((\Sigma^{k_0})))}\right\}.$$
Then, by the construction of the last paragraph applied to $\hat{\f}$, we obtain a finite saturated cover $\hat{\mathcal{U}}=\{\hat{U}_{ij}\}$ of a tubular neighborhood $\mathrm{Tub}_{\hat{r}}(\hat{\Sigma}_{\mathrm{min}})$, with $\hat{r}<r$, such that the cover $\{\hat{V}_{ij}\}$ of $\hat{\Sigma}_{\mathrm{min}}$ by the projections $\hat{V}_{ij}=\rho_{\hat{\Sigma}_{\mathrm{min}}}(\hat{U}_{ij})$ refines $\mathcal{A}$. Define $\mathcal{U}_1$ to be the union of $\mathcal{U}_0$ with the collection of images $\hat{\pi}(\hat{U}_{ij})$ of the elements $\hat{U}_{ij}\in\hat{\mathcal{U}}$ that do not intersect $\mathrm{Tub}_{r}(\hat{\pi}^{-1}((\Sigma^{k_0})))$. This ensures that a non-empty intersection $U$ of elements in $\mathcal{U}_1$ retracts, again by a concatenation of homothetic retractions, to some leaf $L_U\subset U$. In fact, since $\hat{\pi}$ restricts to an isometry $\hat{M}\setminus \mathrm{Tub}_{r}(\hat{\pi}^{-1}((\Sigma^{k_0})))\to M\setminus \mathrm{Tub}_{r}(\Sigma^{k_0})$, if the intersection involves any of the added sets $\hat{\pi}(\hat{U}_{ij})$, then it homothetically retracts to a saturated open set $V$ in $\Sigma^{k_1}$ which is contained in some $\hat{\pi}(\hat{V}_{ij})$. By construction, $V$ in turn homothetically retracts to a leaf $L_U\subset \hat{\pi}(\hat{V}_{ij})$.

By blowing $\hat{M}$ up again along each connected component of $\hat{\Sigma}_{\mathrm{min}}$, the same construction produces an extension of $\mathcal{U}_1$ to a cover $\mathcal{U}_2$ of $\Sigma^{k_0}\sqcup\Sigma^{k_1}\sqcup\Sigma^{k_2}$ by saturated open subsets in $M$ with the desired properties. Continuing this process until $\Sigma^\ell=\Sigma_{\mathrm{reg}}$ one obtains the desired cover $\mathcal{U}$.
\end{proof}

We can now apply the classical Mayer--Vietoris argument to conclude that $H(\f)$ is finite dimensional.

\begin{theorem}\label{theorem: basic cohomology is finite dimensional}
Let $\f$ be a singular Riemannian foliation of a compact manifold $M$. Then $\dim H(\f)<\infty$.
\end{theorem}

\begin{proof}
We proceed by induction on the number $n$ of elements in a finite good cover $\mathcal{U}$ for $\f$. If $n=1$ then $M$ itself admits a foliate deformation retraction to a leaf closure $J=\overline{L}$. By Corollary \ref{corollary: invariance of ebc under homotopies and deformation retracts} (taking $\mathfrak{g}=0$) we have $H(\f)\cong H(\f|_J)$. Since $\f|_J$ is a regular Riemannian foliation of the compact manifold $J$, its basic cohomology is finite dimensional (see \cite[Theorem 0]{kacimi}).

For the induction step, let $\mathcal{U}=\{U_1,\dots, U_n\}$ be a good cover. Defining $U=\bigcup_{i=2}^n U_i$, we easily see that both $\f|_U$ and $\f|_{U_1\cap U}$ are Riemannian foliations admitting a good covers with $n-1$ elements. Thus, by the induction hypothesis, both extremes in the exact sequence
$$H^{k-1}(\f|_{U_1\cap U})\longrightarrow H^{k}(\f) \longrightarrow H^k(\f|_{U_1})\oplus H^k(\f|_{U}),$$
are finite dimensional, for each $k$. Hence $\dim H(\f)\leq\infty$.
\end{proof}


\section{The natural transverse action on singular Killing foliations}

In analogy with the regular case, we define a \textit{singular Killing foliation} as a complete singular Riemannian foliation $\f$ whose Molino sheaf $\mathscr{C}_\f$ is a globally constant sheaf of Lie algebras of germs of transverse Killing vector fields. In other words, $\f$ is a complete singular Riemannian foliation with a globally constant, smooth $\mathscr{C}_\f$. An important class of Killing foliations is that of isometric homogeneous foliations.

\begin{example}[Homogeneous Riemannian foliations are Killing]\label{example: homogeneous Riemannian foliations are Killing}
Let $(M,\f)$ be a complete homogeneous Riemannian foliation, given by the orbits of an isometric action. Then $\f$ is Killing, as it is clear from Example \ref{exe: molino sheaf of homogeneous singular foliation} that $\mathscr{C}_\f$ is a globally constant sheaf of Lie algebras of germs of transverse Killing vector fields.
\end{example}

Notice that if $M$ is simply connected, then $\mathscr{C}_\f$ has trivial monodromy and hence is globally constant. This gives us another important class of examples.

\begin{example}[Orbit-like Riemannian foliations on simply connected manifolds are Killing]
As we saw in Proposition \ref{proposition: strong molino conjecture for orbit-like}, if $\f$ is an orbit-like foliation then $\mathscr{C}_\f$ is smooth, hence $\f$ is Killing when $M$ is simply connected.
\end{example}

As we saw in \ref{section: (Singular) Riemannian foliations}, the Molino sheaf $\mathscr{C}_\alpha$ of the restriction $\f_\alpha$ of $\f$ to a stratum $\Sigma_\alpha$ is a quotient of $\mathscr{C}_{\f}$, hence constant. Moreover, one can change the transverse metric on $(\Sigma_\alpha,\f_\alpha)$ so that it becomes complete, and this in fact does not alter $\mathscr{C}_\alpha$, as it is independent of the transverse metric (see \cite[Section 5.3]{molino}). Therefore, each $\f_\alpha$ is a (regular) Killing foliation. The structural algebra $\mathfrak{a}$ of $\f$ is thus Abelian. Notice that, for a Killing foliation, $\mathfrak{a}$ is well even if $M$ is not connected, since $\mathscr{C}_\f$ is constant. It acts transversely on $\f$ via the isomorphism $\mathfrak{a}\cong \mathscr{C}_\f(M)$, and $\mathfrak{a}\f=\overline{\f}$. From what we saw in Section \ref{section: equivariant basic cohomology}, with this natural transverse $\mathfrak{a}$-action, $\Omega(\f)$ becomes an $\mathfrak{a}^\star$-algebra and we can consider the equivariant cohomology $H_{\mathfrak{a}}(\f)$. Since $\mathfrak{a}\f=\overline{\f}$, it follows that
$$M^{\mathfrak{a}}=\Sigma_{\mathrm{cl}},$$
that is, the $\mathfrak{a}$-fixed locus is the union of closed leaves of $\f$. It is then expected that $H_{\mathfrak{a}}(\f)$ carries information on the closed leaves on $\f$. Before we investigate that, let us establish some basic tools, the first one being the fact that the $\mathfrak{a}$-action behaves well under homothetic transformations.

\subsection{Homothetic invariance of the Molino sheaf}\label{section: homothetic invariance of Molino sheaf}

Let $\mathrm{Tub}_{r}(P)$ be a distinguished tubular neighborhood and $X$ be a smooth vector field on $\mathrm{Tub}_{r}(P)$ which is tangent to $P$. The \textit{linearization of $X$ around $P$} is the vector field $X^\ell$ given by
$$X^\ell=\lim_{\lambda\to 0} \dif h_\lambda^{-1}(X\circ h_\lambda).$$
By choosing adequate coordinates one verifies that $X^\ell$ is in fact well defined, smooth, $h_\lambda$-invariant for all admissible $\lambda$ (including $\lambda=0$) and moreover $X^\ell|_P=X|_P$ (see \cite[Proposition 13]{mendes}). Furthermore, $X^\ell$ is foliate when $X$ is foliate (see \cite[Proposition 2.14]{inagaki}). The next lemma will help us prove that sections of $\mathscr{C}_{\f}$ are invariant under certain homothetic transformations.

\begin{lemma}\label{lemma: invariance of molino sheaf under homothetic transformations}
Let $(M,\f)$ be a complete singular Riemannian foliation with smooth Molino sheaf, $J\subset M$ an $\overline{\f}$-saturated homothety center, and $h_\lambda:\mathrm{Tub}_{r}(J)\to \mathrm{Tub}_{\lambda r}(J)$ a homothetic transformation, with $\lambda>0$. If $\tilde{X}\in\mathfrak{L}$ is a representative of a section $X\in\mathscr{C}_\f(\mathrm{Tub}_r(P))$, for $P\subset J$ an open subset, then $\dif (h_\lambda)_x\tilde{X}_x-\tilde{X}_{h_\lambda(x)}\in T_{h_\lambda(x)}\f$, for all $x\in \mathrm{Tub}_r(P)$.
\end{lemma}

\begin{proof}
It suffices to show this for $\lambda\in(0,1]$. In fact, assuming that, the result for an admissible $\lambda>1$ follows by the chain rule (as $h_\lambda\circ h_{1/\lambda}=\mathrm{id}_{\mathrm{Tub}_r(J)}$), and the fact that homothetic transformations are foliate.

Since $h_t\circ h_s=h_{ts}$, we have that $h_{e^t}(x)$ defines a (local) flow on $\mathrm{Tub}_{r}(J)$. Consider the infinitesimal generator $\tilde{R}$ of this flow, that is
$$\tilde{R}(x)=\left.\od{}{t}h_{e^t}(x)\right|_{t=0}.$$
It is the foliate radial vector field around $J$, and in particular it is tangent to each stratum of $\f$. For a fixed $x\in \mathrm{Tub}_r(P)$, take $r'<r$ with $x\in\mathrm{Tub}_{r'}(J)$ and extend $\tilde{R}$ to a global foliate field by multiplying it with a basic bump function for $\mathrm{Tub}_{r'}(J)$ with support in $\overline{\mathrm{Tub}_{r}(J)}$ and defining $\tilde{R}=0$ outside that set. We will denote the flow of $\tilde{R}$ by $\theta:\mathbb{R}\times M\to M$.

Now since $X$ is a section of $\mathscr{C}_\f$, its restriction $X|_\Sigma$ to the stratum $\Sigma\ni x$ commutes with each global transverse vector field in $\mathfrak{l}(\f|_\Sigma)$ (see \cite[p. 160]{molino}), in particular with the transverse field induced by the restriction $\tilde{R}|_\Sigma$. This means $[\tilde{X}|_\Sigma,\tilde{R}|_\Sigma]$ is tangent to $\f$ on $U=\Sigma\cap\mathrm{Tub}_{r}(P)$. Define $\tilde{Z}(t)=d(\theta_{-t})_{\theta_t(x)}(\tilde{X}_{\theta_t(x)})$. Then
$$\left.\od{}{t}\tilde{Z}(t)\right|_{t=s}=\dif(\theta_{-s})[\tilde{X}|_\Sigma,\tilde{R}|_\Sigma]_{\theta_s(x)}\in T_x\f,$$
since $\theta_{-s}$ is foliate. Moreover, $\tilde{Z}(0)=\tilde{X}_x$, therefore
$$\tilde{Z}(t)-\tilde{X}_x=\int_0^t \dif(\theta_{-s})[\tilde{X}|_\Sigma,\tilde{R}|_\Sigma]_{\theta_s(x)}\;ds\in T_x\f$$
for each $t$. Thus
$$\dif(\theta_t)_x(\tilde{X}_x-\tilde{Z}(t))=\dif(\theta_t)_x\tilde{X}_x-\tilde{X}_{\theta_t(x)}\in T_{\theta_t(x)}\f.$$
By construction, for $t\leq 0$ the flow $\theta_t$ is given by $h_{e^t}$ on $\mathrm{Tub}_{r'}(J)$, hence
$$\dif (h_\lambda)_x\tilde{X}_x-\tilde{X}_{h_\lambda(x)}\in T_{h_\lambda(x)}\f,$$
for $\lambda\in(0,1]$.
\end{proof}

We can now prove that sections of $\mathscr{C}_\f$ admit linearized representatives.

\begin{proposition}[homothetic invariance of the Molino sheaf]\label{proposition: invariance of molino sheaf under homothetic transformations}
Let $(M,\f)$ be a complete singular Riemannian foliation with smooth Molino sheaf, $J\subset M$ an $\overline{\f}$-saturated homothety center, $h_\lambda:\mathrm{Tub}_{r}(J)\to \mathrm{Tub}_{\lambda r}(J)$ a homothetic transformation, and $P\subset J$ an open subset. Then any $X\in\mathscr{C}_\f(\mathrm{Tub}_r(P))$ admits a linearized representative, and hence is $h_\lambda$-invariant (in particular $\rho_P$-invariant).
\end{proposition}

\begin{proof}
Let $\tilde{X}$ be any representative for $X$ on $\mathrm{Tub}_r(P)$. By Lemma \ref{lemma: invariance of molino sheaf under homothetic transformations}, $\dif (h_\lambda)\tilde{X}-\tilde{X}\circ h_\lambda$ is tangent to $\f$, hence $\dif (h_{1/\lambda})\dif (h_\lambda)\tilde{X}-\dif (h_\lambda)\tilde{X}\circ h_\lambda$ is tangent to $\f$ as well. Then
$$\tilde{X}-\lim_{\lambda \to 0} \dif (h_\lambda)\tilde{X}\circ h_\lambda=\tilde{X}-\tilde{X}^\ell$$
must also be tangent to $\f$, and thus $\tilde{X}^\ell$ is a representative for $X$. We already know that $\tilde{X}^\ell$ is $h_\lambda$-invariant for any admissible $\lambda$.
\end{proof}

An $\overline{\f}$-saturated homothety center $J\subset M$ is $\mathfrak{a}$-invariant, since $\mathfrak{a}\f=\overline{\f}$. By Proposition \ref{proposition: invariance of molino sheaf under homothetic transformations}, a homothetic retraction
$$h_J:\mathrm{Tub}_r(J)\times[0,1]\to \mathrm{Tub}_r(J)$$
is $\mathfrak{a}$-equivariant. Let us sum this up in the following corollary.

\begin{corollary}\label{corollary: homothetic retractions are a-equivariant}
Let $(M,\f)$ be a singular Killing foliation. A homothetic retraction $h_J$ with respect to an $\overline{\f}$-saturated homothety center $J\subset M$ is a foliate, $\mathfrak{a}$-equivariant deformation retraction of $\mathrm{Tub}_r(J)$ onto $J$.
\end{corollary}

\subsection{Calculating tools}

Combining Corollary \ref{corollary: homothetic retractions are a-equivariant} and Corollary \ref{corollary: invariance of ebc under homotopies and deformation retracts}, we obtain the following property, which is crucial.

\begin{proposition}[Poincaré lemma for equivariant basic cohomology]\label{proposition: poincare lema for equivariant basic cohomology}
Let $(M,\f)$ be a singular Killing foliation and suppose $J\subset M$ is an $\overline{\f}$-saturated homothety center. Then for any $r>0$ such that $\mathrm{Tub}_r(J)$ is a distinguished tubular neighborhood, the inclusion $J\to \mathrm{Tub}_r(J)$ induces an isomorphism
$$H_{\mathfrak{a}}(\f|_{\mathrm{Tub}_r(J)})\cong H_{\mathfrak{a}}(\f|_{J})$$
of $\sym(\mathfrak{a}^*)$-algebras.
\end{proposition}

Notice that, also by Corollary \ref{corollary: homothetic retractions are a-equivariant}, in the case of a singular Killing foliation $\f$ the good cover $\mathcal{U}=\{U_i\}$ constructed in Proposition \ref{proposition: compact implies finite type} is \textit{equivariant}: any non-empty intersection $U$ of its elements admits a foliate, $\mathfrak{a}$-equivariant, deformation retraction onto the closure of some leaf $L_U\subset U$.

Let us now establish that $H_{\mathfrak{a}}(\f)$ can be calculated via Mayer--Vietoris sequences. We will adapt the arguments from \cite{goertsches} for the regular case.

\begin{lemma}\label{lemma: basic forms are invariant}
Let $\f$ be a singular Killing foliation of $M$ with structural algebra $\mathfrak{a}$ and let $U\subset M$ be an open, $\f$-saturated subset. Then $\Omega(\f|_U)^\mathfrak{a}=\Omega(\f|_U)$, and hence $\Omega_\mathfrak{a}(\f|_U)=\sym(\mathfrak{a}^*)\otimes\Omega(\f_U)$.
\end{lemma}

\begin{proof}
Since the regular locus $\Sigma_\mathrm{reg}$ of $\f$ is open and dense in $M$, it is sufficient to show that $\Omega(\f|_{U\cap\Sigma_\mathrm{reg}})^\mathfrak{a}=\Omega(\f|_{U\cap\Sigma_\mathrm{reg}})$. On $\Sigma_\mathrm{reg}$, the transverse $\mathfrak{a}$-action is nothing but the natural transverse action of the structural algebra of the regular Killing foliation $\f_{\mathrm{reg}}$, so the result follows from \cite[Lemma 3.15]{goertsches}.
\end{proof}

The proof of \cite[Proposition 3.16]{goertsches} now adapts directly to the singular setting: for $\f$-saturated open sets $U,V\subset M$,
$$0\longrightarrow \Omega(\f|_{U\cup V})\stackrel{i_U^*\oplus i_V^*}{\longrightarrow}\Omega(\f|_U)\oplus \Omega(\f|_V)\stackrel{j_U^*-j_V^*}{\longrightarrow} \Omega(\f|_{U\cap V})\longrightarrow 0$$
is a short exact sequence of $\mathfrak{a}^\star$-modules. Since tensoring with $\sym(\mathfrak{a}^*)$ preserves exactness, we obtain, by Lemma \ref{lemma: basic forms are invariant}, the short exact sequence
$$0\longrightarrow \Omega_\mathfrak{a}(\f|_{U\cup V})\longrightarrow \Omega_\mathfrak{a}(\f|_U)\oplus \Omega_\mathfrak{a}(\f|_V)\longrightarrow \Omega_\mathfrak{a}(\f|_{U\cap V})\longrightarrow 0$$
that induces an exact Mayer--Vietoris sequence in equivariant basic cohomology. Let us state this below.

\begin{proposition}[Mayer–Vietoris sequence for equivariant basic cohomology]
Let $\f$ be a singular Killing foliation of $M$ with structural algebra $\mathfrak{a}$ and $U,V\subset M$ saturated open sets. Then the sequence of $\sym(\mathfrak{a}^*)$-modules
$$\cdots\stackrel{\beta}{\longrightarrow}H_\mathfrak{a}(\f|_{U\cup V})\stackrel{i_U^*\oplus i_V^*}{\longrightarrow}H_\mathfrak{a}(\f|_U)\oplus H_\mathfrak{a}(\f|_V)\stackrel{j_U^*-j_V^*}{\longrightarrow} H_\mathfrak{a}(\f|_{U\cap V})\stackrel{\beta}{\longrightarrow}\cdots$$
is exact.
\end{proposition}

\subsection{Borel type localization}

A remarkable feature of the equivariant cohomology of the action of a torus $T$ on a manifold $M$ is that the torsion-free part of the module structure of $H_T(M)$ can be recovered from the fixed point set $M^T$. This property became know as Borel's localization, as he observed close relations between $H_T(M)$ and $H_T(M^T)$ (see \cite[Chapter XII]{borel}) which later led to the formalization of the localization theorem. Our objective in this section is to obtain a version of this result for equivariant basic cohomology of singular Killing foliations. The case of regular Killing foliations, with the canonical action by the structural algebra, was treated in \cite{goertsches}. It was later generalized for arbitrary transverse actions on (regular) Riemannian foliations in \cite{lin}.

This topic is better formulated with the notion of localization of modules, for which we refer to \cite[Chapter 3]{atiyah2}. In fact we will be interested in the localization $S^{-1}H_{\mathfrak{a}}(\f)$ of $H_{\mathfrak{a}}(\f)$ with respect to $S=\sym(\mathfrak{a}^*)\setminus\{0\}$. We can view an element of $S^{-1}H_{\mathfrak{a}}(\f)$ as a fraction $[\omega]/f$, for $[\omega]\in H_{\mathfrak{a}}(\f)$ and $f\in S$. The usual fraction operations turn $S^{-1}H_{\mathfrak{a}}(\f)$ into a vector space over the field $\mathrm{Q}(\mathfrak{a}^*)$ of fractions of $\sym(\mathfrak{a}^*)$, whose dimension is the rank of $H_{\mathfrak{a}}(\f)$. Recall that $[\omega]\in H_{\mathfrak{a}}(\f)$ is a \textit{torsion element} if there exists $f\in S$ such that $f[\omega]=0$. The submodule $\tor(H_{\mathfrak{a}}(\f))$ of torsion elements is the \textit{torsion submodule} of $H_{\mathfrak{a}}(\f)$. A module is a \textit{torsion module} when it is equal to its torsion submodule. In particular, $H_{\mathfrak{a}}(\f)$ is torsion if, and only if, $S^{-1}H_{\mathfrak{a}}(\f)=0$. The following fact will be useful.

\begin{proposition}\label{proposition: cohomology of non closed leaves is torsion}
Let $\f$ be a singular Killing foliation. Then $H_{\mathfrak{a}}(\f|_{\overline{L}})=\sym(\mathfrak{a}_L^*)$ for every $L\in\f$. In particular, if $L$ is not closed, then $H_{\mathfrak{a}}(\f|_{\overline{L}})$ is a torsion module.
\end{proposition}

\begin{proof}
The restriction $\f|_{\overline{L}}$ is a (regular) Killing foliation of the manifold $\overline{L}$, hence it follows from \cite[Corollary 3.21]{goertsches} that $H_{\mathfrak{a}}(\f|_{\overline{L}})=\sym(\mathfrak{a}_L^*)$, which is torsion when $L$ is not closed: if $f\in\sym(\mathfrak{a}^*)$ is any non-zero form that vanishes on $\mathfrak{a}_L^*$, then multiplication by $f$ is the zero map on $\sym(\mathfrak{a}_L^*)$.
\end{proof}

The regular version of Borel's localization theorem states that $S^{-1}H_{\mathfrak{a}}(\f)\cong S^{-1}H_{\mathfrak{a}}(\f|_{M^\mathfrak{a}})$. In the singular setting there is a particular phenomenon that, in principle, prevents us from obtaining the same result for all Killing foliations: unlike the regular case, for some singular Killing foliations the closed locus $M^\mathfrak{a}$ is singular, that is, its connected components may not be submanifolds of $M$.

\begin{example}\label{example: propagation of non minimal closed leaves}
Fix a $\lambda\in\mathbb{R}\setminus\{0\}$ and consider the isometric $\mathbb{R}$-action on $\mathbb{C}\times\mathbb{C}$ given by
$$t(z_1,z_2)=(e^{2\pi\mathrm{i}t}z_1,e^{2\pi\mathrm{i}\lambda t}z_2).$$
Denote by $H<\mathrm{SO}(4)$ the corresponding one-parameter subgroup. As we saw in Example \ref{example: homogeneous Riemannian foliations are Killing}, the homogeneous foliation $\f$ given by its orbits is Killing. Notice that its only singular leaf is the origin $\{0\}$. Since $\f$ is invariant by homotheties with respect to it, it is completely determined by its restriction to $\mathbb{S}^3\subset \mathbb{C}\times\mathbb{C}$, which is a regular Killing foliation with one dimensional leaves (the orbits of the restricted $\mathbb{R}$-action, of course) known as a \textit{generalized Hopf fibration}. We are interested in the case of an irrational $\lambda$, but it is instructive to first analyze rational cases in order to get a better grasp. If $\lambda=1$ one gets the usual Hopf fibration on $\mathbb{S}^3$, whereas if $\lambda=p/q\in\mathbb{Q}$, one gets its orbifold version, whose quotient is the weighted projective space $\mathbb{CP}(p,q)$ (also known as the $(p,q)$-football orbifold): a spherical orbifold with cone singularities of order $p$ and $q$ at the poles corresponding to the exceptional orbits $\mathbb{R}(1,0)$ and $\mathbb{R}(0,1)$. In either case $\f$ is closed, so let us now assume that $\lambda$ is irrational. The exceptional orbits $\mathbb{R}(1,0)$ and $\mathbb{R}(0,1)$ remain closed (two linked circles), but now have infinite holonomy. As we saw in Example \ref{exe: molino sheaf of homogeneous singular foliation}, the closure $\overline{\f}$ is given by the orbits of $\overline{H}<\mathrm{SO}(4)$, which is a maximal torus. A regular $\overline{H}$-orbit is a Clifford torus in $\mathbb{S}^3$, on which $\f$ restricts to an irrational Kronecker foliation. By the homothetic invariance of $\f$ it is clear that the closed locus $M^\mathfrak{a}$ is a double cone with vertex $\{0\}$ and with each nappe ``homothetically spanned'' by an exceptional orbit.
\end{example}

More generally, this phenomenon happens whenever there is a closed $l$-dimensional leaf $L$ near a stratum $\Sigma^k_\alpha$, with $k<l$: since $\mathscr{C}_\f$ is invariant by homothetic transformations (Proposition \ref{proposition: invariance of molino sheaf under homothetic transformations}) $L$ ``propagates'' towards $\Sigma^k_\alpha$, leading to singularities in $M^\mathfrak{a}$. This observation motivates the following definition.

\begin{definition}\label{definition: neat skf}
A singular Killing foliation $(M,\f)$ is \textit{neat} when each connected component $C$ of $M^\mathfrak{a}$ is contained in a single stratum of $\f$.
\end{definition}

Notice that if $\f$ is neat then a connected component $C\subset M^{\mathfrak{a}}$ is a stratum of $\overline{\f}$. Also, if $\Sigma_\alpha$ is the $\f$-stratum containing $C$, then $C$ is a minimal stratum of $\overline{\f}|_{\Sigma_\alpha}$. It follows that each $C\subset M^\mathfrak{a}$ is a closed, saturated, horizontally totally geodesic submanifold of $M$. An important particular case is that of a singular Killing foliation $\f$ with $M^{\mathfrak{a}}\subset \Sigma_{\mathrm{min}}$, in which case $M^{\mathfrak{a}}$, if not empty, is precisely the minimal locus of $\overline{\f}$. Another subclass is that of singular Killing foliations with isolated closed leaves. Notice also that, of course, if $M^{\mathfrak{a}}=\emptyset$ or if $\f$ is regular, then $\f$ is automatically neat.

Let us now prove Borel's localization property for the equivariant basic cohomology of neat singular Killing foliations. We denote $\f^\mathfrak{a}=\f|_{M^{\mathfrak{a}}}$.

\begin{theorem}[Borel localization for equivariant basic cohomology]\label{theorem: borel localization}
Let $\f$ be a neat singular Killing foliation of a compact manifold $M$, with structural algebra $\mathfrak{a}$, and let $i:M^{\mathfrak{a}}\to M$ be the natural inclusion. Then $\ker i^*=\tor(H_{\mathfrak{a}}(\f))$ and $\coker i^*$ is a torsion module. Hence, the localized map
$$S^{-1}i^*:S^{-1}H_{\mathfrak{a}}(\f)\longrightarrow S^{-1}H_{\mathfrak{a}}(\f^\mathfrak{a})$$
is an isomorphism, where $S=\sym(\mathfrak{a}^*)\setminus\{0\}$.
\end{theorem}

\begin{proof}
Let $\mathcal{U}=\{U_i\}$ be a finite equivariant good cover for $(M,\f)$ given by Proposition \ref{proposition: good cover}. Recall that $\mathcal{U}$ is constructed recursively, by a sequence $\mathcal{U}_0\subset \dots\subset \mathcal{U}_\ell=\mathcal{U}$, with each $\mathcal{U}_j$ obtained from a cover $\hat{\mathcal{U}}_j$ of a tubular neighborhood of the minimal locus of the $j$-th blow-up of $\overline{\f}$. Since $\f$ is neat, a connected component $C\subset M^{\mathfrak{a}}$ is a stratum of $\overline{\f}$, hence a minimal stratum of some of its blow-ups. Therefore
$$U=\bigcup_{U_i\cap M^{\mathfrak{a}}\neq \emptyset} U_i=\bigsqcup_k \mathrm{Tub}_{r_k}(C_k),$$
where $C_k$ runs over all connected components of $M^{\mathfrak{a}}$. In particular, we have that the inclusion $i|^U:M^\mathfrak{a}\to U$ induces an isomorphism $H_\mathfrak{a}(\f|_U)\cong H_\mathfrak{a}(\f^\mathfrak{a})$, by Proposition \ref{proposition: poincare lema for equivariant basic cohomology}. Define $V=\bigcup_{U_i\cap M^{\mathfrak{a}}= \emptyset} U_i$, so that $\{U,V\}$ is a cover of $M$ with the property that $\mathcal{U}$ restricts to finite equivariant good covers for both $\f|_V$ and $\f|_{U\cap V}$.

Now we can just follow the standard proofs of the classical Borel localization for torus actions (see, for instance, \cite[Theorem C.21]{guillemin3}, \cite[Theorem 11.4.4]{guillemin} and \cite[Theorem 8.1]{goertsches3}), which we include here for the sake of completeness. First, we claim that $H_{\mathfrak{a}}(\f|_{V})$ and $H_{\mathfrak{a}}(\f|_{U\cap V})$ are torsion modules. In fact, let us prove that if $W\subset M\setminus M^{\mathfrak{a}}$ is any saturated open set admitting a finite equivariant good cover $W=W_1\cup\dots\cup W_k$, then $H_{\mathfrak{a}}(\f|_W)$ is torsion. If $k=1$ then $W$ retracts equivariantly to the closure of a non-closed leaf $L\subset V$, therefore $H_{\mathfrak{a}}(\f|_{V})\cong H_{\mathfrak{a}}(\f|_{\overline{L}})$ is torsion, by Proposition \ref{proposition: cohomology of non closed leaves is torsion} (and Corollary \ref{corollary: invariance of ebc under homotopies and deformation retracts}). For the induction step, define $W'=W_1\cup\dots\cup W_{k-1}$, so $W=W'\cup W_k$. Then, by the induction hypothesis, both extremes of the exact sequence
$$H_{\mathfrak{a}}(\f|_{W'\cap W_k})\longrightarrow H_{\mathfrak{a}}(\f|_W) \longrightarrow H_{\mathfrak{a}}(\f|_W')\oplus H_{\mathfrak{a}}(\f|W_k)$$
are torsion modules.

We have $i^*(\tor(H_\mathfrak{a}(\f)))\subset \tor(H_\mathfrak{a}(\f^\mathfrak{a}))=0$, the last equality following from the fact that $H_{\mathfrak{a}}(\f^\mathfrak{a})=\sym(\mathfrak{a}^*)\otimes H(\f^\mathfrak{a})$ is free (see Example \ref{example: equivariant cohomology of trivial action}). Thus, $\tor(H_{\mathfrak{a}}(\f))\subset \ker i^*$. Conversely, since $i=i_U\circ i|^U$, from the Mayer--Vietoris sequence for the cover $\{U,V\}$ we obtain the exact sequence
\begin{equation}\label{equation: M-V sequence for the cover UV}
\cdots\longrightarrow H_\mathfrak{a}(\f|_{U\cap V})\longrightarrow H_\mathfrak{a}(\f)\stackrel{i^*\oplus i_V^*}{\longrightarrow}H_\mathfrak{a}(\f^\mathfrak{a})\oplus H_\mathfrak{a}(\f|_V) \stackrel{j^*-j_V^*}{\longrightarrow} H_\mathfrak{a}(\f|_{U\cap V})\longrightarrow\cdots.
\end{equation}
One has $\ker(i^*\oplus i_V^*)=\ker i^*\cap\ker i_V^*$, so the sequence
$$H_\mathfrak{a}(\f|_{U\cap V})\longrightarrow \ker i^*\stackrel{i_V^*}{\longrightarrow} H_\mathfrak{a}(\f|_V)$$
of $\sym(\mathfrak{a}^*)$-modules is also exact. As the extremes of this sequence are torsion modules, it follows that $\ker i^*\subset \tor(H_{\mathfrak{a}}(\f))$.

The exactness of \eqref{equation: M-V sequence for the cover UV} also implies that
$$\coker(i^*\oplus i_V^*)=\frac{H_\mathfrak{a}(\f^\mathfrak{a})\oplus H_\mathfrak{a}(\f|_V)}{\image(i^*\oplus i_V^*)}=\frac{H_\mathfrak{a}(\f^\mathfrak{a})\oplus H_\mathfrak{a}(\f|_V)}{\ker(j^*-j_V^*)}\cong \image(j^*-j_V^*)\subset H_\mathfrak{a}(\f|_{U\cap V}),$$
so $\coker(i^*\oplus i_V^*)$ is a torsion module. The projection $H_\mathfrak{a}(\f^\mathfrak{a})\oplus H_\mathfrak{a}(\f|_V)\to H_\mathfrak{a}(\f^\mathfrak{a})$ induces a surjective map $\coker(i^*\oplus i_V^*)\to \coker i^*$, so $\coker i^*$ is torsion as well.

Finally, since localization preserves exactness \cite[Proposition 3.3]{atiyah2} and kills torsion, from the exact sequence
$$0 \longrightarrow \ker i^* \longrightarrow H_\mathfrak{a}(\f)\stackrel{i^*}{\longrightarrow}H_\mathfrak{a}(\f^\mathfrak{a})\longrightarrow \coker i^* \longrightarrow 0$$
it follows that $S^{-1}i^*:S^{-1}H_{\mathfrak{a}}(\f)\to S^{-1}H_{\mathfrak{a}}(\f^\mathfrak{a})$ is an isomorphism.
\end{proof}

The following is a transverse version of \cite[Corollary 5.4]{goertsches}.

\begin{corollary}
Let $\f$ be a neat singular Killing foliation of a compact manifold $M$. Then $H_{\mathfrak{a}}(\f)$ is a torsion module if, and only if, $\f$ has no closed leaves.
\end{corollary}

\begin{proof}
If there are no closed leaves, then $M^{\mathfrak{a}}=\emptyset$ and $H_{\mathfrak{a}}(\f^\mathfrak{a})=0$, so Theorem \ref{theorem: borel localization} implies that $H_{\mathfrak{a}}(\f)$ is torsion.

Conversely, if there is a closed leaf $L\in\f$, then $i^*(1)=1\in H_{\mathfrak{a}}(\f^\mathfrak{a})$. Since $H_{\mathfrak{a}}(\f^\mathfrak{a})\cong \sym(\mathfrak{a}^*)\otimes H(\f^\mathfrak{a})$ is torsion-free, $1\in H_{\mathfrak{a}}(\f)$ must have no torsion.
\end{proof}


\section{Applications of the localization theorem}

In this section we will see that there is a spectral sequence associated to $C_{\mathfrak{a}}(\f)$ that converges to $H_{\mathfrak{a}}(\f)$. This can then be used in association with Theorem \ref{theorem: borel localization} to show that $\chi(\f)=\chi(\f^\mathfrak{a})$, in analogy with the localization of the Euler characteristic of a manifold to the fixed point set of a torus action. In fact, all constructions will be direct adaptations of the classical case of torus actions on manifolds, for which we refer to \cite[Appendix A]{goertsches3} and \cite[Chapter 6]{guillemin}, so we will not delve into too much details of the proofs. These results are also already established for the case or regular Killing foliations in \cite{goertsches}.

\subsection{A spectral sequence for equivariant basic cohomology}

Let $\f$ be a neat singular Killing foliation of $M$. Endow $C_{\mathfrak{a}}(\f)$ with the bigrading
$$C^{k,l}_{\mathfrak{a}}(\f)=\begin{cases}
(\sym^{k/2}(\mathfrak{a}^*)\otimes \Omega^l(\f))^\mathfrak{a}, & \mbox{if } k \mbox{ is even},\\
0, & \mbox{if } k \mbox{ is odd},\end{cases}
$$
so that $C^n_{\mathfrak{a}}(\f)=\bigoplus_{k+l=n} C^{k,l}_{\mathfrak{a}}(\f)$ recovers the grading \eqref{grading of the Cartan complex}. The differential $d_\mathfrak{a}$ splits as $1\otimes d +\delta$, where $(\delta \omega)(X)=-\iota_X(\omega(X))$. The filtration
$$F^kC_{\mathfrak{a}}(\f)=\bigoplus_{\substack{j\geq k\\ l\geq 0}} C^{j,l}_{\mathfrak{a}}(\f),$$
is canonically bounded, hence the associated spectral sequence $(E_r)$ is first quadrant and converges, that is,
$$E_\infty\cong H_\mathfrak{a}(\f)$$
as vector spaces. By Lemma \ref{lemma: basic forms are invariant} we have $\Omega_\mathfrak{a}(\f)=\sym(\mathfrak{a}^*)\otimes\Omega(\f)$, which leads to
$$E_1\cong \sym(\mathfrak{a}^*)\otimes H(\f).$$

Each page $E_r=H(E_{r-1},d_{r-1})$ inherits a $\sym(\mathfrak{a}^*)$-module structure recursively from $E_{r-1}$. On $E_1\cong \sym(\mathfrak{a}^*)\otimes H(\f)$ it is given by the usual product on the first factor. If $M$ is compact, by Theorem \ref{theorem: basic cohomology is finite dimensional} we have $\dim H(\f)<\infty$, hence $E_1$ is finitely generated as an $\sym(\mathfrak{a}^*)$-module. Since $\sym(\mathfrak{a}^*)$ is Noetherian, submodules and quotients of ﬁnitely generated $\sym(\mathfrak{a}^*)$-modules are ﬁnitely generated, thus each $E_r$ is ﬁnitely generated. In particular, let $\{x_i\}\subset E_\infty$ be a finite set of homogeneous generators for $E_\infty$. Then one proves that representatives $y_i\in H_\mathfrak{a}(\f)$ chosen via
$$E_\infty^{k,l}\cong \frac{F^kH^{k+l}_\mathfrak{a}(\f)}{F^{k+1}H^{k+l}_\mathfrak{a}(\f)}$$
generate $H_\mathfrak{a}(\f)$ (see \cite[Lemma A.17]{goertsches3}). Summing up (cf. \cite[Lemma A.18]{goertsches3}):

\begin{proposition} Let $\f$ be a singular Killing foliation of a compact manifold $M$. Then $H_\mathfrak{a}(\f)$ is finitely generated as an $\sym(\mathfrak{a}^*)$-module.
\end{proposition}

As a corollary, it follows that $\rank H_\mathfrak{a}^{\mathrm{even}}(\f)=\rank E_\infty^{\mathrm{even}}$ and $\rank H_\mathfrak{a}^{\mathrm{odd}}(\f)=\rank E_\infty^{\mathrm{odd}}$, with respect to the $\mathbb{Z}_2$-graded decompositions in even and odd degree elements of $H_\mathfrak{a}(\f)$ and $E_\infty$ (see \cite[Corollary A.19]{goertsches3}). This is proved by comparing the respective Poincaré series, and using that $E_\infty\cong H_\mathfrak{a}(\f)$ as vector spaces.

\subsection{Localization of the basic Euler characteristic} We can now directly adapt \cite[Theorem 9.3]{goertsches3} to our transverse setting.

\begin{theorem}[Localization of the basic Euler characteristic]\label{theorem: localization of bec}
Let $\f$ be a neat singular Killing foliation, with structural algebra $\mathfrak{a}$, of a compact manifold $M$. Then
$$\chi(\f)=\chi(\f^\mathfrak{a}).$$
\end{theorem}

\begin{proof}
We have
\begin{align*}
\chi(\f) & = \dim H^{\mathrm{even}}(\f) - \dim H^{\mathrm{odd}}(\f)\\
 & =  \dim_{\mathrm{Q}(\mathfrak{a}^*)} \mathrm{Q}(\mathfrak{a}^*)\otimes H^{\mathrm{even}}(\f) - \dim_{\mathrm{Q}(\mathfrak{a}^*)} \mathrm{Q}(\mathfrak{a}^*)\otimes H^{\mathrm{odd}}(\f).
\end{align*}
Using that $E_1\cong \sym(\mathfrak{a}^*)\otimes H(\f)$, this leads to
$$\chi(\f)=\rank E_1^{\mathrm{even}}-\rank E_1^{\mathrm{odd}}.$$

Localizing each page of the spectral sequence $(E_r)$, we obtain $\mathbb{Z}_2$-graded vector spaces with odd differentials $d_r:E_r\to E_r$, so that $E_{r+1}=H(E_r,d_r)$. It follows that $\rank E_i^{\mathrm{even}}-\rank E_i^{\mathrm{odd}}=\rank E_{i+1}^{\mathrm{even}}-\rank E_{i+1}^{\mathrm{odd}}$, since the Euler characteristic is preserved under taking cohomology in this case (see \cite[Lemma 9.2]{goertsches3}). Therefore
\begin{align*}
\chi(\f) & =\rank E_\infty^{\mathrm{even}}-\rank E_\infty^{\mathrm{odd}}\\
 & = \rank H_\mathfrak{a}^{\mathrm{even}}(\f) - \rank H_\mathfrak{a}^{\mathrm{odd}}(\f).
\end{align*}

Using Theorem \ref{theorem: borel localization} and the fact that $H_{\mathfrak{a}}(\f^\mathfrak{a})=\sym(\mathfrak{a}^*)\otimes H(\f^\mathfrak{a})$, we then have

\begin{align*}
\chi(\f) & = \rank H_\mathfrak{a}^{\mathrm{even}}(\f^\mathfrak{a}) - \rank H_\mathfrak{a}^{\mathrm{odd}}(\f^\mathfrak{a})\\
 & = \dim_{\mathrm{Q}(\mathfrak{a}^*)} \mathrm{Q}(\mathfrak{a}^*)\otimes H^{\mathrm{even}}(\f^\mathfrak{a}) - \dim_{\mathrm{Q}(\mathfrak{a}^*)} \mathrm{Q}(\mathfrak{a}^*)\otimes H^{\mathrm{odd}}(\f^\mathfrak{a}) \\
 & = \dim H^{\mathrm{even}}(\f^\mathfrak{a}) - H^{\mathrm{odd}}(\f^\mathfrak{a})\\
 & = \chi(\f^\mathfrak{a}).\qedhere
\end{align*}
\end{proof}

The following is an obvious consequence.

\begin{corollary}\label{corollary: localization of bec finite case}
Let $\f$ be a singular Killing foliation with finitely many closed leaves, of a compact manifold $M$. Then
$$\#\f^\mathfrak{a}=\chi(\f),$$
that is, the basic Euler characteristic is precisely the number of closed leaves.
\end{corollary}

\subsection{The equivariantly formal case}

As in the classical case of Lie group actions, one can consider the equivariantly formal case, which include the equivariant cohomology of compact symplectic manifolds with Hamiltonian torus actions. In our case, we will say that singular Killing foliation $\f$ is \textit{equivariantly formal} if its equivariant basic cohomology is equivariantly formal, that is,
$$H_\mathfrak{a}(\f)\cong \sym(\mathfrak{a}^*)\otimes H(\f)$$
as $\sym(\mathfrak{a}^*)$-modules (cf. \cite[Section 6.3]{goertsches}). Equivalently, $\f$ is equivariantly formal when $(E_r)$ collapses at the first page (see \cite[Theorem 7.3]{goertsches3}). This is the case, for instance, if $H^{\mathrm{odd}}(\f)=0$ (see \cite[Corollary A.10]{goertsches3}). The following theorem is a singular version of \cite[Theorem 1]{goertsches}.

\begin{theorem}\label{theorem: comparison dimensions}
Let $\f$ be a neat singular Killing foliation, with structural algebra $\mathfrak{a}$, of a compact manifold $M$. Then
$$\dim H(\f^\mathfrak{a})\leq \dim H(\f),$$
and equality holds if, and only if, $\f$ is equivariantly formal.
\end{theorem}

\begin{proof}
The proof of the classical case of torus actions (e.g. \cite[Proposition 9.6]{goertsches3}) again adapts directly. One has
\begin{align*}
H(\f^\mathfrak{a}) & = \rank(\sym(\mathfrak{a}^*)\otimes H(\f^\mathfrak{a}))\\
 & = \rank H_\mathfrak{a}(\f^\mathfrak{a})\\
 & = \rank H_\mathfrak{a}(\f)\\
 & \leq \dim H(\f),
\end{align*}
The last inequality comes from $E_1\cong \sym(\mathfrak{a}^*)\otimes H(\f)$. It is clear that equality holds if $\f$ is equivariantly formal. Conversely, localizing each page of $(E_r)$ one sees that, if some $d_r$ is not zero, there must be a drop in the dimension at the corresponding pages. So if $\dim H(\f^\mathfrak{a})= \dim H(\f)$ (hence $\rank H_\mathfrak{a}(\f)=\rank E_1$), then $d_r=0$ for all $r\geq1$, and the sequence collapses at the first page.
\end{proof}



\section*{Acknowledgements}
I am grateful to Prof. M. Alexandrino for many helpful discussions.


