# Novikov type inequalities for orbifolds

Fabricio Valencia  F. Valencia - Instituto de Matemática e Estatística,
Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090
São Paulo - Brasil.
xx<EMAIL_ADDRESS>
###### Abstract.

We show a natural extension of the Novikov numbers associated to the basic
cohomology class of a closed $1$-form on an orbifold, thus proving
corresponding Novikov inequalities for the compact case.

###### 2020 Mathematics Subject Classification:

22A22, 57R18, 57R70

###### Contents

  1. 1 Introduction
  2. 2 Some algebraic topology ingredients
    1. 2.1 Hurewicz $G$-homomorphism and coverings
    2. 2.2 Groupoid homology with local coefficients
  3. 3 Closed basic 1-forms
    1. 3.1 G-homomorphisms of periods
    2. 3.2 Closed basic 1-forms of Morse type
  4. 4 Novikov numbers for orbifolds
    1. 4.1 Novikov inequalities

##  1\. Introduction

In the seminal works [17, 18] Novikov started a generalization of Morse theory
in which instead of critical points of smooth functions he dealt with closed
1-forms and their zeros, thus obtaining inequalities that generalize the well
known Morse inequalities. In order to do so Novikov defined numbers
$b_{j}(\xi)$ and $q_{j}(\xi)$ which depend on a real cohomology class $\xi\in
H^{1}(M,\mathbb{R})$. Such numbers are known in the literature as the Novikov
Betti and torsion numbers, respectively. It is worth mentioning that in the
special case $\xi=0$, which corresponds to classical Morse theory, the numbers
$b_{j}(\xi)$ recover the usual Betti numbers of the ambient manifold $M$ and
the numbers $q_{j}(\xi)$ agree with the minimal number of generators of the
torsion subgroup $H^{1}(M,\mathbb{Z})$ inside $H^{1}(M,\mathbb{R})$. The
classical Novikov inequalities state that any closed 1-form $\omega$ with
Morse-type zeros has at least $b_{j}(\xi)+q_{j}(\xi)+q_{j-1}(\xi)$ zeros of
index $j$ where $\xi=[\omega]\in H^{1}(M,\mathbb{R})$ is the de Rham
cohomology class of $\omega$. Nowadays, Novikov theory is widely known, as it
provides several applications in geometry, topology, analysis, and dynamics.
The reader is recommended to visit [11] in order to get in touch with the
fundamentals of Novikov theory as well as with some of the classical results
around this beautiful subject.

The main purpose of this paper is to extend some of the ingredients of the
classical Novikov theory to the context of orbifolds, thus providing a way to
generalize the Novikov inequalities for the compact ones. We think of
orbifolds as proper and étale groupoids up to Morita equivalence. In this
spirit, we deal with cohomology classes of closed basic 1-forms on Lie
groupoids whose critical orbits are nondegenerate in the sense of Morse–Bott
theory.

Let $G\rightrightarrows M$ be a Lie groupoid and let
$\Pi_{1}(G)\rightrightarrows M$ denote its fundamental groupoid of
$G$-homotopy classes of $G$-paths. The fundamental group $\Pi_{1}(G,x_{0})$ of
$G$ with respect to a base-point $x_{0}\in M$ is defined to be isotropy group
$\Pi_{1}(G)_{x_{0}}$ which consists of $G$-homotopy classes of $G$-loops at
$x_{0}$. Let us denote by $H_{\bullet}(G,\mathbb{Z})$ the total singular
homology of $G$. There is a canonical way to define a Hurewicz
$G$-homomorphism $h:\Pi_{1}(G,x_{0})\to H_{1}(G,\mathbb{Z})$ which restricts
as an isomorphism to the abelianization of $\Pi_{1}(G,x_{0})$. Let $\omega$ be
a closed basic $1$-form on $M$. That is, a closed $1$-form satisfying
$(t^{\ast}-s^{\ast})(\omega)=0$. For each smooth $G$-path
$\sigma=\sigma_{n}g_{n}\sigma_{n-1}\cdots\sigma_{1}g_{1}\sigma_{0}$ in $M$ we
can naturally define the $G$-path integral
$\int_{\sigma}\omega=\sum_{k=0}^{n}\int_{\sigma_{k}}\omega$. It canonically
determines a group homomorphism $l_{\omega}:\Pi_{1}(G,x_{0})\to(\mathbb{R},+)$
which factors through the Hurewicz $G$-homomorphism $h$ by a uniquely
determined group homomorphism
$\textnormal{Per}_{\xi}:H_{1}(G,\mathbb{Z})\to\mathbb{R}$ which only depends
on the basic cohomology class $\xi:=[\omega]$. This will be called the
$G$-homomorphism of periods of $\xi$. Let Nov denote the Novikov ring of
$\mathbb{R}$ and let us further assume that our Lie groupoid
$G\rightrightarrows M$ is étale and proper with compact orbit space $M/G$.
Thus, we may define a ring homomorphism
$\phi_{\xi}:\mathbb{Z}(\Pi_{1}(G,x_{0}))\to\textbf{Nov}$ by setting
$\phi_{\xi}([\sigma]):=\tau^{\textnormal{Per}_{\xi}(h([\sigma]))}$ for all
$[\sigma]\in\Pi_{1}(G,x_{0})$. It yields a local system $\mathcal{L}_{\xi}$ of
left Nov-modules over $G\rightrightarrows M$ whose total homology groups
$H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ are called Novikov homology
groups of $\xi$. It follows that the homology
$H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ is a finitely generated module
over the ring Nov. Since Nov is a principal ideal domain we have that the
module $H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ is a direct sum of a
free submodule with a torsion submodule. The Novikov Betti number $b_{j}(\xi)$
is defined to be the rank of the free summand of
$H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ and the Novikov torsion number
$q_{j}(\xi)$ is defined to be the minimal number of generators of the torsion
submodule of $H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$.

The critical point set $\textnormal{Crit}(\omega)$ of a closed basic $1$-form
$\omega$ on $M$ is saturated in $M$ so that it is formed by a disjoint union
of groupoid orbits. Therefore, we say that a critical orbit $\mathcal{O}_{x}$
of $\omega$ is nondegenerate if its normal Hessian is a nondegenerate
fiberwise bilinear symmetric form on $\nu(O_{x})$. Accordingly, $\omega$ is
said to be of Morse type if all of its critical orbits are nondegenerate. Let
us fix a groupoid metric on $G\rightrightarrows M$ in the sense of del Hoyo
and Fernandes, visit [9]. The nondegeneracy requirement over $\mathcal{O}_{x}$
imposed above allows us to use the groupoid metric to split
$\nu(\mathcal{O}_{x})$ into the Whitney sum of two subbundles
$\nu_{-}(\mathcal{O}_{x})\oplus\nu_{+}(\mathcal{O}_{x})$ such that normal
Hessian of $\omega$ around $\mathcal{O}_{x}$ is strictly negative on
$\nu_{-}(\mathcal{O}_{x})$ and strictly positive on
$\nu_{+}(\mathcal{O}_{x})$. Let $G_{x}$ be the isotropy group at $x$. From
[19] we know that the normal Hessian is invariant with respect to the normal
representation $G_{x}\curvearrowright\nu(\mathcal{O}_{x})$ so that it
preserves the splitting above since the normal representation is by isometries
in this case. In consequence, we get a normal sub-representation
$G_{x}\curvearrowright\nu_{-}(\mathcal{O}_{x})$. The stacky index of
$\mathcal{O}_{x}$ in $M/G$ is defined to be
$\dim\nu_{-}(\mathcal{O}_{x})/G_{x}=\dim\nu_{-}(\mathcal{O}_{x})-\dim G_{x}$.

Our main result can be stated in the following way:

######  Theorem.

Let $G\rightrightarrows M$ be an étale and proper Lie groupoid such that the
orbit space $M/G$ is compact. Let $\omega$ be a closed basic $1$-form on $M$
of Morse type. If $c_{j}(\omega)$ denotes the number of critical points in
$M/G$ having stacky Morse index $j$ then

$c_{j}(\omega)\geq b_{j}(\xi)+q_{j}(\xi)+q_{j-1}(\xi),$

where $\xi=[\omega]$ is the basic cohomology class of $\omega$.

It is worth mentioning that the natural generalization of the Novikov theory
we have described above provides a tool for using topological methods to study
zeros of symplectic vector fields on symplectic orbifolds.

The paper is structured as follows. In Section 2 we define the fundamental Lie
groupoid of $G$-homotopy classes of $G$-paths and briefly recall the
definition of the total singular homology and cohomology groups of a
topological groupoid. We show a $G$-version of the Hurewicz homomorphism and
exhibit how to construct a groupoid covering space over $G\rightrightarrows M$
out of a fixed subgroup in $\Pi_{1}(G,x_{0})$. This is the content of
Propositions 2.2 and 2.4, respectively. Motivated by the notion of local
coefficients system for topological spaces we introduce a corresponding notion
as well as its associated homology in the groupoid setting. Indeed, we define
a double chain complex out of a local system of modules over a Lie groupoid,
thus proving that its associated total homology is Morita invariant, compare
Lemma 2.7 and Proposition 2.8. We also show in Proposition 2.9 a $G$-version
of the Eilenberg homology isomorphism. In Section 3 we study closed basic
$1$-forms. We define the $G$-homomorphism of periods associated to the basic
cohomology class $\xi$ of a closed basic $1$-form $\omega$ on $M$ and explore
some of its elementary properties. In particular, we characterize when $\xi$
is an integral class and describe the groupoid covering space associated to
the kernel of the $G$-homomorphism of periods of $\xi$, see Propositions 3.4
and 3.5. We introduce closed basic $1$-forms of Morse type and prove that such
a notion is Morita invariant, see Proposition 3.8. This gives rise to a notion
of stacky closed $1$-form of Morse type over the differentiable stack $[M/G]$
presented by $G\rightrightarrows M$. Finally, in Section 4 we present three
different manners to define the Novikov numbers $b_{j}(\xi)$ and $q_{j}(\xi)$
associated to the basic cohomology class $\xi$ over an orbifold. As a result
we show the Novikov inequalities for compact orbifolds, compare Theorem 4.6,
and quickly explain how to use this result in order to find a lower bound for
the amount of zeros of certain symplectic vector fields on symplectic
orbifolds.

Acknowledgments: Part of this work was carried out during a visit to the
Dipartimento di Matematica, Università degli Studi di Salerno, Fisciano, Italy
in 2023. I am very thankful for the hospitality and support that the Geometry
Group gave me while being there. I have benefited from several conversations
with Juan Camilo Arias, Antonio Maglio, Cristián Ortiz and Luca Vitagliano, so
that I am grateful for all their comments and suggestions that improved this
work. Valencia was supported by Grants 2020/07704-7 and 2022/11994-6 Sao Paulo
Research Foundation - FAPESP.

##  2\. Some algebraic topology ingredients

Before defining the $G$-homomorphism of periods of a closed basic $1$-form as
well as its corresponding Novikov numbers we have to fill out some algebraic
topology gaps concerning the fundamental Lie groupoid of $G$-homotopy classes
of $G$-paths which, to our knowledge, do not seem to have been described
before in the literature; see for instance [16, s. 3.3] and [4, c. G; s. 3].
Let $G\rightrightarrows M$ be a Lie groupoid [8, 16]. Throughout this paper
the structural maps of $G$ will be denoted by $(s,t,m,u,i)$ where $s,t:G\to M$
are the maps respectively indicating the source and target of the arrows,
$m:G^{(2)}\to G$ stands for the partial composition of arrows, $u:M\to G$ is
the unit map, and $i:G\to G$ is the map determined by the inversion of arrows.
A _$G$ -path_ in $M$ is a sequence
$\sigma:=\sigma_{n}g_{n}\sigma_{n-1}\cdots\sigma_{1}g_{1}\sigma_{0}$ where
$\sigma_{0},\cdots,\sigma_{n}:[0,1]\to M$ are (piecewise smooth) paths in $M$
and $g_{1},\cdots,g_{n}$ are arrows in $G$ such that
$g_{j}:\sigma_{j-1}(1)\to\sigma_{j}(0)$ for all $j=1,\cdots,n$. We shall say
that $\sigma$ is a $G$-path of _order_ $n$ from $\sigma_{0}(0)$ to
$\sigma_{n}(1)$. Our groupoid $G$ is said to be _$G$ -connected_ if for any
two points $x,y\in M$ there exists a $G$-path from $x$ to $y$. We will always
assume that the groupoids we are working with are $G$-connected unless
otherwise stated. If
$\sigma^{\prime}:=\sigma_{n}^{\prime}g_{n}^{\prime}\sigma_{n-1}^{\prime}\cdots\sigma_{1}^{\prime}g_{1}^{\prime}\sigma_{0}^{\prime}$
is another $G$-path with $\sigma_{0}^{\prime}(0)=\sigma_{n}(1)$ then we can
_concatenate_ $\sigma$ and $\sigma^{\prime}$ into a new $G$-path

$\sigma\ast\sigma^{\prime}=\sigma_{n}^{\prime}g_{n}^{\prime}\sigma_{n-1}^{\prime}\cdots\sigma_{1}^{\prime}g_{1}^{\prime}\sigma_{0}^{\prime}1_{\sigma_{n}(1)}\sigma_{n}g_{n}\sigma_{n-1}\cdots\sigma_{1}g_{1}\sigma_{0}.$

We define an equivalence relation in the set of $G$-paths which is generated
by the following _multiplication equivalence_

$\sigma_{n}g_{n}\cdots\sigma_{j+1}g_{j+1}\sigma_{j}g_{j}\sigma_{j-1}\cdots
g_{1}\sigma_{0}\quad\sim\quad\sigma_{n}g_{n}\cdots\sigma_{j+1}g_{j+1}g_{j}\sigma_{j-1}\cdots
g_{1}\sigma_{0},$

if $\sigma_{j}$ is the constant path for any $0<j<n$, and _concatenation
equivalence_

$\sigma_{n}g_{n}\cdots g_{j+1}\sigma_{j}g_{j}\sigma_{j-1}g_{j-1}\cdots
g_{1}\sigma_{0}\quad\sim\quad\sigma_{n}g_{n}\cdots
g_{j+1}\sigma_{j}\cdot\sigma_{j-1}g_{j-1}\cdots g_{1}\alpha_{0},$

if $g_{j}=1_{\alpha_{j-1}(1)}$ for any $0<j<n$ where
$\sigma_{j}\cdot\sigma_{j-1}$ stands to denote the standard concatenation of
the paths $\sigma_{j}$ and $\sigma_{j-1}$. A _deformation_ between two
$G$-paths $\sigma$ and $\sigma^{\prime}$ of the same order $n$ from $x$ to $y$
consist of homotopies $D_{j}:[0,1]\times[0,1]\to M$ from $\sigma_{j}$ to
$\sigma_{j}^{\prime}$ for $j=0,1,\cdots,n$ and paths $d_{i}:[0,1]\to G$ from
$g_{j}$ to $g_{j}^{\prime}$ for $j=1,\cdots,n$ such that $s\circ
d_{j}=D_{j-1}(\cdot,1)$ and $t\circ d_{j}=D_{j}(\cdot,0)$ for $j=1,\cdots,n$
verifying $D_{0}([0,1],0)=x$ and $D_{n}([0,1],1)=y$. That is, a deformation as
a continuous family of $G$-paths of order $n$ from $x$ to $y$ which may be
written as $D_{n}(\tau,\cdot)d_{n}(\tau)\cdots d_{1}(\tau)D_{0}(\tau,\cdot)$
for $\tau\in[0,1]$. Accordingly, two $G$-paths with fixed endpoints in $M$ are
said to be _$G$ -homotopic_ if it is possible to pass from one to another by a
sequence of equivalences and deformations. With the multiplication induced by
concatenation it follows that the $G$-homotopy classes of $G$-paths form a Lie
groupoid over $M$ which is called the _fundamental groupoid_ of $G$. This
shall be denoted by $\Pi_{1}(G)\rightrightarrows M$, see [16, s. 3.3]. The
_fundamental group_ of $G$ with respect to a base-point $x_{0}\in M$ is the
isotropy group $\Pi_{1}(G,x_{0}):=\Pi_{1}(G)_{x_{0}}$. Note that it consists
of $G$-homotopy classes of $G$-loops at $x_{0}$ which are by definition the
$G$-homotopy classes of $G$-paths from $x_{0}$ to $x_{0}$. It is simple to
check that for any two different points $x_{0},y_{0}\in M$ it holds that
$\Pi_{1}(G,x_{0})$ and $\Pi_{1}(G,y_{0})$ are isomorphic by $G$-connectedness.
Every Lie groupoid morphism $\phi_{1}:G\to G^{\prime}$ covering $\phi_{0}:M\to
M^{\prime}$ induces another Lie groupoid morphism
$(\phi_{1})_{\ast}:\Pi_{1}(G)\to\Pi_{1}(G^{\prime})$ by mapping $[\sigma]$ to
$[(\phi_{1})_{\ast}(\sigma)]$, where
$(\phi_{1})_{\ast}(\sigma)=(\phi_{0})_{\ast}(\sigma_{n})\phi_{1}(g_{n})(\phi_{0})_{\ast}(\sigma_{n-1})\cdots(\phi_{0})_{\ast}(\sigma_{1})\phi_{1}(g_{0})(\phi_{0})_{\ast}(\sigma_{0})$.
This also covers $\phi_{0}:M\to M^{\prime}$ so that it induces a Lie group
homomorphism between the fundamental groups
$(\phi_{1})_{\ast}:\Pi_{1}(G,x_{0})\to\Pi_{1}(G^{\prime},\phi_{0}(x_{0}))$. As
an important feature we have that Morita equivalent groupoids have isomorphic
fundamental groupoids, see [16, p. 195]. For specific details and results
concerning the fundamental groupoid of a Lie groupoid the reader is
recommended to visit [16, s. 3.3] and [4, c. G; s. 3].

Let us now consider the nerve of $G\rightrightarrows M$. This is formed by
manifolds $\\{G^{(n)}\\}_{n\in\mathbb{N}}$ with $G^{(0)}=M$, $G^{(1)}=G$ and
$G^{(n)}=\\{(g_{1},\cdots,g_{n}):s(g_{j})=t(g_{j+1})\\}$. We also have the
face maps $d_{k}^{n}:G^{(n)}\to G^{(n-1)}$ given by the surjective submersions
$d^{1}_{0}=t$, $d^{1}_{1}=s$ and

$d_{k}^{n}(g_{1},\cdots,g_{n})=\left\\{\begin{array}[]{lcc}(g_{2},\cdots,g_{n})&\textnormal{if}&k=0\\\
(g_{1},\cdots,g_{i}g_{i+1},\cdots,g_{n})&\textnormal{if}&0<k<n\\\
(g_{1},\cdots,g_{n-1})&\textnormal{if}&k=n.\end{array}\right.$

These maps satisfy the simplicial relations $d_{k}^{n-1}\circ
d_{k^{\prime}}^{n}=d_{k^{\prime}-1}^{n-1}\circ d_{k}^{n}$ for $k<k^{\prime}$.
We define the singular double chain complex of $G\rightrightarrows M$ as
follows. This is determined by the data
$\\{C_{\bullet}(G^{(\bullet)}),\partial,d\\}$ where $C_{q}(G^{(n)})$ is given
by the set of singular $q$-chains on $G^{(n)}$, $\partial:C_{q}(G^{(n)})\to
C_{q}(G^{(n-1)})$ is the simplicial boundary operator
$\partial=\sum_{k=0}^{n}(-1)^{k}d_{k}^{n}$, and $d:C_{q}(G^{(n)})\to
C_{q-1}(G^{(n)})$ is the usual singular boundary operator
$d=\sum_{j=0}^{q}(-1)^{j}d_{j}$. We can then consider the associated total
complex $\\{\tilde{C}_{\bullet}(G),\delta\\}$ where
$\tilde{C}_{v}(G)=\bigoplus_{q+n=v}C_{q}(G^{(n)})$ with total differential
operator $\delta:\tilde{C}_{v}(G)\to\tilde{C}_{v-1}(G)$ defined as

$\delta(\gamma)=(-1)^{q+n}\partial(\gamma)+(-1)^{q}d(\gamma),\qquad\gamma\in
C_{q}(G^{(n)}).$

The associated total singular homology will be denoted by
$H_{\bullet}(G,\mathbb{Z})$. It is worth mentioning that Morita equivalent
groupoids have isomorphic total singular homologies, visit for instance [1,
7]. By following a similar approach it may be easily verified that the total
singular cohomology $H^{\bullet}(G,\mathbb{Z})$ of $G\rightrightarrows M$ can
be obtained by dualizing the total singular complex $\tilde{C}_{\bullet}(G)$
in order to define another total complex $\tilde{C}^{\bullet}(G)$ and then
considering its associated cohomology. Also, by tensoring
$C_{\bullet}(G)\otimes_{\mathbb{Z}}\mathbb{R}$ and
$C^{\bullet}(G)\otimes_{\mathbb{Z}}\mathbb{R}$ we get the total singular
homology and cohomology groups $H_{\bullet}(G,\mathbb{R})$ and
$H^{\bullet}(G,\mathbb{R})$, respectively. Furthermore, via a spectral
sequence argument it follows that the classical de Rham isomorphism for
manifolds extends to define an isomorphism between the de Rham cohomology
$H^{\bullet}_{dR}(G)$ of $G$ defined through the Bott-Shulman-Stasheff total
complex and the singular cohomology $H^{\bullet}(G,\mathbb{R})$. Here, the
cohomology $H^{\bullet}_{dR}(G)$ is defined as the total cohomology obtained
from the double cochain complex
$\\{\Omega_{\bullet}(G^{(\bullet)}),\overline{\partial},d_{dR}\\}$ where
$\Omega_{q}(G^{(n)})$ is given by the differential $q$-forms on $G^{(n)}$,
$\overline{\partial}:\Omega_{q}(G^{(n)})\to\Omega_{q}(G^{(n+1)})$ is the
simplicial boundary operator
$\overline{\partial}=\sum_{k=0}^{n}(-1)^{k}(d_{k}^{n})^{\ast}$, and
$d_{dR}:\Omega_{q}(G^{(n)})\to\Omega_{q+1}(G^{(n)})$ is the usual the de Rham
differential. As expected, all of these (co)homologies are Morita invariant.
For more details visit [1].

###  2.1. Hurewicz $G$-homomorphism and coverings

Let $G\rightrightarrows M$ be a $G$-connected Lie groupoid and fix $x_{0}\in
M$. Consider $\Pi_{1}(G,x_{0})$ and $H_{1}(G,\mathbb{Z})$ the fundamental
group of $G$ with respect to the base-point $x_{0}$ and the first total
singular homology group of $G$, respectively. It is simple to check that every
$G$-loop at $x_{0}$ canonically defines a $1$-cycle in
$\tilde{C}_{1}(G)=C_{0}(G)\oplus C_{1}(M)$. This suggests us to define the map
$h:\Pi_{1}(G,x_{0})\to H_{1}(G,\mathbb{Z})$ which sends the $G$-homotopy class
$[\sigma]$ of a $G$-loop $\sigma$ at $x_{0}$ to its corresponding homology
class $|\sigma|$.

######  Lemma 2.1.

Let $\sigma$ and $\sigma^{\prime}$ be two $G$-paths. The following assertions
hold true:

  * •

$\sigma+\sigma^{-1}$ is equivalent to a boundary in $\tilde{C}_{1}(G)$, and

  * •

if $\sigma_{0}^{\prime}(0)=\sigma_{n}(1)$ then
$\sigma\ast\sigma^{\prime}-\sigma-\sigma^{\prime}$ is also equivalent to a
boundary in $\tilde{C}_{1}(G)$.

###### Proof.

Firstly, for every path $\sigma_{j}$ in $\sigma$ we may argue as in Lemma 3.2
from [5, p. 173] and for every arrow $g_{j}$ in $\sigma$ we consider the pair
of composable arrows $(g_{j},g_{j}^{-1})$. Secondly, observe that the
expression $\sigma\ast\sigma^{\prime}-\sigma-\sigma^{\prime}$ equals

$\sigma_{n}\ast\sigma_{0}^{\prime}-\sigma_{n}-\sigma_{0}^{\prime}+\textnormal{boundary}.$

So, the last assertion follows directly as a consequence of Lemma 3.1 in [5,
p. 173]. ∎

######  Proposition 2.2 (Hurewicz $G$-homomorphism).

The map $h$ is a well defined group homomorphism which restricts to the
abelianization
$\Pi_{1}^{ab}(G,x_{0})=\Pi_{1}(G,x_{0})/[\Pi_{1}(G,x_{0}),\Pi_{1}(G,x_{0})]$
as an isomorphism.

###### Proof.

We shall follow the usual strategy used to prove the classical version of this
result, see for instance [5, c.4; s.3]. Throughout the proof we shall fix
$\sigma=\sigma_{n}g_{n}\sigma_{n-1}\cdots\sigma_{1}g_{1}\sigma_{0}$ a $G$-loop
at $x_{0}$. In order to see that $h$ is well defined let us consider
$\sigma^{\prime}=\sigma_{n}^{\prime}g_{n}^{\prime}\sigma_{n-1}^{\prime}\cdots\sigma_{1}^{\prime}g_{1}^{\prime}\sigma_{0}^{\prime}$
another $G$-loop at $x_{0}$ which is in the $G$-homotopy class of $\sigma$. If
$\sigma$ and $\sigma^{\prime}$ are equivalent then the assertion is trivial.
Let us suppose then that there is a $G$-homotopy
$D(\tau,\cdot)=D_{n}(\tau,\cdot)d_{n}(\tau)\cdots
d_{1}(\tau)D_{0}(\tau,\cdot)$ from $\sigma$ to $\sigma^{\prime}$. We split the
square $[0,1]\times[0,1]$ into two 2-simplices with vertices
$\\{(0,0),(0,1),(1,1)\\}$ and $\\{(0,0),(1,0),(1,1)\\}$, respectively, so that
every homotopy $D_{j}$ can be regarded as the sum of two singular 2-simplices
over $M$. The diagonal from $(0,0)$ to $(1,1)$ will be denoted by $\lambda$.
Therefore, by using Lemma 2.1 we get:

$\displaystyle\delta D$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n}(t\circ
d_{j}-s\circ d_{j})+\sum_{j=0}^{n}(s\circ
d_{j+1}-D_{j}|_{\lambda}-\sigma_{j})-\sum_{j=0}^{n}(\sigma_{j}^{\prime}-D_{j}|_{\lambda}-t\circ
d_{j})$ $\displaystyle-$
$\displaystyle\sum_{j=1}^{n}(g_{j}^{\prime}-g_{j})+\textnormal{constant}=\sigma-\sigma^{\prime}+\textnormal{boundary},$

where $s\circ d_{n+1}:=D_{n}|_{[0,1]\times 1}=x_{0}\ \textnormal{and}\ t\circ
d_{0}:=D_{0}|_{[0,1]\times 0}=x_{0}$. That is, $\sigma$ and $\sigma^{\prime}$
are homologous which means that $h$ is well defined. Again, as consequence of
Lemma 2.1 we have that if $[\sigma],[\sigma^{\prime}]\in\Pi_{1}(G,x_{0})$ then

$h([\sigma][\sigma^{\prime}])=h([\sigma\ast\sigma^{\prime}])=|\sigma\ast\sigma^{\prime}|=|\sigma|+|\sigma^{\prime}|=h([\sigma])+h([\sigma]),$

thus obtaining that $h$ is a group homomorphism. Note that if $\sigma^{-1}$
denotes the inverse $G$-loop of $\sigma$ then $h([\sigma^{-1}])=-|\sigma|$
since $h$ is a homomorphism. Hence, by Lemma 2.1 it follows that the
commutator subgroup of $\Pi_{1}(G,x_{0})$ lies inside $\textnormal{ker}(h)$ so
that we get another well defined group homomorphism
$\tilde{h}:\Pi_{1}^{ab}(G,x_{0})\to H_{1}(G,\mathbb{Z})$.

Since our groupoid is $G$-connected we have that for each $x\in M$ there
exists some $G$-path $\lambda_{x}$ from our base-point $x_{0}$ to $x$. The
constant path at $x$ is denoted by $c_{x}$. Let us construct the inverse group
homomorphism of $\tilde{h}$. It is clear that we may think of every arrow
$g\in G$ as a $G$-path connecting the constant paths at $s(g)$ and $t(g)$.
Also, it is obvious that every path $\gamma:[0,1]\to M$ is equivalent to the
$G$-path $c_{\gamma(0)}1_{\gamma(0)}\gamma 1_{\gamma(1)}c_{\gamma(1)}$. In
consequence, after possibly reversing orientations and fixing an order, each
singular groupoid $1$-chain $\alpha\in\tilde{C}_{1}(G)$ can be thought of as a
formal sum of not necessarily different $G$-paths $\alpha=\sum\alpha_{j}$. We
define the homomorphism $l:\tilde{C}_{1}(G)\to\Pi_{1}^{ab}(G,x_{0})$ as
$l(\alpha)=[\sum_{\ast}\lambda_{\alpha_{j}(1)}^{-1}*\alpha_{j}*\lambda_{\alpha_{j}(0)}]$,
where $\sum_{\ast}$ denotes the concatenation of all the $G$-loops at $x_{0}$
given by $\lambda_{\alpha_{j}(1)}^{-1}*\alpha_{j}*\lambda_{\alpha_{j}(0)}$.
Note that the definition of $l$ does not depend on the choice of the $G$-paths
$\lambda_{\alpha_{j}(0)}$ and $\lambda_{\alpha_{j}(1)}$ since we are going
into $\Pi_{1}^{ab}(G,x_{0})$ instead of $\Pi_{1}(G,x_{0})$. We claim that $l$
takes the boundaries in $\tilde{C}_{1}(G)$ into $1\in\Pi_{1}^{ab}(G,x_{0})$.
Indeed, it is clear that it suffices to check this assertion when we apply the
boundary operator $\delta$ over a pair of composable arrows $(g,g^{\prime})\in
G^{(2)}$, a path $\tilde{\sigma}:[0,1]\to G$ and a singular 2-simplex
$\Sigma:\triangle_{2}\to M$. Firstly,
$\delta(g,g^{\prime})=g^{\prime}-g^{\prime}g+g$ so that

$\displaystyle l(\delta(g,g^{\prime}))$ $\displaystyle=$ $\displaystyle
l(g^{\prime})l(g)l(g^{\prime}g)^{-1}=[\lambda_{s(g^{\prime})}^{-1}*g^{\prime}*\lambda_{t(g)}][\lambda_{t(g)}^{-1}*g*\lambda_{s(g)}][\lambda_{s(g^{\prime})}^{-1}*g^{\prime}g*\lambda_{s(g)}]^{-1}$
$\displaystyle=$
$\displaystyle[\lambda_{s(g^{\prime})}^{-1}*g^{\prime}g(g^{\prime}g)^{-1}*\lambda_{s(g^{\prime})}]]=[\textnormal{constant}]=1.$

Secondly,
$\delta\tilde{\sigma}=t\circ\tilde{\sigma}-s\circ\tilde{\sigma}-(\tilde{\sigma}(1)-\tilde{\sigma}(0))$.
Note that

$l(t\circ\tilde{\sigma}+\tilde{\sigma}(0)-s\circ\tilde{\sigma}-\tilde{\sigma}(1))=l(t\circ\tilde{\sigma})l(\tilde{\sigma}(0))l(s\circ\tilde{\sigma})^{-1}l(\tilde{\sigma}(1))^{-1}\\\
$
$=[\lambda_{t\circ\tilde{\sigma}(1)}^{-1}*t\circ\tilde{\sigma}*\lambda_{t\circ\tilde{\sigma}(0)}][\lambda_{t\circ\tilde{\sigma}(0)}^{-1}*\tilde{\sigma}(0)*\lambda_{s\circ\tilde{\sigma}(0)}][\lambda_{s\circ\tilde{\sigma}(1)}^{-1}*s\circ\tilde{\sigma}*\lambda_{s\circ\tilde{\sigma}(0)}]^{-1}[\lambda_{t\circ\tilde{\sigma}(1)}^{-1}*\tilde{\sigma}(1)*\lambda_{s\circ\tilde{\sigma}(1)}]^{-1}$
$=[\lambda_{t\circ\tilde{\sigma}(1)}^{-1}*(t\circ\tilde{\sigma})\tilde{\sigma}(0)(s\circ\tilde{\sigma})^{-1}\tilde{\sigma}(1)^{-1}*\lambda_{t\circ\tilde{\sigma}(1)}]=1,$

since $\tilde{\sigma}(1)$ is $G$-homotopic to the $G$-path
$(t\circ\tilde{\sigma})\tilde{\sigma}(0)(s\circ\tilde{\sigma})^{-1}$, see [16,
p. 191]. Thirdly, the case of the singular $2$-simplex $\Sigma$ over $M$
follows directly by Lemma 3.5 in [5, p. 174].

The fact we just proved implies that $l$ descends to a well defined group
homomorphism $\tilde{l}:H_{1}(G,\mathbb{Z})\to\Pi_{1}^{ab}(G,x_{0})$.
Furthermore, if $\sigma$ is a $G$-loop at $x_{0}$ then

$(\tilde{l}\circ\tilde{h})([\sigma])=\tilde{l}(|\sigma|)=[\lambda_{x_{0}}^{-1}*\sigma*\lambda_{x_{0}}]=[\sigma],$

since $\lambda_{x_{0}}$ may be chosen to be just the constant $G$-path at
$x_{0}$. Let us now look at the opposite composition. Observe that the
assignment $x\mapsto\lambda_{x}$ allows us to send singular groupoid
$0$-simplices into singular groupoid $1$-simplices and this can be clearly
extended to a homomorphism $\lambda:\tilde{C}_{0}(G)\to\tilde{C}_{1}(G)$.
Suppose that $\alpha=\sum\alpha_{j}$ is a singular groupoid 1-chain. Then, by
Lemma 2.1 it holds that

$\displaystyle(\tilde{h}\circ l)(\alpha)$ $\displaystyle=$
$\displaystyle\tilde{h}([\sum_{\ast}\lambda_{\alpha_{j}(1)}^{-1}*\alpha_{j}*\lambda_{\alpha_{j}(0)}])=\sum|\lambda_{\alpha_{j}(1)}^{-1}*\alpha_{j}*\lambda_{\alpha_{j}(0)}|$
$\displaystyle=$
$\displaystyle\sum|\lambda_{\alpha_{j}(1)}^{-1}+\alpha_{j}+\lambda_{\alpha_{j}(0)}|=\sum|\lambda_{\alpha_{j}(0)}+\alpha_{j}-\lambda_{\alpha_{j}(1)}|$
$\displaystyle=$
$\displaystyle|\sum\alpha_{j}+\lambda_{\sum(\alpha_{j}(1)-\alpha_{j}(0))}|=|\alpha+\lambda_{\delta\alpha}|.$

Therefore, if $\alpha$ is a singular groupoid $1$-cycle it follows that
$(\tilde{h}\circ l)(\alpha)=|\alpha|$. In particular,
$(\tilde{h}\circ\tilde{l})(|\alpha|)=|\alpha|$, as desired. ∎

Let us now consider the notion of covering space over a groupoid as defined in
[16, s. 3.3] and [4, c. G; s. 3]. A _covering space_ $E$ over
$G\rightrightarrows M$ is a covering space $p:E\to M$ equipped with a right
$G$-action $E\times_{M}G\to E$ along $p$. Morphisms between two covering
spaces $E$ and $F$ over $G$ are equivariant maps $f:E\to F$. It is clear that
any such morphism is necessarily a covering projection. We will denote by
$\Gamma^{G}(E)$ the set of _equivariant automorphisms_ of $E$. The map $p$
extends to a Lie groupoid morphism $p:E\rtimes G\to G$ from the action
groupoid $E\rtimes G$ into $G$ defined by $p(e,g)=g$, which covers the
covering map $E\to M$. We say that $E$ is _universal_ if the action groupoid
$E\rtimes G\rightrightarrows E$ is $(E\rtimes G)$-connected and the
fundamental groupoid $\Pi_{1}(E\rtimes G,e_{0})$ is trivial for one (hence
all) base-point $e_{0}\in E$. The latter kind of Lie groupoids shall be called
_simply $G$-connected_.

######  Example 2.3.

There is an explicit way to construct the universal covering space over
$G\rightrightarrows M$ when is $G$-connected. Namely, the target fiber
$\Pi_{1}(G)(-,x_{0})$ at $x_{0}\in M$ of the fundamental groupoid $\Pi_{1}(G)$
of $G$ becomes a covering space over $M$ by restricting the source map
$s:\Pi_{1}(G)(-,x_{0})\to M$. In fact, this map is a left principal
$\Pi_{1}(G,x_{0})$-bundle. Also, there is a natural right $G$-action on
$\Pi_{1}(G)(-,x_{0})$ along $s$ given by $([\sigma],g)\to[\sigma gc_{s(g)}]$,
where $c_{s(g)}$ is the constant map at $s(g)$, which makes it into a covering
space over $G$. This is actually a universal covering space over $G$, see [4,
p. 612-613].

It is well known that $G$-paths have a unique path lifting property, visit
[16, p. 191]. Let $e_{0}$ be a base-point in $E$, denote $x_{0}=p(e_{0})$, and
suppose that
$\sigma=\sigma_{n}g_{n}\sigma_{n-1}\cdots\sigma_{1}g_{1}\sigma_{0}$ is a
$G$-path from $x_{0}$ to $x$. Then there are unique paths
$\tilde{\sigma_{n}},\tilde{\sigma_{n-1}},\cdots,\tilde{\sigma_{0}}$ in $E$
with $p_{\ast}(\tilde{\sigma_{j}})=\sigma_{j}$, $\tilde{\sigma_{0}}(0)=e_{0}$
and $\tilde{\sigma_{i}}(0)g_{j}=\tilde{\sigma_{j-1}}(1)$. By setting
$\tilde{\sigma_{j}}(0)=e_{j}$ and $(e_{j},g_{j}):e_{j}g_{j}\to e_{j}$ the
arrows in $E\rtimes G$ it follows that
$\tilde{\sigma}=\tilde{\sigma_{n}}(e_{n},g_{n})\tilde{\sigma_{n-1}}\cdots(e_{1},g_{1})\tilde{\sigma_{0}}$
is the unique $(E\rtimes G)$-path starting at $e_{0}$ which projects onto
$\sigma$. Since $G$-homotopic paths are in this way lifted to $(E\rtimes
G)$-homotopic paths, it holds that any covering space over $G$ with a base-
point $e_{0}$ as above has a natural fiber-wise action of $\Pi_{1}(G,x_{0})$.
Indeed, if $[\sigma]$ is the $G$-homotopy class of a $G$-loop at $x_{0}$ then
we define $e_{0}\cdot[\sigma]:=\tilde{\sigma_{n}}(1)$. This defines a right
action of $\Pi_{1}(G,x_{0})$ on $p^{-1}(x_{0})$. As a consequence of the
previous lifting property we get that the induced group homomorphism
$\Pi_{1}(E\rtimes G,e_{0})\to\Pi_{1}(G,x_{0})$ is injective, see [4, p. 611].

Let us assume that the action groupoid $E\rtimes G\rightrightarrows E$ is
$(E\rtimes G)$-connected. Thus, it is simple to check that the action of
$\Pi_{1}(G,x_{0})$ on $p^{-1}(x_{0})$ is transitive which implies that
$p^{-1}(x_{0})$ is homogeneous. That is, it is isomorphic to the quotient
$\Pi_{1}(G,x_{0})/p_{\ast}(\Pi_{1}(E\rtimes G,e_{0}))$ since the isotropy at
$e_{0}$ agrees with $p_{\ast}(\Pi_{1}(E\rtimes G,e_{0}))$ by definition.
Observe that every element $f\in\Gamma^{G}(E)$ induces an automorphism of
$p^{-1}(x_{0})$ when seeing it as a $\Pi_{1}(G,x_{0})$-space. Indeed, if
$F:E\rtimes G\to E\rtimes G$ is the groupoid automorphism $F(e,g)=(f(e),g)$
induced by $f$ then note that the $(E\rtimes G)$-path
$F_{\ast}(\tilde{\sigma})$ has initial point $f(e_{0})$ and final point
$f(e_{0}\cdot[\sigma])$ and satisfies
$p_{\ast}(F_{\ast}(\tilde{\sigma}))=\sigma$, which means that it is also a
lift of $\sigma$. Therefore, $f(e_{0})\cdot[\sigma]=f(e_{0}\cdot[\sigma])$ by
uniqueness.

Just as in the classical case it follows that if our covering space $E$ over
$G$ is universal then $\Gamma^{G}(E)\cong\Pi_{1}(G,x_{0})$, see for instance
[4, p. 612]. This important fact together with the previous comments allow us
to prove the following result.

######  Proposition 2.4.

If $H$ is a subgroup of $\Pi_{1}(G,x_{0})$ then there exists a covering
$r:F\to M$ over $G$ such that $r_{\ast}(\Pi_{1}(F\rtimes G,e_{0}))=H$.

###### Proof.

Let us fix a universal covering $p:E\to M$ over $G$, see Example 2.3. As
$\Gamma^{G}(E)\cong\Pi_{1}(G,x_{0})$ and $\Gamma^{G}(E)$ acts on the fibers
$p^{-1}(x_{0})$ without any fixed points then we may assume that
$\Pi_{1}(G,x_{0})$ acts on it without fixed points, see [4, p. 612]. Choose
$e_{0}\in p^{-1}(x_{0})$ and define the subgroup $H^{\prime}$ of
$\Gamma^{G}(E)$ as follows: $f\in H^{\prime}$ if and only if there exists
$[\sigma]\in H$ such that $f(e_{0})=e_{0}\cdot[\sigma]$. It is simple to check
that $H$ and $H^{\prime}$ are isomorphic. As $H^{\prime}$ is a subgroup of
$\Gamma^{G}(E)$ it is actually a properly discontinuous subgroup of
diffeomorphisms of $E$. Let us define $F$ as the quotient manifold
$E/H^{\prime}$ and denote by $q:E\to F$ the corresponding canonical
projection. It is well known that this is also a covering space. Denote by
$r:F\to M$ the induced smooth map defined as $r([e])=p(e)$. As $r\circ q=p$ it
follows that $r$ is also a covering space. The right action of $G$ along
$p:E\to M$ induces a well defined right action of $G$ along $q:E\to F$ as
$[e]\cdot g:=[e\cdot g]$ which, in turn, induces another right action of $G$
along $r:F\to M$ by the same expression since the elements of $\Gamma^{G}(E)$
are $G$-equivariant. Therefore, we get that $\Pi_{1}(G,x_{0})$ acts
transitively on the right of the fibers $r^{-1}(x_{0})$. Let
$\tilde{x_{0}}=q(e_{0})\in r^{-1}(x_{0})$. Finally, by the construction of $F$
it holds that the isotropy group of the right $\Pi_{1}(G,x_{0})$-action on
$r^{-1}(x_{0})$ corresponding to $\tilde{x_{0}}$ is precisely the subgroup
$H$. That is, $r_{\ast}(\Pi_{1}(F\rtimes G,e_{0}))=H$. ∎

###  2.2. Groupoid homology with local coefficients

Motivated by the notion of local coefficients system for topological spaces
(see for instance [26, c. 6]), we introduce a corresponding notion as well as
its associated homology in the groupoid setting. Let $R$ be a ring. A _local
system of $R$-modules_ over $G\rightrightarrows M$ is defined as a function
which assigns to any point $x_{0}\in M$ a left $R$-module $\mathcal{L}_{x}$
and to any continuous $G$-path $\sigma$ from $x$ to $y$ an $R$-homomorphism
$\sigma_{\\#}:\mathcal{L}_{y}\to\mathcal{L}_{x}$ such that the following
conditions are satisfied:

  * •

if $\sigma$ and $\sigma^{\prime}$ are $G$-homotopic then
$\sigma_{\\#}=\sigma^{\prime}_{\\#}$,

  * •

if $\sigma$ is the constant $G$-path at $x_{0}$ then
$\sigma_{\\#}:\mathcal{L}_{x_{0}}\to\mathcal{L}_{x_{0}}$ is the identity map,
and

  * •

if $\sigma_{0}^{\prime}(0)=\sigma_{n}(1)$ then
$(\sigma\ast\sigma^{\prime})_{\\#}=\sigma_{\\#}\circ\sigma^{\prime}_{\\#}$.

Compare with [6, 22]. Note that for any two point $x,y\in M$ it holds that
$\mathcal{L}_{x}$ and $\mathcal{L}_{y}$ are isomorphic by $G$-connectedness.
In particular, any $G$-loop $\sigma$ at $x_{0}$ determines an automorphism
$\sigma_{\\#}:\mathcal{L}_{x}\to\mathcal{L}_{x}$ which depends only on its
$G$-homotopy class. This implies that the correspondence
$[\sigma]\mapsto\sigma_{\\#}$ induces an action of $\Pi_{1}(G,x_{0})$ on
$\mathcal{L}_{x_{0}}$, thus turning $\mathcal{L}_{x_{0}}$ into a left module
over the group ring $R[\Pi_{1}(G,x_{0})]$. A _homomorphism_ between two local
systems of $R$-modules $\mathcal{L}$ and $\mathcal{L}^{\prime}$ over $G$ is
just natural transformation $\Phi:\mathcal{L}\to\mathcal{L}^{\prime}$. Namely,
for each $x_{0}\in M$ we have a homomorphism
$\Phi_{x_{0}}:\mathcal{L}_{x_{0}}\to\mathcal{L}^{\prime}_{x_{0}}$ such that
for each $G$-path $\sigma$ from $x$ to $y$ it holds
$(\sigma_{\\#})^{\prime}\circ\Phi_{y}=\Phi_{x}\circ\sigma_{\\#}$. If each
$\Phi_{x}$ is an isomorphism then we say that $\mathcal{L}$ and
$\mathcal{L}^{\prime}$ are _isomorphic_.

######  Lemma 2.5.

Suppose that our groupoid is $G$-connected.

  * •

Let $\mathcal{L}$ and $\mathcal{L}^{\prime}$ be two local systems of
$R$-modules over $G\rightrightarrows M$ and let
$\phi:\mathcal{L}_{x_{0}}\to\mathcal{L}^{\prime}_{x_{0}}$ be an isomorphism.
Then there exists a unique isomorphism
$\Phi:\mathcal{L}\to\mathcal{L}^{\prime}$ such that $\Phi_{x_{0}}=\phi$.

  * •

Let $\mathcal{L}_{0}$ be an $R$-module acted upon by $\Pi_{1}(G,x_{0})$. Then
there exists a local system of $R$-modules $\mathcal{L}$ such that
$\mathcal{L}_{x_{0}}=\mathcal{L}_{0}$ and which induces the given action of
$\Pi_{1}(G,x_{0})$ on $\mathcal{L}_{x_{0}}$.

###### Proof.

The proofs of these statements are straightforward adaptations of the proofs
of Theorems 1.11 and 1.12 in [26, p. 263], so that they are left as an
exercise to the reader. ∎

######  Example 2.6.

Suppose that $\alpha:\mathbb{Z}[\Pi_{1}(G,x_{0})]\to R$ is a ring
homomorphism. We may view $R$ as a left $\mathbb{Z}[\Pi_{1}(G,x_{0})]$-module
with the action $[\sigma]\cdot r:=r\alpha([\sigma])$. Such an action commutes
with the standard left $R$-module structure on $R$. Therefore, by Lemma 2.5 it
follows that $\alpha$ determines a local system of $\mathcal{L}_{\alpha}$ of
$R$-modules over $G\rightrightarrows M$. Note that each module
$(\mathcal{L}_{\alpha})_{x}$ is isomorphic to $R$.

Let $\mathcal{L}$ be a local system of $R$-modules over $G$ and consider the
nerve structure $\\{G^{(n)},d^{n}_{k}\\}$ of $G$. Let $\lambda_{n}:G^{(n)}\to
G^{(0)}$ denote the “last vertex” map
$\lambda_{n}(g_{1},\cdots,g_{n})=s(g_{n})$ with
$\lambda_{0}=\textnormal{id}_{G^{(0)}}$. We define
$C^{q}(G^{(n)},\mathcal{L})$ as the set of all functions $c$ with the
following properties:

  1. $a.$

for any $n$-singular $q$-simplex $\Sigma:\triangle_{q}\to G^{(n)}$ the value
$c(\Sigma)$ is defined and belongs to the $R$-module
$\mathcal{L}_{\lambda_{n}(\Sigma(a_{0}))}$. Here we assume that $a_{0}$ is the
first vertex of the standard simplex $\triangle_{q}$ with vertices
$a_{0},a_{1},\cdots,a_{q}$, and

  2. $b.$

the set of $n$-singular $q$-simpleces $\Sigma$ such that $c(\Sigma)\neq 0$ is
finite.

The elements of $C_{q}(G^{(n)},\mathcal{L})$ are called _$n$ -singular
$q$-chains with coefficients in $\mathcal{L}$_. Note that any chain $c\in
C_{q}(G^{(n)},\mathcal{L})$ can be formally written as a finite sum of the
form $c=\sum_{j}v_{j}\cdot\Sigma_{j}$ where $\Sigma_{j}:\triangle_{q}\to
G^{(n)}$ are $n$-singular $q$-simplices and
$v_{j}\in\mathcal{L}_{\lambda_{n}(\Sigma_{j}(a_{0}))}$. That is,
$c(\Sigma_{j})=v_{j}$ and $c(\Sigma)=0$ for any $n$-singular $q$-simplex
$\Sigma$ different from $\Sigma_{j}$. Let $c=v\cdot\Sigma$ be an elementary
$n$-chain (i.e. the sum above contains only one term). On the one hand, we
define the boundary homomorphism $\partial:C_{q}(G^{(n)},\mathcal{L})\to
C_{q}(G^{(n-1)},\mathcal{L})$ as

$\partial
c=\partial(v\cdot\Sigma)=\sum_{k=0}^{n-1}(-1)^{k}v\cdot(d_{k}^{n}\circ\Sigma)+(-1)^{n}(g^{\Sigma}_{n})^{-1}_{\\#}(v)\cdot(d^{n}_{n}\circ\Sigma).$

Here $g^{\Sigma}_{n}$ denotes the arrow determined by the $n$-projection of
$\Sigma(a_{0})\in G^{(n)}$ onto $G^{(1)}$ and $(g^{\Sigma}_{n})^{-1}_{\\#}$ is
the inverse of the isomorphism
$(g^{\Sigma}_{n})_{\\#}:\mathcal{L}_{t(g^{\Sigma}_{n})}\to\mathcal{L}_{s(g^{\Sigma}_{n})}$
which is induced by the arrow $g^{\Sigma}_{n}$, viewed as a $G$-path from
$s(g^{\Sigma}_{n})=\lambda_{n}(\Sigma(a_{0}))$ to
$t(g^{\Sigma}_{n})=s(g^{\Sigma}_{n-1})=\lambda_{n-1}(d^{n}_{n}\circ\Sigma(a_{0}))$.
It is well known that $\partial^{2}=0$, compare [6, 7, 22]. On the other hand,
we also define the boundary operator
$\overline{d}:C_{q}(G^{(n)},\mathcal{L})\to C_{q-1}(G^{(n)},\mathcal{L})$ as

$\overline{d}c=\overline{d}(v\cdot\Sigma)=\sigma_{\\#}(v)\cdot
d_{0}(\Sigma)+\sum_{j=1}^{q}(-1)^{j}v\cdot d_{j}(\Sigma).$

In this case $d$ denotes the usual homology boundary operator and
$\sigma:[0,1]\to G^{(0)}$ is the path from $\lambda_{n}(\Sigma(a_{1}))$ to
$\lambda_{n}(\Sigma(a_{0}))$ defined by
$\sigma(\tau)=\lambda_{n}(\Sigma((1-\tau)a_{1}+\tau a_{0}))$, which has
corresponding isomorphism
$\sigma_{\\#}:\mathcal{L}_{\lambda_{n}(\Sigma(a_{0}))}\to\mathcal{L}_{\lambda_{n}(\Sigma(a_{1}))}$.
This operator also satisfies $\overline{d}^{2}=0$, see [26, p. 266].

######  Lemma 2.7.

The boundary operators $\partial$ and $\overline{d}$ commute.

###### Proof.

On the one side,

$\displaystyle\overline{d}(\partial(c))$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}(-1)^{k}(\sigma_{k})_{\\#}(v)\cdot
d_{0}(d_{k}^{n}\circ\Sigma))+\sum_{k=0}^{n-1}\sum_{j=1}^{q}(-1)^{k+j}v\cdot
d_{j}(d_{k}^{n}\circ\Sigma))$ $\displaystyle+$
$\displaystyle(-1)^{n}((\sigma_{n})_{\\#}((g^{\Sigma}_{n})^{-1}_{\\#}(v)))(d_{0}(d^{n}_{n}\circ\Sigma))+\sum_{j=1}^{q}(-1)^{j+n}(g^{\Sigma}_{n})^{-1}_{\\#}(v)\cdot
d_{j}(d_{n}^{n}\circ\Sigma)),$

where $\sigma_{k}(\tau)=(\lambda_{n-1}\circ
d_{k}^{n})(\Sigma((1-\tau)a_{1}+\tau a_{0}))$ for all $k=0,1,\cdots,n$. On the
other side,

$\displaystyle\partial(\overline{d}(c))$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}(-1)^{k}\sigma_{\\#}(v)\cdot d_{k}^{n}\circ
d_{0}(\Sigma)+\sum_{j=1}^{q}\sum_{k=0}^{n-1}(-1)^{j+k}v\cdot d_{k}^{n}\circ
d_{j}(\Sigma)$ $\displaystyle+$
$\displaystyle(-1)^{n}(g^{d_{0}(\Sigma)}_{n})^{-1}_{\\#}(\sigma_{\\#}(v))\cdot
d^{n}_{n}\circ
d_{0}(\Sigma)+\sum_{j=1}^{q}(-1)^{j+n}(g^{d_{j}(\Sigma)}_{n})^{-1}_{\\#}(v)\cdot
d_{n}^{n}\circ d_{j}(\Sigma).$

Firstly, note that for all $k=0,1,\cdots,n-1$ it holds that
$\lambda_{n-1}\circ d_{k}^{n}=\lambda_{n}$ so that the first two expressions
in the right hand side of the above equalities agree. The second two
expressions are exactly the same. Secondly, as
$d_{j}(\Sigma)(a_{0})=\Sigma(a_{0})$ for all $j=1,\cdots,q$ then
$g^{d_{j}(\Sigma)}_{n}=g^{\Sigma}_{n}$ which implies that the last expressions
also agree. It remains to prove that
$(\sigma_{n})_{\\#}\circ(g^{\Sigma}_{n})^{-1}_{\\#}=(g^{d_{0}(\Sigma)}_{n})^{-1}_{\\#}\circ(\sigma)_{\\#}$.
Observe that
$\lambda_{n-1}(d^{n}_{n}\circ\Sigma(a_{1}))=\lambda_{n-1}(\Sigma(a_{1}))$ so
that $\sigma_{n}\ast(g^{\Sigma}_{n})^{-1}$ and
$(g^{d_{0}(\Sigma)}_{n})^{-1}\ast\sigma$ are two $G$-paths from
$\lambda_{n-1}(\Sigma(a_{1}))$ to $\lambda_{n}(\Sigma(a_{0}))$. Define the
path $\gamma:[0,1]\to G^{(1)}$ as
$\gamma(\tau)=(\textnormal{pr}_{n}(\Sigma((1-\tau)a_{1}+\tau a_{0})))^{-1}$.
This is such that $\gamma(0)=(g^{d_{0}(\Sigma)}_{n})^{-1}$ and
$\gamma(1)=(g^{\Sigma}_{n})^{-1}$. Also, $t\circ\gamma=\sigma$ and
$s\circ\gamma=\sigma_{n}$. Therefore, since $\gamma(1)$ is $G$-homotopic to
the $G$-path $(s\circ\gamma)^{-1}\gamma(0)(t\circ\gamma)$ (see [16, p. 191]),
it holds that the $G$-paths above are $G$-homotopic and the result follows as
desired. ∎

The total homology associated to the double chain complex
$\\{C_{\bullet}(G^{(\bullet)},\mathcal{L}),\partial,\overline{d}\\}$ will be
denoted by $H_{\bullet}^{\textnormal{tot}}(G,\mathcal{L})$ and called the
_groupoid homology of $G\rightrightarrows M$ with local coefficients in
$\mathcal{L}$_. Let $\phi:(G\rightrightarrows
M)\to(G^{\prime}\rightrightarrows M^{\prime})$ be a Lie groupoid morphism and
suppose that $\mathcal{L}$ is a local system of $R$-modules over $G^{\prime}$.
By using the induced Lie groupoid morphism
$\phi_{\ast}:(\Pi_{1}(G)\rightrightarrows
M)\to(\Pi_{1}(G^{\prime})\rightrightarrows M^{\prime})$ it is possible to
define another local system of $R$-modules $\phi^{\ast}\mathcal{L}$ over $G$
by setting $(\phi^{\ast}\mathcal{L})_{x}=\mathcal{L}_{\phi_{0}(x)}$. Let
$c=\sum_{j}v_{j}\cdot\Sigma_{j}$ be an $n$-singular $q$-chain in
$C_{q}(G^{(n)},\phi^{\ast}\mathcal{L})$. As
$v_{j}\in(\phi^{\ast}\mathcal{L})_{\lambda_{n}(\Sigma_{j})(a_{0})}=\mathcal{L}_{\lambda^{\prime}_{n}(\phi_{n}\circ\Sigma_{j})(a_{0})}$
then we have a well defined map
$\phi_{\ast}:C_{q}(G^{(n)},\phi^{\ast}\mathcal{L})\to
C_{q}(G^{\prime(n)},\mathcal{L})$ given as
$\phi_{\ast}(\sum_{j}v_{j}\cdot\Sigma_{j})=\sum_{j}v_{j}\cdot(\phi_{n}\circ\Sigma_{j})$.
It is simple to check that $\phi_{\ast}$ commutes with both pairs of boundary
operators $\partial,\overline{d}$ and
$\partial^{\prime},\overline{d}^{\prime}$ so that we have a well defined group
homomorphism
$\phi_{\ast}:H_{\bullet}^{\textnormal{tot}}(G,\phi^{\ast}\mathcal{L})\to
H_{\bullet}^{\textnormal{tot}}(G^{\prime},\mathcal{L})$. It is well known that
if $\phi$ is a Morita map then horizontal homologies defined by $\partial$ and
$\partial^{\prime}$ are isomorphic, see [6, 7] and [16, p. 214]. Thus, by
using a usual argument of spectral sequences (see for instance [3, p. 108]),
we conclude:

######  Proposition 2.8.

A Morita map $\phi:G\to G^{\prime}$ induces an isomorphism
$\phi_{\ast}:H_{\bullet}^{\textnormal{tot}}(G,\phi^{\ast}\mathcal{L})\to
H_{\bullet}^{\textnormal{tot}}(G^{\prime},\mathcal{L})$.

Let us fix a universal covering $p:E\to M$ over $G$ and consider the action
groupoid $E\rtimes G\rightrightarrows E$, see Example 2.3. Recall that $p$
extends to a Lie groupoid morphism $p:E\rtimes G\to G$, defined by $p(e,g)=g$,
which covers the covering map $E\to M$. Pick $x_{0}\in M$. We know that
$\Gamma^{G}(E)\cong\Pi_{1}(G,x_{0})$ where the isomorphism is as follows. The
group $\Gamma^{G}(E)$ acts simply transitively on the fiber $p^{-1}(x_{0})$.
Take $e_{0}\in p^{-1}(x_{0})$. Given $f\in\Gamma^{G}(E)$, let $\tilde{\sigma}$
be a $E\rtimes G$-path joining $f(e_{0})$ with $e_{0}$. Define the map
$\beta:\Gamma^{G}(E)\to\Pi_{1}(G,x_{0})$ as
$\beta(f)=[p_{\ast}(\tilde{\sigma})]$. Observe that $\beta$ does not depend on
$\tilde{\sigma}$ since $\Pi_{1}(E\rtimes G,e_{0})$ is trivial. This is the
isomorphism we are interested in. For every $[\sigma]\in\Pi_{1}(G,x_{0})$ we
denote by $\beta_{[\sigma]}=\beta^{-1}([\sigma])$ the corresponding element in
$\Gamma^{G}(E)$. That is, $[\sigma]$ can be presented by
$p_{\ast}(\tilde{\sigma})$ where $\tilde{\sigma}$ is any $E\rtimes G$-path
from $e_{0}$ to $\beta_{[\sigma]}(e_{0})$.

Let us now consider nerve structure $\\{E^{(n)},(d^{n}_{k})^{E}\\}$ of the
action groupoid $E\rtimes G\rightrightarrows E$ as well as its total singular
groupoid homology. The group $\Gamma^{G}(E)$ acts naturally on
$C_{q}(E^{(n)},\mathbb{Z})$ as follows. Let $f_{1}:E\rtimes G\to E\rtimes G$
be the groupoid automorphism $f_{1}(e,g)=(f_{0}(e),g)$ induced by
$f_{0}=f\in\Gamma^{G}(E)$. So, $f\cdot\Sigma=f_{n}\circ\Sigma$ for
$\Sigma:\triangle_{q}\to E^{(n)}$ where $f_{n}:E^{(n)}\to E^{(n)}$ is the
induced map along the nerve. Let $\mathcal{L}$ be a local system of
$R$-modules over $G$. Denote by $\mathcal{L}_{0}:=\mathcal{L}_{x_{0}}$. It is
clear that $\Pi_{1}(G,x_{0})$ acts on $\mathcal{L}_{0}$ from the left. We
transform this action into a right action of $\Gamma^{G}(E)$ on
$\mathcal{L}_{0}$ by setting $v\cdot\beta_{[\sigma]}=(\sigma_{\\#})^{-1}(v)$.
Let us denote by
$\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{q}(E^{(n)},\mathbb{Z})$ the quotient
group of the tensor product $\mathcal{L}_{0}\otimes C_{q}(E^{(n)},\mathbb{Z})$
by the subgroup $Q_{q}(\mathcal{L}_{0},E^{(n)})$ generated by all elements of
the form $v\cdot f\otimes\Sigma-v\otimes f\cdot\Sigma$ with
$v\in\mathcal{L}_{0}$, $f\in\Gamma^{G}(E)$, and $\Sigma\in
C_{q}(E^{(n)},\mathbb{Z})$. Observe that, after naturally extending them to
the tensor product, the simplicial boundary operator $\partial$ maps
$Q_{q}(\mathcal{L}_{0},E^{(n)})$ onto $Q_{q}(\mathcal{L}_{0},E^{(n-1)})$ and
the homology boundary operator $d$ maps $Q_{q}(\mathcal{L}_{0},E^{(n)})$ onto
$Q_{q-1}(\mathcal{L}_{0},E^{(n)})$ so that they pass to the quotient, thus
giving rise to well defined boundary operators
$\partial:\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{q}(E^{(n)},\mathbb{Z})\to\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{q}(E^{(n-1)},\mathbb{Z})$
and
$d:\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{q}(E^{(n)},\mathbb{Z})\to\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{q-1}(E^{(n)},\mathbb{Z})$.
Hence, we have obtained a double chain complex
$\\{\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{\bullet}(E^{(\bullet)},\mathbb{Z}),\partial,d\\}$
whose associated total cohomology will be denoted by
$H_{\bullet}(\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}E,\mathbb{Z})$. We are now
in conditions to prove our $G$-version of the well known _Eilenberg homology
isomorphism_ , namely:

######  Proposition 2.9 (Eilenberg $G$-isomorphism).

There exists a chain isomorphism between
$\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{\bullet}(E^{(\bullet)},\mathbb{Z})$
and $C_{\bullet}(G^{(\bullet)},\mathcal{L})$. In particular, the total
homologies $H_{\bullet}(\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}E,\mathbb{Z})$
and $H_{\bullet}^{\textnormal{tot}}(G,\mathcal{L})$ are isomorphic.

###### Proof.

Let us first exhibit isomorphisms
$\tilde{p}:\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{q}(E^{(n)},\mathbb{Z})\to
C_{q}(G^{(n)},\mathcal{L})$. It is clear that the simply $E\rtimes
G$-connectedness implies that for each $e\in E$ there exists a unique
$E\rtimes G$-homotopy class $[\xi_{e}]$ presented by any $E\rtimes G$-path
$\xi_{e}$ from $e$ to $e_{0}$. Denote by $p_{n}:E^{(n)}\to G^{(n)}$ the map
induced by $p$ along the nerves. Let $v_{0}\in\mathcal{L}_{0}$,
$\Sigma:\triangle_{q}\to E^{(n)}$, $\Lambda=p_{n}\circ\Sigma$,
$e=\lambda^{E}_{n}(\Sigma(a_{0}))$, $x=\lambda^{G}_{n}(\Lambda(a_{0}))=p(e)$
and define $\tilde{p}_{o}:\mathcal{L}_{0}\otimes C_{q}(E^{(n)},\mathbb{Z})\to
C_{q}(G^{(n)},\mathcal{L})$ by

$\tilde{p}_{o}(v_{0}\otimes\Sigma)=v\cdot\Lambda,$

where $v=(p_{\ast}(\xi_{e}))_{\\#}(v_{0})\in\mathcal{L}_{x}$. Analogously to
how it was proven in Theorem 3.4 from [26, p. 279] it is simple to check that
$\tilde{p}_{o}Q_{q}(\mathcal{L}_{0},E^{(n)})=0$, so that $\tilde{p}_{o}$
induces a homomorphism
$\tilde{p}:\mathcal{L}_{0}\otimes_{\Gamma^{G}(E)}C_{q}(E^{(n)},\mathbb{Z})\to
C_{q}(G^{(n)},\mathcal{L})$ which is actually an isomorphism. It remains to
verify that these maps determine a chain map between the double chain
complexes involved.

Firstly, recall that $d_{j}(\Sigma(a_{0}))=\Sigma(a_{0})$ for all $1\leq j\leq
q$ so that

$\tilde{p}_{o}(d_{j}(v_{0}\otimes\Sigma))=\tilde{p}_{o}(v_{0}\otimes
d_{j}(\Sigma))=v\cdot(p_{n}\circ(d_{j}(\Sigma)))=v\cdot
d_{j}(\Lambda)=\overline{d}_{j}(\tilde{p}_{o}(v_{0}\otimes\Sigma)).$

As usual, denote by $\sigma(\tau)=\lambda_{n}^{E}(\Sigma((1-\tau)a_{1}+\tau
a_{0}))$. Note that
$e^{\prime}=\lambda_{n}^{E}(d_{0}(\Sigma)(a_{0}))=\lambda_{n}^{E}(\Sigma(a_{1}))$,
thus obtaining $\xi_{e^{\prime}}=\sigma\ast\xi_{e}$. Hence

$\tilde{p}_{o}(d_{0}(v_{0}\otimes\Sigma))=\tilde{p}_{o}(v_{0}\otimes
d_{0}(\Sigma))=v^{\prime}\cdot(p_{n}\circ(d_{0}(\Sigma)))=v^{\prime}\cdot
d_{0}(\Lambda),$

where

$v^{\prime}=(p_{\ast}(\xi_{e^{\prime}}))_{\\#}(v_{0})=(p_{\ast}(\sigma)\ast
p_{\ast}(\xi_{e}))_{\\#}(v_{0})=(p_{\ast}(\sigma))_{\\#}(v).$

But $p\circ\lambda_{n}^{E}=\lambda_{n}^{G}\circ p_{n}$ so that
$p_{\ast}(\sigma)(\tau)=\lambda_{n}^{G}(\Lambda((1-\tau)a_{1}+\tau a_{0}))$.
That is,
$\tilde{p}_{o}(d_{0}(v_{0}\otimes\Sigma))=\overline{d}_{0}(\tilde{p}_{o}(v_{0}\otimes\Sigma))$.

Secondly, observe that for $0\leq k\leq n-1$ we have

$\displaystyle\tilde{p}_{o}(\partial_{k}^{E}(v_{0}\otimes\Sigma))$
$\displaystyle=$
$\displaystyle\tilde{p}_{o}(v_{0}\otimes(d_{k}^{n})^{E}\circ\Sigma)=v\cdot(p_{n-1}\circ(d_{k}^{n})^{E}\circ\Sigma)$
$\displaystyle=$ $\displaystyle v\cdot((d_{k}^{n})^{G}\circ
p_{n}\circ\Sigma)=v\cdot(d_{k}^{n})^{G}\circ\Lambda=\partial_{k}^{G}(\tilde{p}_{o}(v_{0}\otimes\Sigma)).$

We denote by $e^{\Sigma}_{n}$ the arrow determined by the $n$-projection of
$\Sigma(a_{0})\in E^{(n)}$ onto $E\rtimes G$. Observe that
$e^{\prime}=\lambda_{n-1}^{E}((d_{n}^{n})^{E}\circ\Sigma)(a_{0}))=s(e^{\Sigma}_{n-1})=t(e^{\Sigma}_{n})$
which implies that $\xi_{e^{\prime}}=(e^{\Sigma}_{n})^{-1}\ast\xi_{e}$. Thus,

$\tilde{p}_{o}(\partial_{n}^{E}(v_{0}\otimes\Sigma))=\tilde{p}_{o}(v_{0}\otimes(d_{n}^{n})^{E}\circ(\Sigma))=v^{\prime}\cdot(p_{n-1}\circ((d_{n}^{n})^{E}\circ(\Sigma)))=v^{\prime}\cdot(d_{n}^{n})^{G}\circ\Lambda,$

where

$v^{\prime}=(p_{\ast}(\xi_{e^{\prime}}))_{\\#}(v_{0})=(p_{\ast}((e^{\Sigma}_{n})^{-1})\ast
p_{\ast}(\xi_{e}))_{\\#}(v_{0})=(p(e^{\Sigma}_{n})^{-1})_{\\#}(v)=((g^{\Lambda}_{n})^{-1})_{\\#}(v),$

since $p\circ\textnormal{pr}_{n}^{E}=\textnormal{pr}_{n}^{G}\circ p_{n}$. In
consequence,
$\tilde{p}_{o}(\partial_{n}^{E}(v_{0}\otimes\Sigma))=\partial_{n}^{G}(\tilde{p}_{o}(v_{0}\otimes\Sigma))$.
This completes the proof. ∎

######  Remark 2.10.

It is worth mentioning that similarly to how they were defined the total
homology groups $H_{\bullet}^{\textnormal{tot}}(G,\mathcal{L})$ it is possible
to define total cohomology groups
$H^{\bullet}_{\textnormal{tot}}(G,\mathcal{L})$ as well as to prove a
$G$-version of the Eilenberg cohomology isomorphism. This can be done by
mimicking our approach together with the classical constructions, see [11, p.
14].

##  3\. Closed basic 1-forms

We say that a Lie groupoid $G\rightrightarrows M$ is _proper_ if the
source/target map $(s,t):G\to M\times M$ is proper. In this case the groupoid
orbits $\mathcal{O}_{x}$ are embedded in $M$, the isotropy groups $G_{x}$ are
compact, and the orbit space $M/G$ is Hausdorff, second-countable, and
paracompact, visit [8]. Let $G\rightrightarrows M$ be a proper groupoid and
denote by $X=M/G$ its corresponding orbit space. A differential form $\omega$
on $M$ is said to be _basic_ if $s^{\ast}\omega=t^{\ast}\omega$, see [20, 25].
The set of basic forms will be denoted by
$\Omega_{\textnormal{bas}}^{\bullet}(G)$. It is clear that the de Rham
exterior differential on $\Omega^{\bullet}(M)$ restricts to
$\Omega_{\textnormal{bas}}^{\bullet}(G)$, thus yielding the so-called _basic
cohomology_ $H_{\textnormal{bas}}^{\bullet}(G,\mathbb{R})$ of $G$. Such a
cohomology is Morita invariant and also satisfies that
$H^{\bullet}(X,\mathbb{R})\cong H_{\textnormal{bas}}^{\bullet}(G,\mathbb{R})$,
where $H^{\bullet}(X,\mathbb{R})$ denotes the singular cohomology of $X$.

###  3.1. G-homomorphisms of periods

Let $\omega$ be a closed basic $1$-form on $G$ and let $\xi$ denote the basic
cohomology class $[\omega]\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$. We are
interested in studying some features of $\xi$ by defining its $G$-homomorphism
of periods as well as its corresponding covering space. In order to do so we
will make use of the topological ingredients described in the previous
section. For each smooth $G$-path
$\sigma=\sigma_{n}g_{n}\sigma_{n-1}\cdots\sigma_{1}g_{1}\sigma_{0}$ in $M$ we
define the $G$-path integral

$\int_{\sigma}\omega=\sum_{k=0}^{n}\int_{\sigma_{k}}\omega.$

######  Lemma 3.1.

Let $[\sigma]$ denote the $G$-homotopy class of $\sigma$. Then the expression

$\int_{[\sigma]}\omega=\int_{\sigma}\omega,$

is well defined.

###### Proof.

Let us pick another $G$-path $\sigma^{\prime}$ being $G$-homotopic to
$\sigma$. If $\sigma$ and $\sigma^{\prime}$ are equivalent then the assertion
is trivial. Suppose then that there is a smooth $G$-homotopy
$D(\tau,\cdot)=D_{n}(\tau,\cdot)d_{n}(\tau)\cdots
d_{1}(\tau)D_{0}(\tau,\cdot)$ from $\sigma$ to $\sigma^{\prime}$ with fixed
endpoints $x$ and $y$. It suffices to check that the expression

$I_{\tau}=\int_{D(\tau,\cdot)}\omega=\sum_{k=0}^{n}\int_{0}^{1}\omega_{D_{k}(\tau,\nu)}(D_{k}^{\prime}(\tau,\nu))d\nu,$

does not depend on $\tau$ by differentiating it with respect to $\tau$.
However, this computation can be verified in local coordinates as in the
classical case by using the fact that $\omega$ is a closed basic $1$-form and
the following identities are satisfied: $D_{k}(\tau,0)=t(d_{k}(\tau))$,
$D_{k-1}(\tau,1)=s(d_{k}(\tau))$ for all $k=1,\cdots,n$ and $D_{0}(\tau,0)=x$,
$D_{n}(\tau,1)=y$ for all $\tau\in[0,1]$.

∎

######  Remark 3.2.

Suppose that $\omega$ and $\omega^{\prime}$ are basic-cohomologous closed
basic 1-forms. That is, there is a basic smooth function $f:M\to\mathbb{R}$
such that $\omega-\omega^{\prime}=df$. By arguing as in Lemma 3.5 below it is
simple to check that
$\int_{\sigma}(\omega-\omega^{\prime})=\int_{\sigma}df=f(\sigma_{n}(1))-f(\sigma_{0}(0))$
since $f$ is basic. In consequence, the expression
$\int_{[\sigma]}\xi=\int_{\sigma}\omega$ is also well defined when $\sigma$ is
a $G$-loop.

The first interesting consequence of the previous result is the following.

######  Proposition 3.3.

If $G\rightrightarrows M$ is simply $G$-connected then
$H_{\textnormal{bas}}^{1}(G,\mathbb{R})=0$.

###### Proof.

Let $\omega$ be a closed basic $1$-form and define
$f(x)=\int_{\lambda_{x}}\omega$ where $\lambda_{x}$ is any $G$-path joining
$x_{0}$ with $x$. This function is smooth and well defined since
$\Pi_{1}(G,x_{0})$ is trivial. Let $\gamma:[1,3]\to M$ be a smooth path such
that $\gamma(2)=x$ and $\gamma^{\prime}(2)=X_{x}\in T_{x}M$. Let
$\lambda_{\gamma(1)}$ be a fixed $G$-path from $x_{0}$ to $\gamma(1)$ and
consider the $G$-path
$\lambda_{\gamma(\tau)}=\gamma|_{[1,\tau]}1_{\gamma(1)}\lambda_{\gamma(1)}$
for $1\leq\tau\leq 3$. Observe that
$f(\gamma(\tau))=f(\gamma(1))+\int_{1}^{\tau}\omega_{\gamma(\nu)}(\gamma^{\prime}(\nu))d\nu$.
Thus

$df_{x}(X_{x})=\frac{d}{d\tau}\left(f(\gamma(1))+\int_{1}^{\tau}\omega_{\gamma(\nu)}(\gamma^{\prime}(\nu))d\nu\right)|_{\tau=2}=\frac{d}{d\tau}\left(\int_{1}^{\tau}\omega_{\gamma(\nu)}(\gamma^{\prime}(\nu))d\nu\right)|_{\tau=2}=\omega_{x}(X_{x}).$

Let us now check that $f$ is basic. If $g\in G$ then it follows that
$\lambda_{t(g)}=c_{t(g)}g\lambda_{s(g)}$, where $c_{t(g)}$ denotes the
constant path at $t(g)$ and $\lambda_{s(g)}$ is any $G$-path from $x_{0}$ to
$s(g)$, is a $G$-path joining $x_{0}$ with $t(g)$. Note that by definition
$\int_{\lambda_{t(g)}}\omega=\int_{\lambda_{s(g)}}\omega$ since the $G$-path
$c_{t(g)}g$ does not contribute to the $G$-path integral of the left hand
side. Hence, $f(s(g))=f(t(g))$ as desired.

∎

If $\Pi_{1}(G,x_{0})$ is the fundamental group of $G$ at the base-point
$x_{0}\in M$ then from Lemma 3.1 we get a well defined group homomorphism
$l_{\omega}:\Pi_{1}(G,x_{0})\to(\mathbb{R},+)$ by sending
$[\sigma]\mapsto\int_{\sigma}\omega$. Since $\mathbb{R}$ is abelian it follows
that $l_{\omega}$ factors through the Hurewicz $G$-homomorphism $h$ from
Proposition 2.2 by a uniquely determined group homomorphism
$\textnormal{Per}_{\xi}:H_{1}(G,\mathbb{Z})\to\mathbb{R}$ which only depends
on the cohomology class $[\omega]\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$,
see Remark 3.2. This will be called the _$G$ -homomorphism of periods_ of
$\omega$.

Let $d\theta$ denote the angle form on $S^{1}$. Here
$\theta=\frac{1}{2\pi}\phi$ where $\phi$ is the angle multi-valued function on
$S^{1}$. This is a closed 1-form without zeroes which can not be presented as
the differential of a smooth function. The latter fact is consequence of the
Stokes Theorem and the fact that $\int_{S^{1}}d\theta=1$. It is clear that if
$f:M\to S^{1}$ is basic then $f^{\ast}(d\theta)$ becomes a closed basic
$1$-form on $M$. Let us characterize the closed basic 1-forms that can be
obtained in this way.

######  Proposition 3.4.

Let $\omega$ be a closed basic 1-form on $M$. Then $\omega=f^{\ast}(d\theta)$
where $f:M\to S^{1}$ is a smooth basic function if and only if the cohomology
class $\xi=[\omega]\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$ is integral,
that is, $\xi\in
H_{\textnormal{bas}}^{1}(G,\mathbb{Z})=H_{\textnormal{bas}}^{1}(G,\mathbb{R})\cap
H^{1}(M,\mathbb{Z})$.

###### Proof.

We will mainly follow the classical proof of this result as in [11, p. 37].
Suppose that $\omega=f^{\ast}(d\theta)$ with $f:M\to S^{1}$ basic. Note that
$f$ induces a Lie groupoid morphism $F:(G\rightrightarrows
M)\to(S^{1}\rightrightarrows S^{1})$ where $F$ is either given by $s^{\ast}f$
or $t^{\ast}f$. Therefore, if
$\sigma=\sigma_{n}g_{n}\sigma_{n-1}\cdots\sigma_{1}g_{1}\sigma_{0}$ is a
$G$-loop then

$f_{\ast}(\sigma)=f_{\ast}(\sigma_{n})1_{f_{\ast}(\sigma_{n})(0)}f_{\ast}(\sigma_{n-1})\cdots
f_{\ast}(\sigma_{1})1_{f_{\ast}(\sigma_{1})(0)}f_{\ast}(\sigma_{0}),$

turns out to be equivalent to a usual loop on $S^{1}$ (actually, we obtain a
branch of loops formed by $f_{\ast}(\sigma_{j})$ with $j=0,1,\cdots,n$). In
consequence, the number

$\int_{\sigma}\omega=\sum_{k=0}^{n}\int_{\sigma_{k}}f^{\ast}(d\theta)=\sum_{k=0}^{n}\int_{f_{\ast}(\sigma_{k})}d\theta=\int_{f_{\ast}(\sigma)}d\theta\in\mathbb{Z},$

is an integer since it agrees with the sum of the degrees of the loops
$f_{\ast}(\sigma_{j})$ on $S^{1}$, which clearly computes the degree of the
whole branch loop $f_{\ast}(\sigma)$. Thus, we get that the $G$-homomorphism
of periods of any closed basic $1$-form $\omega=f^{\ast}(d\theta)$ with
$f:M\to S^{1}$ basic takes integral values so that its associated cohomology
class lies inside $H_{\textnormal{bas}}^{1}(G,\mathbb{Z})$.

Conversely, let us now suppose that all the $G$-periods associated to $\xi$
are integral. Fix a base point $x_{0}\in M$ and define
$f(x)=\exp\left(2\pi\sqrt{-1}\int_{\lambda_{x}}\omega\right)$, where
$\lambda_{x}$ is any $G$-path joining $x_{0}$ with $x$. Note that the
definition of $f$ does not depend on $\lambda_{x}$ since if
$\lambda^{\prime}_{x}$ is another $G$-path from $x_{0}$ to $x$ then for the
$G$-loop at $x_{0}$ we get $\sigma=(\lambda^{\prime}_{x})^{-1}\ast\lambda_{x}$
and
$\int_{\lambda_{x}}\omega-\int_{\lambda^{\prime}_{x}}\omega=\int_{\sigma}\omega\in\mathbb{Z}$.
By performing similar computations as those in the proof of Proposition 3.3 it
is simple to check that $f$ is a smooth basic function satisfying
$\omega=f^{\ast}(d\theta)$. ∎

It follows from Proposition 2.4 that we can consider a covering space
$p:M_{\xi}\to M$ over $G$ which corresponds to the kernel of the
$G$-homomorphism of periods $\Pi_{1}(G,x_{0})\to\mathbb{R}$ of the cohomology
class $\xi$. Every $(M_{\xi}\rtimes G)$-loop in $M_{\xi}$ project by $p$ to a
$G$-loop in $M$ with trivial periods with respect to $\omega$. Therefore, it
follows that the pullback basic 1-form $p^{\ast}\omega$ is basic exact. That
is, there is a basic function $f:M_{\xi}\to\mathbb{R}$ such that
$p^{\ast}\omega=df$. Recall that the free abelian group of equivariant
covering transformations $\Gamma^{G}(E)$ acts by the left on $M_{\xi}$. Thus,
the cohomology class $\xi$ determines an injective group homomorphism
$\alpha_{\xi}:\Gamma^{G}(E)\to\mathbb{R}$ with image equal to the group of
periods. Indeed, we define $\alpha_{\xi}$ through the composition
$\textnormal{Per}_{\xi}\circ\beta^{-1}$ where
$\beta:\Gamma^{G}(E)\to\Pi_{1}(G,x_{0})$ as
$\beta(f)=[p_{\ast}(\tilde{\sigma})]$ is the isomorphism previously described.
Therefore, for $\varphi\in\Gamma^{G}(E)$ we choose $x\in M_{\xi}$ and any
$(M_{\xi}\rtimes G)$-path $\tilde{\sigma}$ from $x$ to $\varphi(x)$. Note that
$p_{\ast}(\tilde{\sigma})$ defines a $G$-loop at $p(x)$, so that it also
defines a homology class in $|p_{\ast}(\tilde{\sigma})|\in
H_{1}(G,\mathbb{Z})$. Hence, since
$H^{1}_{dR}(G)\cong\textnormal{Hom}(H_{1}(G,\mathbb{R}),\mathbb{R})$ and
$H_{\textnormal{bas}}^{1}(G,\mathbb{R})\hookrightarrow H^{1}_{dR}(G)$ it makes
sense to set

$\alpha_{\xi}(\varphi)=\langle\xi,|p_{\ast}(\tilde{\sigma})|\rangle=\int_{p_{\ast}(\tilde{\sigma})}\omega\in\mathbb{R}.$
(1)

Such an expression does not depend on $x$ nor $\tilde{\sigma}$. It is clear
that $df=p^{\ast}\omega$ is invariant by the action of $\Gamma^{G}(E)$ but $f$
is not, in fact:

######  Lemma 3.5.

The following formula holds true

$f(\varphi(x))=f(x)+\alpha_{\xi}(\varphi),$

for all $x\in M_{\xi}$ and $\varphi\in\Gamma^{G}(E)$.

###### Proof.

Pick a $(M_{\xi}\rtimes G)$-path
$\tilde{\sigma}=\tilde{\sigma}_{n}\tilde{g}_{n}\tilde{\sigma}_{n-1}\cdots\tilde{\sigma}_{1}\tilde{g}_{1}\tilde{\sigma}_{0}$
from $x$ to $\varphi(x)$. Then, since $f$ is basic we get

$\displaystyle\int_{\tilde{\sigma}}p^{\ast}\omega$ $\displaystyle=$
$\displaystyle\int_{\tilde{\sigma}}df=\sum_{k=0}^{n}(f(\tilde{\sigma_{k}}(1))-f(\tilde{\sigma_{k}}(0)))$
$\displaystyle=$ $\displaystyle
f(\tilde{\sigma_{n}}(1))+\sum_{k=1}^{n}(f(t(\tilde{g}_{k}))-f(s(\tilde{g}_{k})))-f(\tilde{\sigma_{0}}(0))=f(\varphi(x))-f(x).$

But
$\int_{\tilde{\sigma}}p^{\ast}\omega=\int_{p_{\ast}\tilde{\sigma}}\omega=\alpha_{\xi}(\varphi)$,
so that the formula follows. ∎

###  3.2. Closed basic 1-forms of Morse type

Let $\omega$ be a closed basic $1$-form on $G$ and let $\xi$ denote the basic
cohomology class $[\omega]\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$. Note
that $\xi=0$ if and only if there exists a basic smooth function
$f:M\to\mathbb{R}$ such that $\omega=df$. Therefore, in this case the Morse
theoretical features of $\omega$ on $X$ are the same as those described by
means of $f$ in [19]. We will be mainly interested in studying the case
$\xi\neq 0$.

######  Lemma 3.6.

The critical point set of $\omega\in\Omega_{\textnormal{bas}}^{1}(G)$ is
saturated in $M$. In particular, if $\omega_{1}=s^{\ast}\omega=t^{\ast}\omega$
then we have a topological subgroupoid
$\textnormal{Crit}(\omega_{1})\rightrightarrows\textnormal{Crit}(\omega)$ of
$G\rightrightarrows M$.

###### Proof.

This easily follows from the fact that both $t$ and $s$ are surjective
submersions and
$\textnormal{Crit}(\omega_{1})=s^{-1}\textnormal{Crit}(\omega)=t^{-1}\textnormal{Crit}(\omega)$.
∎

If $\omega$ is a closed basic 1-form then by the Poincaré Lemma [20, Lem. 8.5]
it follows that for each groupoid orbit $\mathcal{O}$ there exists an open
neighborhood $\mathcal{O}\subset U\subset M$ and a basic smooth function
$f_{U}\in\Omega_{\textnormal{bas}}^{0}(G|_{U})$ such that $\omega|_{U}=df$. If
$U$ is connected then the function $f_{U}$ is determined by $\omega|_{U}$
uniquely up to a constant. In particular,
$\textnormal{Crit}(\omega|_{U})=\textnormal{Crit}(f_{U})$. The _normal
Hessian_ of $\omega$ along a critical orbit $\mathcal{O}$ is defined to the
the normal Hessian of $f_{U}$ along $\mathcal{O}$. Thus:

######  Definition 3.7.

A critical orbit $\mathcal{O}$ of $\omega$ is said to be _nondegenerate_ if
and only if $f_{U}$ is nondegenerate in the sense of Bott. Accordingly, we say
that $\omega$ is _Morse_ if all of its critical orbits are nondegenerate.

The notion of Morse–Bott function is classical and was initially introduced by
Bott in [2]. Our first key observation is that the previous notion is Morita
invariant.

######  Proposition 3.8.

Suppose that $G$ an $G^{\prime}$ are Morita equivalent Lie groupoids. If
$G^{\prime}$ admits a Morse closed basic $1$-form then so does $G$.

###### Proof.

Let $P$ be a principal bi-bundle between $G$ and $G^{\prime}$ with anchor maps
$a_{l}:P\to M$ and $a_{r}:P\to M^{\prime}$, see [8]. If $\omega^{\prime}$ is a
Morse closed basic $1$-form on $G^{\prime}$ then there exists a unique closed
basic $1$-form $\omega$ on $G$ such that
$a_{l}^{\ast}(\omega)=a_{r}^{\ast}(\omega^{\prime})$, see for instance [20,
25]. This establishes a correspondence between critical orbits since both
$a_{l}$ and $a_{r}$ are surjective submersions, compare Lemma 5.11 in [21].
Furthermore, if $\mathcal{O^{\prime}}$ and $\mathcal{O}$ are related critical
orbits then there are connected neighborhoods $\mathcal{O^{\prime}}\subseteq
U^{\prime}\subseteq M^{\prime}$ and $\mathcal{O}\subseteq U\subseteq M$
together with basic functions $f^{\prime}_{U^{\prime}}$ and $f_{U}$ such that
$a_{l}^{\ast}(f_{U})=a_{r}^{\ast}(f^{\prime}_{U^{\prime}})$. Hence, if
$\mathcal{O^{\prime}}$ is nondegenerate then so is $\mathcal{O}$ since both
$a_{l}$ and $a_{r}$ are surjective submersions. ∎

Observe that if $\omega$ is a basic $1$-form on $G$ then the expression
$\overline{\omega}([x])=\omega(x)$ for $[x]\in X$ is well defined. In
particular, this fact allows us to define a _stacky closed $1$-form_ on the
differentiable stack $[M/G]$ presented by $G\rightrightarrows M$ as an element
$\overline{\omega}$ presented by a closed basic $1$-form $\omega$ on $G$. In
consequence, we say that $[x]$ is a critical point of $\overline{\omega}$ if
and only if $\mathcal{O}_{x}$ is a critical orbit of $\omega$. Also, a
critical point $[x]$ is nondegenerate if and only if $\mathcal{O}_{x}$ is
nondegenerate for $\omega$.

######  Definition 3.9.

A stacky closed $1$-form $\overline{\omega}$ on $[M/G]$ is _Morse_ if all of
its critical points are nondegenerate. That is, it is presented by a Morse
closed basic $1$-form on $G$.

It is well known that if $G\rightrightarrows M$ is a Lie groupoid then its
_tangent groupoid_ $TG\rightrightarrows TM$ is obtained by applying the
tangent functor to each of its structural maps. If $\mathcal{O}_{x}\subset M$
is an orbit then we can restrict the groupoid structure to
$G_{\mathcal{O}_{x}}=s^{-1}(\mathcal{O}_{x})=t^{-1}(\mathcal{O}_{x})$, thus
obtaining a Lie subgroupoid
$G_{\mathcal{O}_{x}}\rightrightarrows\mathcal{O}_{x}$ of $G\rightrightarrows
M$. Furthermore, the Lie groupoid structure of $TG\rightrightarrows TM$
induces a Lie groupoid
$\nu(G_{\mathcal{O}_{x}})\rightrightarrows\nu(\mathcal{O}_{x})$ on the normal
bundles, having the property that all of its structural maps are fiberwise
isomorphisms. In particular, we have that
$\overline{dt}\circ\overline{ds}^{-1}:s^{*}\nu(\mathcal{O}_{x})\to
t^{*}\nu(\mathcal{O}_{x})$ defines a representation
$(G_{\mathcal{O}_{x}}\rightrightarrows\mathcal{O}_{x})\curvearrowright(\nu(\mathcal{O}_{x})\to\mathcal{O}_{x})$.
As a consequence, for every $x\in M$ the isotropy group $G_{x}$ has a
canonical representation on the normal fiber $\nu_{x}(\mathcal{O}_{x})$ called
the _normal representation_ of $G_{x}$ on the normal direction.

Let $\mathcal{O}_{x}$ be a nondegenerate critical orbit of $\omega$ and let
$\mathcal{O}_{x}\subset U\subset M$ and $f_{U}:U\to\mathbb{R}$ respectively be
an open neighborhood and basic smooth function such that $\omega|_{U}=f_{U}$.
Let us also fix a groupoid metric on $G\rightrightarrows M$ in the sense of
del Hoyo and Fernandes, visit [9]. Since the normal Hessian
$\textnormal{Hess}(f_{U})$ is nondegenerate it follows that by using the
groupoid metric the normal bundle $\nu(\mathcal{O}_{x})$ splits into the
Whitney sum of two subbundles
$\nu_{-}(\mathcal{O}_{x})\oplus\nu_{+}(\mathcal{O}_{x})$ such that
$\textnormal{Hess}(f_{U})$ is strictly negative on $\nu_{-}(\mathcal{O}_{x})$
and strictly positive on $\nu_{+}(\mathcal{O}_{x})$. Let $G_{x}$ be the
isotropy group at $x$. From [19] we know that $\textnormal{Hess}(f_{U})$ is
invariant with respect to the normal representation
$G_{x}\curvearrowright\nu(\mathcal{O}_{x})$ so that it preserves the splitting
above since the normal representation is by isometries in this case. In
consequence, we get a normal sub-representation
$G_{x}\curvearrowright\nu_{-}(\mathcal{O}_{x})$. The _stacky index_ of $[x]$
is defined to be
$\dim\nu_{-}(\mathcal{O}_{x})/G_{x}=\dim\nu_{-}(\mathcal{O}_{x})-\dim G_{x}$.

##  4\. Novikov numbers for orbifolds

A Lie groupoid $G\rightrightarrows M$ is said to be _étale_ if either $s$ or
$t$ is a local diffeomorphism. From now on we assume that our Lie groupoid
$G\rightrightarrows M$ is étale and proper (i.e. it presents an orbifold) and
that the orbit space $X$ is compact. From [20] we know that $X$ can be
triangulated so that the basic cohomology
$H_{\textnormal{bas}}^{\bullet}(G,\mathbb{R})$ becomes a finite dimensional
vector space. Moreover, in this specific case we also get that
$H_{dR}^{\bullet}(G)\cong H_{\textnormal{bas}}^{\bullet}(G,\mathbb{R})$, see
[23]. Thus, $H_{dR}^{\bullet}(G)\cong H^{\bullet}(X,\mathbb{R})$. In
particular, it follows that we may identify the total singular homology
$H_{1}(G,\mathbb{Z})$ of $G$ with the singular homology
$H_{\bullet}(X,\mathbb{Z})$ of $X$. That is, we may assume from now on that
$H_{1}(G,\mathbb{Z})$ is a finitely generated $\mathbb{Z}$-module since $X$ is
compact.

Let $\xi\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$ be the cohomology class of
a closed basic 1-form $\omega$ on $M$ and let $\textnormal{Per}_{\xi}$ denote
its corresponding $G$-homomorphism of periods. Observe that in the particular
case of proper and étale Lie groupoids any group homomorphism
$H_{1}(G,\mathbb{Z})\to\mathbb{R}$ can be realized as the homomorphism of
periods of a closed basic $1$-form. We define the _rank_ of the basic
cohomology class $\xi$ as the rank of the image of $\textnormal{Per}_{\xi}$.
This number will be denoted by $\textnormal{rank}(\xi)$. Note that by arguing
as in the proof Proposition 3.3 we may prove that $\textnormal{rank}(\xi)=0$
if and only if there is a basic function $f:M\to\mathbb{R}$ such that
$\omega=df$. That is, $\textnormal{rank}(\xi)=0$ if and only if $\xi=0$.

######  Remark 4.1.

If we consider a covering space $p:M_{\xi}\to M$ over $G$ which corresponds to
the kernel of the $G$-homomorphism of periods then the expression (1) also
shows that the rank of the group $\Gamma^{G}(M_{\xi})$ equals the rank of the
cohomology class $\xi$.

######  Proposition 4.2.

The set of classes in $H_{\textnormal{bas}}^{1}(G,\mathbb{R})$ having rank $1$
is dense.

###### Proof.

The proof of this result is similar to [11, Cor. 2.2; p. 38] when considering
instead $H_{\textnormal{bas}}^{1}(G,\mathbb{R})$ and
$H_{\textnormal{bas}}^{1}(G,\mathbb{Z})$. ∎

We are now in conditions to define the Novikov Betti and torsion numbers
associated to the basic cohomology class $\xi$ by mimicking the three
definitions introduced in [11, s. 1.5]. Let $\Gamma\subset\mathbb{R}$ be an
additive subgroup. The _Novikov ring_ Nov$(\Gamma)$ consists of formal power
series of the form $x=\sum_{j=0}^{\infty}n_{j}\tau^{\gamma_{j}}$ where the
coefficients are integers $n_{j}\in\mathbb{Z}$ and the exponents
$\gamma_{j}\in\Gamma$ are real numbers forming a decreasing sequence
converging to $-\infty$, visit [11, c. 1] for further details concerning the
ring structure as well as the properties of Nov$(\Gamma)$. When
$\Gamma=\mathbb{R}$ we denote Nov$(\mathbb{R})$ just by Nov. For any basic
cohomology class $\xi\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$ we may define
a ring homomorphism $\phi_{\xi}:\mathbb{Z}(\Pi_{1}(G,x_{0}))\to\textbf{Nov}$
by setting $\phi_{\xi}([\sigma]):=\tau^{\langle\xi,|\sigma|\rangle}$ for all
$[\sigma]\in\Pi_{1}(G,x_{0})$. As consequence of Example 2.6 we get that
$\phi_{\xi}$ determines a local system $\mathcal{L}_{\xi}$ of left Nov-modules
over $G\rightrightarrows M$. The groups
$H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ are called _Novikov homology
groups_ of $\xi$. It follows that the homology
$H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ is a finitely generated module
over the ring Nov. Since Nov is a principal ideal domain we have that the
module $H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ is a direct sum of a
free submodule with a torsion submodule. The _Novikov Betti number_
$b_{j}(\xi)$ is defined to be the rank of the free summand of
$H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$ and the _Novikov torsion
number_ $q_{j}(\xi)$ is defined to be the minimal number of generators of the
torsion submodule of $H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$.

######  Remark 4.3.

On the one hand, if $G\rightrightarrows M$ is étale and proper and $p:E\to M$
is a covering space over $G$ (e.g. the universal covering from Example 2.3),
then the action groupoid $E\rtimes G\rightrightarrows E$ is étale and proper
as well. That is, it also represents an orbifold. On the other hand, note that
if $\xi=0$ then $\textnormal{Per}_{\xi}=0$ which implies that the homomorphism
$\phi_{\xi}$ takes values within the subring $\mathbb{Z}\subset\textbf{Nov}$,
meaning that $\mathcal{L}_{\xi}$ is trivial. Therefore, by looking at the
groupoid homology with local coefficients as the groupoid equivariant homology
described in Proposition 2.9 it follows that the total chain complex
$\textbf{Nov}\otimes_{\mathbb{Z}(\Pi_{1}(G,x_{0}))}\tilde{C}_{\bullet}(E)$
agrees in this case with
$\textbf{Nov}\otimes_{\mathbb{Z}}\tilde{C}_{\bullet}(G)$. In consequence, from
[11, Lem. 1.12] we get that $b_{j}(\xi)$ coincides with the Betti number
$\textnormal{rank}H_{j}(G,\mathbb{Z})$ and $q_{j}(\xi)$ equals the minimal
number of generators of the torsion subgroup of $H_{j}(G,\mathbb{Z})$. But
$H_{j}(G,\mathbb{Z})\cong H_{j}(X,\mathbb{Z})$, so that we recover the
corresponding numerical invariants of the orbit space $X$.

There are other two possible ways to define the Novikov numbers associated to
$\xi$. Let $\mathbb{Z}[\textbf{Nov}(\Gamma)]$ be the group ring consisting
just of finite sums $x$ as above. This is a subring of $\textbf{Nov}(\Gamma)$.
Let $S\subset\mathbb{Z}[\textbf{Nov}(\Gamma)]$ denote the subset consisting of
the elements in $\mathbb{Z}[\textbf{Nov}(\Gamma)]$ with leading term $1$, so
that we have a canonical inclusion of the localized ring
$\mathcal{R}(\Gamma):=S^{-1}\mathbb{Z}[\textbf{Nov}(\Gamma)]$ into the Novikov
ring $\textbf{Nov}(\Gamma)$. The ring $\mathcal{R}(\Gamma)$ will be called the
_rational part_ of $\textbf{Nov}(\Gamma)$.

Firstly, consider the $G$-homomorphism of periods
$\textnormal{Per}_{\xi}:H_{1}(G,\mathbb{Z})\to\mathbb{R}$ and denote by
$\Gamma_{\xi}$ its image inside $\mathbb{R}$. It follows that $\Gamma_{\xi}$
is a finitely generated free abelian group. The class $\xi$ determines a ring
homomorphism
$\psi_{\xi}:\mathbb{Z}(\Pi_{1}(G,x_{0}))\to\mathcal{R}(\Gamma_{\xi})$ defined
as $\psi_{\xi}([\sigma]):=\tau^{\langle\xi,|\sigma|\rangle}$ for all
$[\sigma]\in\Pi_{1}(G,x_{0})$. As above, the homomorphism $\psi_{\xi}$ gives
rise to a local system of left $\mathcal{R}(\Gamma_{\xi})$-modules
$\mathcal{M}_{\xi}$ over $G\rightrightarrows M$ and the homology
$H_{j}^{\textnormal{tot}}(G,\mathcal{M}_{\xi})$ is a finitely generated module
over the principal ideal $\mathcal{R}(\Gamma_{\xi})$. From [11, Cor. 1.12] we
obtain that $\textbf{Nov}(\Gamma_{\xi})$ is flat over
$\mathcal{R}(\Gamma_{\xi})$ so that we may get an isomorphism
$H_{j}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})\cong\textbf{Nov}(\Gamma_{\xi})\otimes_{\mathcal{R}(\Gamma_{\xi})}H_{j}^{\textnormal{tot}}(G,\mathcal{M}_{\xi})$.
This immediately implies that
$\textnormal{rank}H_{j}^{\textnormal{tot}}(G,\mathcal{M}_{\xi})$ equals
$b_{j}(\xi)$ and the minimal number of generators of the torsion submodule of
$H_{j}^{\textnormal{tot}}(G,\mathcal{M}_{\xi})$ agrees with $q_{j}(\xi)$.

Secondly, consider the covering space $p:M_{\xi}\to M$ over $G$ which
corresponds to the kernel of the $G$-homomorphism of periods
$\textnormal{Per}_{\xi}$, see Proposition 2.4. It is simple to see that a
$G$-loop $\sigma$ in $M$ lifts to another $(M_{\xi}\rtimes G)$-loop in
$M_{\xi}$ if and only if
$\textnormal{Per}_{\xi}(|\sigma|)=\langle\xi,|\sigma|\rangle=0$, where
$|\sigma|\in H_{1}(G,\mathbb{R})$ denotes the corresponding homology class of
the $G$-loop $\sigma$. Thus, by the isomorphism theorem it follows that the
group of covering transformations $\Gamma^{G}(M_{\xi})$ can be naturally
identified with $L_{\xi}=H_{1}(G,\mathbb{Z})/\textnormal{ker}({\xi})$ and
$\textnormal{Per}_{\xi}$ yields an isomorphisms between the groups $L_{\xi}$
and $\Gamma_{\xi}$. Observe that after fixing a base for the free abelian
group $L_{\xi}$ we may identify the group ring
$\Lambda_{\xi}=\mathbb{Z}[L_{\xi}]$ with the ring of Laurent integral
polynomials $\mathbb{Z}[T_{1},\cdots,T_{r},T_{1}^{-1},\cdots,T_{r}^{-1}]$. Let
us denote by $w_{1},\cdots,w_{r}$ the weights of the variables
$T_{1},\cdots,T_{r}$ which are determined by the $G$-homomorphism of periods
$\textnormal{Per}_{\xi}:L_{\xi}\to\Gamma_{\xi}$. These weights are linearly
independent over $\mathbb{Z}$ so that we may define the weight of a monomial
$T_{1}^{n_{1}}\cdots T_{r}^{n_{r}}$ as $\sum n_{j}w_{j}$. Denote by
$S_{\xi}\subset\Lambda_{\xi}$ the set consisting of the Laurent polynomials
such that the monomial of maximal weight appearing in them has coefficient
$1$. This is a multiplicative subset and the localized ring
$S_{\xi}^{-1}\Lambda_{\xi}$ is a principal ideal domain since it is isomorphic
to the rational subring $\mathcal{R}(\Gamma_{\xi})$ of the Novikov ring, see
[11, s. 1.3] for specific details.

Let $p_{1}:E\to M$ be the universal covering over $G$ (see Example 2.3) and
let $p_{2}:F\to M$ be the covering space over $G$ corresponding to the kernel
$\textnormal{ker}(\xi)\subset H_{1}(G,\mathbb{Z})$, after mapping it to
$\Pi_{1}(G,x_{0})$ by using the Hurewicz $G$-homomorphism (see Proposition
2.4). These covering spaces give rise to the action groupoids $E\rtimes
G\rightrightarrows E$ and $F\rtimes G\rightrightarrows F$ which are also étale
and proper. By viewing at the groupoid homology with local coefficients in
$\mathcal{M}_{\xi}$ as the groupoid equivariant homology described in
Proposition 2.9 we get isomorphisms among the total chain complexes

$\mathcal{R}(\Gamma_{\xi})\otimes_{\mathbb{Z}(\Pi_{1}(G,x_{0}))}\tilde{C}_{\bullet}(E)\cong
S_{\xi}^{-1}\Lambda_{\xi}\otimes_{\Lambda_{\xi}}\tilde{C}_{\bullet}(F)\cong
S_{\xi}^{-1}\tilde{C}_{\bullet}(F).$

It is important to notice that in the previous identifications we used the
isomorphism $\textnormal{Per}_{\xi}:L_{\xi}\to\Gamma_{\xi}$. As localization
is an exact functor we obtain that
$H_{\bullet}^{\textnormal{tot}}(G,\mathcal{M}_{\xi})\cong
S_{\xi}^{-1}H_{\bullet}(F,\mathbb{Z})$. Therefore, the Novikov Betti number
$b_{j}(\xi)$ coincides with $\textnormal{rank}H_{\bullet}(F,\mathbb{Z})$ and
the Novikov torsion number $q_{j}(\xi)$ equals the minimal number of
generators of the torsion submodule of the
$S_{\xi}^{-1}\Lambda_{\xi}$-submodule $S_{\xi}^{-1}H_{\bullet}(F,\mathbb{Z})$.

######  Remark 4.4.

The reader probably already noticed that the definitions of the Novikov
numbers in the context of orbifolds provided above became both natural and
straightforward after having described the algebraic topology notions from
Sections 2 and 3. It is left as an exercise to the reader to verify that
similar results as those in Sections 1.5 and 1.6 from [11] may be adapted in
our context without so many changes along the proofs. In particular, we have
that if $\xi_{1},\xi_{2}\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$ are two
basic cohomology classes such that
$\textnormal{ker}(\xi_{1})=\textnormal{ker}(\xi_{2})$ then
$b_{j}(\xi_{1})=b_{j}(\xi_{2})$ for all $j$. Also,
$q_{j}(\xi_{1})=q_{j}(\lambda\xi_{2})$ for all $\lambda\in\mathbb{R}$ with
$\lambda>0$.

######  Remark 4.5.

Since the Lie groupoids we are working with above are all of them étale and
proper it follows that after naturally adapting Corollaries 4.13 and 4.14 from
[16, p. 224] to the homology case we may think of the total homologies
$H_{\bullet}^{\textnormal{tot}}(G,\mathcal{L}_{\xi})$,
$H_{\bullet}^{\textnormal{tot}}(G,\mathcal{M}_{\xi})$,
$H_{\bullet}(E,\mathbb{Z})$ and $H_{\bullet}(F,\mathbb{Z})$ as being
respectively identified with the usual homologies of the orbit spaces
$H_{\bullet}(X,\pi_{\ast}(\mathcal{L}_{\xi}))$,
$H_{\bullet}(X,\pi_{\ast}(\mathcal{M}_{\xi}))$, $H_{\bullet}(E/E\rtimes
G,\mathbb{Z})$ and $H_{\bullet}(F/F\rtimes G,\mathbb{Z})$ where $\pi:M\to X$
denotes the canonical orbit projection.

###  4.1. Novikov inequalities

Let us now prove the Novikov inequalities for orbifolds. This can be done by
following the strategy described in [11, s. 2.3] step by step. It is worth
mentioning that the inequalities below depend at some point of the usual Morse
inequalities for orbifolds which were already proven in [12] (see also [19]).
Although the ideas of the proof are natural and straightforward adaptations of
the classical ones, we will bring enough details in order to use most of the
machinery introduced in the previous sections.

######  Theorem 4.6.

Let $G\rightrightarrows M$ be an étale and proper Lie groupoid such that the
orbit space $X$ is compact. Let $\omega$ be a Morse closed basic $1$-form on
$M$. If $c_{j}(\omega)$ denotes the number of critical points in $M/G$ having
stacky Morse index $j$ then

$c_{j}(\omega)\geq b_{j}(\xi)+q_{j}(\xi)+q_{j-1}(\xi),$ (2)

where $\xi=[\omega]\in H^{1}_{\textnormal{bas}}(G,\mathbb{R})$ is the basic
cohomology class of $\omega$.

###### Proof.

It suffices to prove the inequalities under the additional assumption that the
basic cohomology class $\xi$ is integral. That is, $\xi\in
H_{\textnormal{bas}}^{1}(G,\mathbb{Z})$. The latter requirement is equivalent
to asking that $\xi$ has rank $1$. Indeed, basic cohomology classes $\xi$ of
rank 1 are real multiples of integral basic cohomology classes, namely,
$\xi=\lambda\xi_{0}$ where $\xi\in H_{\textnormal{bas}}^{1}(G,\mathbb{Z})$ and
$\lambda$ is a nonzero real number. It is because in this specific case the
image of the $G$-homomorphism of periods $\textnormal{Per}_{\xi}$ is a cyclic
subgroup in $\mathbb{R}$ so that all periods are integral multiples of a
minimal period $\lambda\in\mathbb{R}_{>0}$. In other words, the
$\lambda^{-1}\xi$ has all integral periods and it belongs to
$H_{\textnormal{bas}}^{1}(G,\mathbb{Z})$. Therefore, if $\xi=\lambda\xi_{0}$
has rank $1$ and $\omega$ is a Morse closed basic 1-form in the class $\xi$
then $\omega_{0}=\lambda^{-1}\omega$ is another Morse closed basic 1-form in
the class $\xi_{0}$ having the same zeroes. In consequence,
$b_{j}(\xi)=b_{j}(\xi_{0})$ and $q_{j}(\xi)=q_{j}(\xi_{0})$ by Remark 4.4.

Assume for a moment that the Novikov inequalities (2) hold true for basic
cohomology classes of rank $1$. The argument to prove that the previous
assumption is enough to ensure that the inequalities hold true for every basic
cohomology class of rank $>1$ is similar to that in Lemma 2.5 from [11] after
considering instead basic cohomology. We sketch the argument here for the sake
of completeness as well as the role it plays in our proof. Suppose that
$\xi\in H_{\textnormal{bas}}^{1}(G,\mathbb{R})$ is a basic cohomology class of
rank $>1$ and let $\omega$ be a Morse closed basic $1$-form in $\xi$. Let
$S\subset X$ denote the set of zeroes of $\overline{\omega}$. This is a finite
set since $X$ is compact. Consider the vector subspace $N_{\xi}=\\{\eta\in
H_{\textnormal{bas}}^{1}(G,\mathbb{R}):\eta|_{\ker(\xi)}=0\\}$. By similar
arguments as those used in the proof of Theorem 1.44 from [11] and by
Proposition 4.2 it follows that there exists a sequence of rank 1 basic
cohomology classes $\xi_{n}\in N_{\xi}$ such that $\xi_{n}\to\xi$ as $n$ goes
to $\infty$, $b_{j}(\xi_{n})=b_{j}(\xi)$ and $q_{j}(\xi_{n})=q_{j}(\xi)$ for
all $j$ and $n$. Let us fix a basis $\eta_{1},\cdots,\eta_{r}$ of $N_{\xi}$
whose elements are respectively represented by closed basic $1$-forms
$\omega_{1},\cdots,\omega_{r}$. Note that since $G$ is proper and
$\eta_{k}|_{\ker(\xi)}=0$ we may ensure that there are open neighborhoods
$U_{k}\subset M$ such that $U_{k}/G_{U_{k}}\subset X$ are open neighborhoods
of $S$ and $\overline{\omega}_{k}$ vanishes identically on $U_{k}/G_{U_{k}}$
for all $k=1,\cdots,r$. It is clear that we can rewrite
$\xi=\sum_{k=1}^{r}a_{k}\eta_{k}$ and $\xi_{n}=\sum_{k=1}^{r}a_{k,n}\eta_{k}$
with $a_{k},a_{n,k}\in\mathbb{R}$ such that $a_{n,k}\to a_{k}$ as $n$ goes to
$\infty$. Let us define
$\omega_{n}=\omega-\sum_{k=1}^{r}(a_{k}-a_{n,k})\omega_{k}$. This is a basic
closed $1$-form for which there is an open neighborhood $U\subset M$ made out
of the $U_{k}^{\prime}s$ above such that $S\subset U/G_{U}$ and
$\omega-\overline{\omega}_{n}$ vanishes identically on $U/G_{U}$. Furthermore,
for $n$ large enough it follows that $\omega_{n}$ has no critical points
outside $U$. That is, $c_{j}(\omega_{n})=c_{j}(\omega)$ for any $j$ provided
that $n$ is large enough. Note that from the defining formula above it follows
that the basic cohomology class $[\omega_{n}]$ agrees with $\xi_{n}$ and they
have rank $1$. In consequence, we get that if the Novikov inequalities hold
true for $[\omega_{n}]$ then they must hold true also for $\xi$.

Suppose that $\xi$ has rank $1$. Let us take a covering space $p:M_{\xi}\to M$
over $G$ which corresponds to the kernel of the $G$-homomorphism of periods
$H_{1}(G,\mathbb{Z})\to\mathbb{R}$ of the cohomology class $\xi$, see
Proposition 2.4. In this case $M_{\xi}$ has an infinite cyclic group of
equivariant covering transformations $\Gamma^{G}(M_{\xi})$ whose generator is
denoted by $T$. We already know that the pullback $p^{\ast}(\omega)$ is a
closed basic-exact $1$-form so that there exists a basic function
$f:M_{\xi}\to\mathbb{R}$, uniquely determined up to constant, such that
$p^{\ast}(\omega)=df$. From Lemma 3.5 it follows that $f(Tx)-f(x)=c$ is a
constant for all $x\in M_{\xi}$. The number $c$ equals the minimal period of
$\omega$ so that we may assume $c=\pm 1$ in our case since $\xi$ is integral.
Assume that the generator $T$ is chosen so that $f(Tx)-f(x)=-1$ for all $x\in
M_{\xi}$. Otherwise, we may take $T^{-1}$ instead of $T$.

Let us consider the action groupoid $M_{\xi}\rtimes G\rightrightarrows
M_{\xi}$, the stacky function $\overline{f}:[M_{\xi}/M_{\xi}\rtimes
G]\to\mathbb{R}$ determined by $f$ and the Lie groupoid morphism
$p:M_{\xi}\rtimes G\to G$ induced by the covering space $M_{\xi}\to M$ over
$G$. We denote by $X_{\xi}$ the orbit space $M_{\xi}/M_{\xi}\rtimes G$ and by
$\overline{T}:X_{\xi}\to X_{\xi}$ and $\overline{p}:X_{\xi}\to X$ the induced
maps between the orbit spaces. On the one hand, observe that the formula
$\overline{f}(\overline{T}[x])-\overline{f}([x])=-1$ holds true for all
$[x]\in X_{\xi}$. On the other hand, the critical points of $f$ are precisely
the elements in $p^{-1}(x)$ of the zeroes $x\in M$ of $\omega$ so that the
critical points of the stacky function $\overline{f}$ in $X_{\xi}$ are given
by the elements in $\overline{p}^{-1}([x])$ of the zeroes $[x]$ of
$\overline{\omega}$ in $X$. Pick a regular value $b\in\mathbb{R}$ of
$\overline{f}$ and set $V=\overline{f}^{-1}(b)$,
$N=\overline{f}^{-1}([b,b+1])$, and $Y=\overline{f}^{-1}((-\infty,b+1])$. Note
that the projection $\overline{p}$ determines a one-to-one correspondence
between the critical points of $\overline{f}|_{N}$ and the zeroes of
$\overline{\omega}$. Furthermore, $\overline{f}|_{N}:N\to[b,b+1]$ is a stacky
Morse function since $p_{\xi}$ is a local diffeomorphism and $\omega$ is of
Morse type. Therefore, $c_{j}(\overline{f}|_{N})=c_{j}(\omega)$ for all
$j=0,1,2,\cdots$. The homeomorphism $\overline{T}$ maps $Y$ into itself. As
$X_{\xi}$ can be triangulated (see [20]), it follows that we may fix a
triangulation of $V$ which in turn induces another triangulation of
$\overline{T}^{-1}V$, thus obtaining a simplicial isomorphism
$\overline{T}:\overline{T}^{-1}V\to V$. Let us choose a triangulation of $N$
in such a way $V$ and $\overline{T}^{-1}V$ are sub-complexes. So, after
applying the homeomorphism $\overline{T}$ we can get a triangulation of the
whole $Y$ so that $\overline{T}:Y\to Y$ is represented by a simplicial map. In
other words, we have obtained a chain complex $\overline{C}_{\bullet}(Y)$ of
simplicial chains which actually is a complex of finitely generated
$\mathbb{Z}[\overline{T}]$-modules.

The standard Morse inequalities for orbifolds were proved in Theorem 7.11 from
[12]. Hence, by mimicking the analysis of the Betti numbers
$b_{j}(\overline{C},\mathfrak{p})$ associated to different prime ideals
$\mathfrak{p}\subset\mathbb{Z}[\overline{T}]$, exactly to how it is done in
the remaining part of the proof of Theorem 2.4 in [11], we can get the
inequalities

$\sum_{k=0}^{j}(-1)^{k}c_{j-k}(\omega)\geq
q_{j}(\xi)+\sum_{k=0}^{j}(-1)^{k}b_{j-k}(\xi),$

which are slightly stronger than the Novikov inequalities we wanted to prove.
∎

Let us quickly explain how to apply Theorem 4.6 to find a lower bound for the
numbers of zeros of certain symplectic vector fields on symplectic orbifolds.
Let $G\rightrightarrows M$ be an étale and proper Lie groupoid with compact
orbit space $X$ and let $\rho:A\to TM$ denote its Lie algebroid. The set of
_basic vector fields_ $\mathfrak{X}_{\textnormal{bas}}(G)$ is by definition
the quotient

$\mathfrak{X}_{\textnormal{bas}}(G)=\frac{\\{(v_{1},v_{0})\in\mathfrak{X}(G)\times\mathfrak{X}(M):ds(v_{1})=v_{0}\circ
s,\ dt(v_{1})=v_{0}\circ t\\}}{\\{(v_{1},v_{0}):ds(v_{1})=v_{0}\circ s,\
dt(v_{1})=v_{0}\circ t,\ v_{1}\in(\ker(ds)+\ker(dt))\\}}.$

That is, a basic vector field is not strictly a vector field, but a pair of
equivalence classes of vector fields. It is simple to see that a basic vector
field $v=(v_{1},v_{0})$ is determined by its 2st component $v_{0}$. By
Proposition 5.3.12 from [13] we know that Morita equivalent étale a proper
groupoids have isomorphic spaces of basic vector fields so that we may think
of $\mathfrak{X}_{\textnormal{bas}}(G)$ as the space of vector field on the
orbifold presented by $G\rightrightarrows M$. Note that if we identify the
basic forms $\Omega_{\textnormal{bas}}^{\bullet}(G)$ with the set of pairs
$\\{(\theta_{1},\theta_{0})\in\Omega^{\bullet}(G)\times\Omega^{\bullet}(M):s^{\ast}(\theta_{0})=\theta_{1}=t^{\ast}(\theta_{0})\\}$
then we have contraction operations and Lie derivatives
$\iota:\mathfrak{X}_{\textnormal{bas}}(G)\times\Omega_{\textnormal{bas}}^{\bullet}(G)\to\Omega_{\textnormal{bas}}^{\bullet-1}(G)$
and
$\mathcal{L}:\mathfrak{X}_{\textnormal{bas}}(G)\times\Omega_{\textnormal{bas}}^{\bullet}(G)\to\Omega_{\textnormal{bas}}^{\bullet}(G)$
respectively defined by

$\iota_{v}\theta=(\iota_{\tilde{v_{1}}}\theta_{1},\iota_{\tilde{v_{0}}}\theta_{0})\quad\textnormal{and}\quad\mathcal{L}_{v}\theta=(\mathcal{L}_{\tilde{v_{1}}}\theta_{1},\mathcal{L}_{\tilde{v_{0}}}\theta_{0}),$

where $(\tilde{v_{1}},\tilde{v_{0}})\in\mathfrak{X}(G)\times\mathfrak{X}(M)$
is a representative of $v$. These expressions do not depend on the choice of
$(\tilde{v_{1}},\tilde{v_{0}})$, see [13]. By following [13, 14] we have that
a _symplectic form_ on $G\rightrightarrows M$ is by definition a closed basic
$2$-form $\Omega$ on $M$ which is _nondegenerate_ in the sense that
$\ker(\Omega)=\textnormal{im}(\rho)$. This nondegeneracy requirement implies
that the contraction with a symplectic form $\Omega$ induces a linear
isomorphism
$\Omega^{\flat}:\mathfrak{X}_{\textnormal{bas}}(G)\to\Omega_{\textnormal{bas}}^{1}(G)$.
Such a notion is also Morita invariant so that it yields a well defined notion
of symplectic form over an orbifold. We say that a basic vector field $v$ is
_symplectic_ if $\mathcal{L}_{\tilde{v_{0}}}\Omega=0$. Note that after using
the Cartan formula for the Lie derivative the latter requirement is equivalent
to asking that $\omega=\iota_{\tilde{v_{0}}}\Omega$ is a closed basic
$1$-form. But we already know that if $\omega$ is a closed $1$-form then the
formula $\omega=\iota_{\tilde{v_{0}}}\Omega$ defines a basic vector field $v$
which must be symplectic since $\omega$ is closed. Therefore, it follows that
there is a one-to-one correspondence between the closed basic 1-forms and
basic symplectic vector fields.

Motivated by Proposition 2.6 in [14] we define the _critical point set of a
basic vector field_ $v$ as the critical point set of $\tilde{v_{0}}$ viewed as
a section of the vector bundle $TM/\textnormal{im}(\rho)\to M$. It is simple
to check that such a definition does not depend on $\tilde{v_{0}}$. Because of
the nondegeneracy condition imposed over $\Omega$ it holds automatically that
the critical points of $v$ and $\omega=\iota_{\tilde{v_{0}}}\Omega$ agree.
Hence, on symplectic orbifolds, the problem of estimating from below the
number of zeros of closed basic 1-forms is equivalent to finding a lower bound
for the numbers of zeros of basic symplectic vector fields. In consequence,
the natural generalization of the Novikov theory we have developed in this
paper provides a tool for using topological methods to study zeros of
symplectic vector fields on symplectic orbifolds. Furthermore, it also opens
new research directions for many important physical models which can be
described by the Hamiltonian formalism over orbifolds allowing closed basic
1-forms as their Hamiltonians. This can be done in the same spirit that it was
studied by Novikov in [18].

## References

  * [1] K. Behrend: Cohomology of stacks, Intersection theory and moduli, ICTP Lect. Notes, XIX. Abdus Salam Int. Cent. Theoret. Phys., Trieste (2004), 249–294.
  * [2] R. Bott: Nondegenerate critical manifolds, Ann. of Math. (2), 60 (1954), 248–261.
  * [3] R. Bott, L. Tu: Differential forms in algebraic topology, Graduate Texts in Mathematics vol 82, Springer-Verlag, New York-Berlin (1982).
  * [4] M. R. Bridson, A. Haefliger: _Metric spaces of non-positive curvature_ , Springer-Verlag, Berlin 319 (1999).
  * [5] G. E. Bredon: _Topology and geometry_ , Graduate Texts in Mathematics, Springer-Verlag, New York, 139 (1997).
  * [6] M. Crainic: _Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes,_ Commentari Math. Helvetici, 78 (2003), 681–721.
  * [7] M. Crainic, I. Moerdijk: _A homology theory for étale groupoids,_ J. Reine Angew. Math., 521 (2000), 25–46.
  * [8] M. del Hoyo: _Lie groupoids and their orbispaces_ , Port. Math., 70 (2) (2012), 161–209.
  * [9] M. del Hoyo, R. Fernandes: _Riemannian metrics on Lie groupoids_ , J. Reine Angew. Math., 735 (2018), 143–173.
  * [10] M. del Hoyo, R. Fernandes: _Riemannian metrics on differentiable stacks_ , Math. Z., 292 (1-2) (2019), 103–132.
  * [11] M. Farber: _Topology of closed one-forms_ , Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, (2004).
  * [12] R. Hepworth: Morse inequalities for orbifold cohomology, Algebr. Geom. Topol., 9 (2) (2009), 1105–1175.
  * [13] B. Hoffman, R. Sjamaar: _Stacky Hamiltonian actions and symplectic reduction_ , Int. Math. Res. Not. IMRN, 2021 (20) (2021), 15209–15300.
  * [14] E. Lerman, A. Malkin: _Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds_ , Adv. Math., (229) (2012) n. 2, 984–1000.
  * [15] Y. Lin, R. Sjamaar, Convexity properties of presymplectic moment maps, J. Symplectic Geom., 17 (4) (2019), 1159–1200.
  * [16] I. Moerdijk, J. Mrcun: _Lie groupoids, sheaves and cohomology in: Poisson Geometry, Deformation Quantisation and Group Representations_ , London Math. Soc. Lecture Note Ser., Vol. 323, Cambridge Univ. Press, Cambridge, (2005) 145–272.
  * [17] S. P. Novikov: _Multi-valued functions and functionals. An analogue of Morse theory_ , Soviet Math. Doklady, 24 (1981), 222–226.
  * [18] S. P. Novikov: _The Hamiltonian formalism and a multi-valued analogue of Morse theory_ , Russian Math. Surveys, 37:5 (1982), 1–56.
  * [19] C. Ortiz, F. Valencia: _Morse theory on Lie groupoids_ , preprint: arXiv:math.DG/2207.07594, (2022).
  * [20] M. J. Pflaum, H. Posthuma, X. Tang: _Geometry of orbit spaces of proper Lie groupoids_ , J. Reine Angew. Math., 694 (2014), 49–84.
  * [21] H. Posthuma, X. Tang, K. Wang: _Resolutions of proper Riemannian Lie groupoids_ , Int. Math. Res. Not. IMRN, (2021) no. 2, 1249–1287.
  * [22] J-L. Tu: _Groupoid cohomology and extensions_ , Trans. Amer. Math. Soc., 358 (2006) no. 11, 4721–4747.
  * [23] J-L. Tu, P. Xu: _Chern character for twisted $K$-theory of orbifolds_, Adv. Math., 207 (2006) no. 2, 455–483.
  * [24] J-L. Tu, P. Xu, C. Laurent-Gengoux: _Twisted $K$-theory of differentiable stacks_, Ann. Sci. École Norm. Sup. (4), 37 (2004) no. 6, 841–910.
  * [25] J. Watts: _The Orbit Space and Basic Forms of a Proper Lie Groupoid_. In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics(). Birkhäuser, Cham, (2022) 513–523.
  * [26] G. W. Whitehead: _Elements of homotopy theory_ , Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 61 (1978).