# An Integral Euler Cycle in Normally Complex Orbifolds and
$\mathbb{Z}$-valued Gromov–Witten type invariants

Shaoyun Bai  Department of Mathematics, Princeton University, Princeton, NJ
08544, USA<EMAIL_ADDRESS>and  Guangbo Xu  Department of
Mathematics, Texas A&M University, College Station, Texas 77843, USA
<EMAIL_ADDRESS>
###### Abstract.

We define an integral Euler cycle for a vector bundle $E$ over an effective
orbifold $X$ for which $(E,X)$ is (stably) normally complex. The
transversality is achieved by using Fukaya–Ono’s “normally polynomial
perturbations” [FO97] and Brett Parker’s generalization [Par13] to “normally
complex perturbations.” One immediate application in symplectic topology is
the definition of integer-valued genus-zero Gromov–Witten type invariants for
general compact symplectic manifolds using the global Kuranishi chart
constructed by Abouzaid–McLean–Smith [AMS21].

###### Contents

  1. 1 Introduction
  2. 2 Normally complex orbifolds and bundles
  3. 3 Whitney stratifications on the variety $Z$
  4. 4 Pseudocycles in Thom–Mather stratified spaces
  5. 5 Construction of the integral Euler cycle
  6. 6 Stable complex derived orbifold bordism
  7. A Whitney stratifications

##  1\. Introduction

Let $(X,\omega)$ be a closed symplectic manifold and suppose $J:TX\rightarrow
TX$ is an almost complex structure tamed by $\omega$. Given a homology class
$A\in H_{2}(X,\mathbb{Z})$ and an integer $g\geq 0$, the moduli space of
stable $J$-holomorphic maps of class $A$ and genus $g$ with $n$ marked points
$\overline{\mathcal{M}}_{g,n}(X,J,A)$ carries a rational virtual fundamental
class $[\overline{\mathcal{M}}_{g,n}(X,J,A)]^{\text{vir}}$ as constructed by
Fukaya–Ono [FO99] using the theory of Kuranishi structures, along with other
constructions by Li–Tian [LT98], Ruan [Rua99], Hofer–Wysocki–Zehnder [HWZ17],
McDuff–Wehrheim [MW17], Pardon [Par16], and many others. The Gromov–Witten
invariants are defined by pairing
$[\overline{\mathcal{M}}_{g,n}(X,J,A)]^{\text{vir}}$ with cohomology classes
obtained from pulling back classes in
$H^{*}(\overline{\mathcal{M}}_{g,n};\mathbb{Q})$ and $H^{*}(X;\mathbb{Q})$
under the forgetful map and the evaluation maps.

The Gromov–Witten invariants are in general $\mathbb{Q}$-valued because of the
presence of non-trivial automorphisms of stable maps, though genus-$0$
Gromov–Witten invariants are $\mathbb{Z}$-valued for semi-positive symplectic
manifolds as proved by Ruan–Tian [RT95] and studied extensively in [MS04].
Technically speaking, rational numbers show up due to the use of abstract
multi-valued perturbations of the $\overline{\partial}_{J}$-equations as from
[FO99, LT98, HWZ17, MW17], or because Poincaré duality type constructions only
hold over $\mathbb{Q}$ rather than $\mathbb{Z}$ for orbifolds as used in
[Rua99, Par16] (see also [LT98, Remark 4] for a discussion about when the
invariants are integers).

Motivated by seeking for $\mathbb{Z}$-valued Gromov–Witten type invariants,
this paper explains how to construct integral “Euler classes” for an orbifold
complex vector bundle over an orbifold endowed with a “normal complex
structure” following a proposal of Fukaya–Ono [FO97] and the subsequent
development by Parker [Par13]. Combined with recent advances in algebraic
topology [Par19, Par20] and regularization of moduli spaces of $J$-holomorphic
maps [AMS21], we can study generalized bordism theories over orbispaces and
define $\mathbb{Z}$-valued Gromov–Witten type invariants for general
symplectic manifolds.

###  1.1. Statement of main results

Recall that an orbifold is called _effective_ if the local uniformizer group
from any orbifold chart acts faithfully.

######  Theorem 1.1.

Suppose $X$ is a compact effective orbifold without boundary endowed with an
almost complex structure and let $\mathcal{E}\rightarrow X$ be a complex
orbibundle. Denote by $X^{\rm free}\subset X$ the suborbifold consisting of
points with trivial isotropy group. Then there exist _normally complex_
(Definition 2.27) smooth sections $s:X\rightarrow\mathcal{E}$ which is
_strongly transverse_ (Definition 5.1) such that $s^{-1}(0)\cap X^{\rm free}$
defines a pseudocycle (Definition 4.7). Moreover, given a pair of such
sections $s_{1}$ and $s_{2}$, the pseudocycles $s_{1}^{-1}(0)\cap X^{\rm
free}$ and $s_{2}^{-1}(0)\cap X^{\rm free}$ are cobordant. Therefore, homology
class defined by the cobordism class of the pseudocycle

$\chi_{\rm free}^{{\rm FOP}}(X,\mathcal{E}):=[s^{-1}(0)\cap X^{\rm free}]\in
H_{*}(X;\mathbb{Z}),$ (1.1)

which we call by the Fukaya–Ono–Parker–Euler class, is an invariant of
${\mathcal{E}}\to X$.

######  Remark 1.2.

Zinger [Zin08] proved that any (oriented) pseudocycle in a smooth manifold
defines an integral homology class and the space of pseudocycles modulo
cobordism is natually isomorphic to the integral homology. In Section 4, we
show that a pseudocycle in any _Thom–Mather stratified space_ (including any
orbifold equipped with the _isotropy stratification_ , see Section 2.2),
defines an integral homology class.

Theorem 1.1 is a special case of Theorem 5.3 in Section 5.2, and we comment on
how it generalizes Theorem 1.1. Firstly, the existence of (almost) complex
structures can be weakened into the existence of _normal complex structures_.
Given $x\in X$, the tangent space
$T_{x}X=(T_{x}X)^{\text{triv}}\oplus(T_{x}X)^{\text{non}}$ is a representation
of the isotropy group $G_{x}$ and a normal complex structure on $X$ is a
family of complex structures on the direct sum of nontrivial $G_{x}$ isotypic
pieces $(T_{x}X)^{\text{non}}$ satisfying certain compatibility conditions,
see Section 2.5. In particular, the Fukaya–Ono–Parker–Euler class could have
an odd degree. Secondly, the compactness assumption on $X$ can be dropped as
follows. Instead of considering a pair $(X,\mathcal{E})$, we can start with a
triple $(X,\mathcal{E},s)$ where $s:X\rightarrow\mathcal{E}$ is a smooth
section of an complex orbibundle $\mathcal{E}\rightarrow X$ over an arbitrary
effective normally complex orbifold $X$, but the zero locus $s^{-1}(0)$ is
required to be compact. Then we can construct strongly transverse normally
complex perturbations of $s$ “relative $\infty$” to define an element in
$\chi_{\text{free}}^{\text{FOP}}(X,\mathcal{E},s)\in H_{*}(X;\mathbb{Z})$
independent of the perturbation. Thirdly, a strongly transverse normally
complex section actually defines an Euler-type class of $\mathcal{E}$ for any
given stratum of the isotropy stratification, and the class
$\chi_{\text{free}}^{\text{FOP}}(X,\mathcal{E})$ is the one associated to the
stratum consisting of points with trivial isotropy group. These classes could
be viewed as refinements of the rational Euler class $\chi(X,\mathcal{E})\in
H_{*}(X;\mathbb{Q})$ which could be constructed by multi-valued perturbations.

The first main application of Theorem 5.3 concerns about the algebraic
topology of bordisms of (derived) orbifolds. To state this result, we
introduce the following definition. Note that we do not impose the
effectiveness on $D$.

######  Definition 1.3.

A _derived orbifold chart_ (with boundary) is a triple $(D,E,s)$ where $D$ is
a smooth orbifold (with boundary), $E\rightarrow D$ is a smooth orbibundle and
$s:D\rightarrow E$ is a smooth section. $(D,E,s)$ is said to be compact if
$s^{-1}(0)$ is compact.

Following [Joy07, Par20], we can study geometric bordism type invariants
constructed from derived orbifold charts. Let $Y$ be an orbispace, i.e. a
topological stack locally modeled on the quotient of a topological space by a
finite group (see Definition 6.4), then one can consider compact derived
orbifold charts of the form $(D,E,s)$ together with a map $f:D\rightarrow Y$.
We introduce an equivalence relation among such quadruples $(D,E,s,f)$
generated by the following relations:

  1. (1)

(Restriction) $(D,E,s,f)\sim(D^{\prime},E^{\prime},s^{\prime},f^{\prime})$ if
$D^{\prime}\subset D$ is an open subset with $s^{-1}(0)\subset D^{\prime}$ and
$E^{\prime}=E|_{D^{\prime}}$, $s^{\prime}=s|_{D^{\prime}}$, and
$f^{\prime}=f|_{D^{\prime}}$.

  2. (2)

(Stabilization) $(D,E,s,f)\sim(D^{\prime},E^{\prime},s^{\prime},f^{\prime})$
if $D^{\prime}$ is the total space of a vector bundle $\pi_{F}:F\to D$,
$E^{\prime}=\pi_{F}^{*}E\oplus\pi_{F}^{*}F$,
$s^{\prime}=\pi_{F}^{*}s\oplus\tau_{F}$ where $\tau_{F}:F\to\pi_{F}^{*}F$ is
the tautological section, and $f^{\prime}=f\circ\pi_{F}$.

  3. (3)

(Cobordism) $(D,E,s,f)\sim(D^{\prime},E^{\prime},s^{\prime},f^{\prime})$ if
there is a bordism between them (see Section 6.3).

In fact, we restrict our attention to derived orbifold charts $(D,E,s)$
endowed with a _stable complex structure_ , which is defined to be a stable
complex structure on the virtual orbibundle $TD-E$. Then stable complex
derived orbifold charts mapped to $Y$ modulo the above equivalence relation
defines the _stable complex derived orbifold bordism_ of $Y$, written as
$\overline{\Omega}_{*}^{\mathbb{C},\text{der}}(Y)$. This is a generalization
of the stable complex bordism group over manifolds. A more detailed and
precise exposition is presented in Section 6. In particular, any map $f:M\to
Y$ from a closed almost complex manifold $M$ represents an element of
$\overline{\Omega}_{*}^{{\mathbb{C}},{\rm der}}(Y)$.

######  Theorem 1.4.

(= Theorem 6.21) Denote by $\Gamma$ the set of equivalence classes of triples
$[G,V,W]$ where $G$ is a finite group and $V,W$ are finite-dimensional complex
$G$-representations which do not contain any trivial summand, and where the
equivalence is induced from stabilization:

$(G,V,W)\sim(G,V\oplus R,W\oplus R).$

For any $[\gamma]\in\Gamma$, there is a well-defined map

$\mathcal{FOP}_{[\gamma]}:\overline{\Omega}^{\mathbb{C},{\rm
der}}_{*}(Y)\rightarrow H_{*-\dim V+\dim W}(Y;\mathbb{Z})$ (1.2)

which restricts to the pushforward of the usual fundamental class for the
trivial $[\gamma]$ and elements represented by maps from compact almost
complex manifolds.

We call the set $\Gamma$ from the above theorem the set of _stabilized
isotropy types_. In fact, together with a relative version of the map
$\mathcal{FOP}_{[\gamma]}$, we can obtain a natural transformation between two
(generalized) homology theories for (orbi)spaces. Theorem 1.4 is already
nontrivial even when $G$ is the trivial group, as the map
$\mathcal{FOP}_{[\gamma]}$ can be used to endow any compact stably complex
orbifold which is not necessarily effective an integral “fundamental class”.
It is well-known to experts that the set
$\overline{\Omega}^{\mathbb{C},\text{der}}_{*}(Y)$ could be very large and
complicated. However, we expect that the integral shadow (1.2) is more
amenable to manipulate but still contains considerable amount of information.

Our second application of Theorem 5.3 is the construction of
$\mathbb{Z}$-valued Gromov–Witten type invariants in symplectic topology as
proposed by Fukaya–Ono [FO97]. Besides the transversality issue resolved in
proving Theorem 5.3, an accompanying fundamental difficulty in the
construction is the lack of smoothness of the moduli space of pseudo-
holomorphic curves. One possible approach for a systematic solution to the
smoothness problem is to either apply the polyfold theory of
Hofer–Wysocki–Zehnder [HWZ07] or the delicate exponential decay estimates of
Fukaya–Oh–Ohta–Ono [FOOO16]. Instead, in this paper we appeal to a recent work
of Abouzaid–McLean–Smith [AMS21] which ingeniously constructed a global
Kuranishi chart with a smoothing on the genus zero Gromov–Witten type moduli
space using a geometric perturbation. Using the language of derived orbifold
charts, one main result from [AMS21] can be formulated as follows.

######  Proposition 1.5 ([AMS21, Proposition 5.35, 6.31]).

Let $(X,\omega)$ be a closed symplectic manifold and $J$ be a compatible
almost complex structure. Let $A\in H_{2}(X;\mathbb{Z})$ be a homology class
and denote by $\overline{\mathcal{M}}_{0,n}(X,J,A)$ the moduli space of
genus-$0$ stable $J$-holomorphic maps to $X$ with $n\geq 0$ marked points.
Then after choosing certain auxiliary data, there exists a smooth stable
complex derived orbifold chart $(D,E,s)$ along with a map

$\widetilde{\text{forget}}\times\widetilde{\text{ev}}:D\rightarrow\overline{\mathcal{M}}_{0,n}\times
X^{n}$

such that the zero locus $s^{-1}(0)$ is isomorphic to
$\overline{\mathcal{M}}_{0,n}(X,J,A)$ and the restriction of
$\widetilde{\text{forget}}\times\widetilde{\text{ev}}$ along $s^{-1}(0)$
coincides with the product of the forgetful map and the evaluation map.
Moreover, for different choices of $J$ and auxiliary data, the corresponding
derived orbifold charts together with
$\widetilde{\text{forget}}\times\widetilde{\text{ev}}$ define the same element
in $\overline{\Omega}^{\mathbb{C},{\rm
der}}_{*}(\overline{\mathcal{M}}_{0,n}\times X^{n})$.

######  Remark 1.6.

The paper [AMS21] uses the notion of _global Kuranishi charts_ instead of
derived orbifold charts. A topological/smooth global Kuranishi chart is a
quadruple $(G,D,E,s)$ where $G$ is a compact Lie group, $D$ is a
topological/smooth manifold endowed with an almost free continuous/smooth
$G$-action, $E$ is a $G$-equivariant vector bundle over $D$, and $s$ is a
$G$-equivariant continuous/smooth section of $E\rightarrow D$. The equivalence
relations introduced in the definition of
$\overline{\Omega}^{\mathbb{C},\text{der}}_{*}$ have counterparts in this
context, but there is an additional equivalence relation between global
Kuranishi charts, namely identifying $(G,D,E,s)$ with $(G^{\prime}\times
G,P,q^{*}E,q^{*}s)$ where $q:P\rightarrow D$ is a $G$-equivariant principal
$G^{\prime}$-bundle. This change-of-group operation should be interpreted as
choosing different global quotient presentation of the orbifold $[D/G]$, so
the arguments in [AMS21] indeed prove the statements in the form of
Proposition 1.5.

######  Remark 1.7.

In the forthcoming appendix written by Z. Zhou, it is shown that the
statements from Proposition 1.5 hold for moduli spaces of higher genus stable
pseudo-holomorphic maps as well. Unlike the construction in [AMS21] which
depends on smoothing theory, the results in the appendix are proved using
polyfolds.

######  Theorem 1.8.

Suppose $(X,\omega)$ is a closed symplectic manifold and $A\in
H_{2}(X;\mathbb{Z})$. Fix a non-negative integer $n$. Given any
$[\gamma]\in\Gamma$ as in Theorem 1.4, there is a well-defined integral
homology class

$[\overline{\mathcal{M}}_{0,n}(X,J,A)]^{\text{vir}}_{[\gamma]}\in
H_{*}(\overline{\mathcal{M}}_{0,n}\times X^{n};\mathbb{Z})$ (1.3)

defined by virtually “counting” $J$-holomorphic maps in
$\overline{\mathcal{M}}_{0,n}(X,J,A)$ in the stratum indexed by $[\gamma]$.

###### Proof.

This is a combination of Theorem 1.4 and Proposition 1.5. ∎

By pairing with classes in
$H^{*}(\overline{\mathcal{M}}_{0,n};\mathbb{Z})\otimes_{\mathbb{Z}}H^{*}(X;\mathbb{Z})^{\otimes
n}$, we can define $\mathbb{Z}$-valued Gromov–Witten type invariants indexed
by $\Gamma$. When $(X,\omega)$ is semi-positive and $[\gamma]$ is the trivial
stabilized isotropy type, as argued in [LT98, Remark 4], our invariants should
match the usual genus zero Gromov–Witten invariants.

######  Remark 1.9.

The main theorem in [AMS21] asserts that if $P\rightarrow S^{2}$ is a smooth
Hamiltonian fibration over the $2$-sphere with fiber given by a closed
symplectic manifold $(X,\omega)$, there is an isomorphism between
$H^{*}(P;\mathbb{Z})$ and
$H^{*}(X;\mathbb{Z})\otimes_{\mathbb{Z}}H^{*}(S^{2};\mathbb{Z})$. Via using
the virtual class indexed by the trivial group, our construction should
provide an alternative proof of this result.

###  1.2. A brief discussion of the proof

Our results have their roots in a brilliant proposal of Fukaya–Ono [FO97]. We
explain Fukaya–Ono’s idea in technical terms and discuss how to realize their
proposal using extra technical inputs.

We consider the following simplified situation. Let $G$ be a finite group and
let $V,W$ be a pair of finite-dimensional _complex_ $G$-representations. Let
us also assume that the $G$ acts on $V$ freely except at the $0$, whose
stabilizer is given by $G$. Suppose $M$ is a smooth manifold and we consider
the derived orbifold chart of the form

$([(M\times V)/G],\underline{W},s)$

where $G$ acts on $M\times V$ by extending the $G$-action on $V$ trivially
along the $M$-factor, $\underline{W}$ is the orbibundle induced from the
trivial bundle $M\times V\times W$, and $s$ is a smooth section with
$s^{-1}(0)$ compact. Furthermore, we assume that the isotypic decomposition of
$W$ do not contain the trivial summand. Under these assumptions, the smooth
section $s:[(M\times V)/G]\rightarrow\underline{W}$ can be identified with a
map with target in the space of smooth $G$-equivariant maps between $V$ and
$W$

$f:M\rightarrow C^{\infty}(V,W)^{G},$ (1.4)

where for each $p\in M$, $f(p)$ is the restriction of $s$ to the fiber
$\\{p\\}\times V$, so that

$s_{f}(p,v)=f(p)(v)\in W,\ \forall(p,v)\in M\times V.$

We wish to define a (relative) homology class of expected degree ${\rm
dim}M+{\rm dim}V-{\rm dim}W$ represented by $s_{f}^{-1}(0)/G$ for a transverse
section $s_{f}$. In general, due to the failure of equivariant transversality,
such a transverse section does not exist. Indeed $s_{f}$ necessarily vanishes
along $M$ whose dimension could be much higher than the expected dimension.

Fukaya–Ono proposed a solution by considering _normally complex polynomial
perturbations_. Namely, under the correspondence as in (1.4), we consider
perturbations of $s$ by sections of the form

$P:M\rightarrow{\rm Poly}_{d}^{G}(V,W)$

where ${\rm Poly}_{d}^{G}(V,W)\subset C^{\infty}(V,W)^{G}$ is the space of
$G$-equivariant complex polynomial maps of degree at most $d$. When $V$ is a
_faithful_ $G$-representation and $d$ is large enough, the arguments in [FO97]
show that for a generic choice of an element $P\in\text{Maps}(M,{\rm
Poly}_{d}^{G}(V,W))$ the zero locus along the stratum with trivial isotropy
group under the $G$-action

${\mathcal{M}}_{P}^{\rm free}:=s_{P}^{-1}(0)\cap(M\times V^{\text{free}})/G$
(1.5)

is a smooth manifold of the expected dimension. Most crucially, because $V$
and $W$ are both complex, the set-theoretic boundary of it has codimension at
least $2$. Therefore, using the notion of pseudocycles or any other
essentially equivalent language, an integral homology class can be defined.

However, the right notion of transversality is not only that the free part
(1.5) is cut out transversely, but also that its boundary (strata) are cut out
transversely. Two technical difficulties arise then. First, although the
interior transversality behaves well when we increase the cut-off degree $d$,
it was not clear if one has a proper notion of boundary transversality which
also behaves well with respect to the change of degrees. More seriously, when
the action of $G$ on $V$ has additional orbit types, the pointwise
transversality condition depends crucially on the stabilizers. It was not
clear if the transversality at one point implies the transversality at nearby
points with smaller stabilizers. More precisely, for any nontrivial subgroup
$H<G$, let $V_{H}\subset V$ be the space of $H$-fixed points and $N_{H}\subset
V$ the complement of $V_{H}$; write $v\in V$ as
$(v^{\prime},v^{\prime\prime})$. Then a normally polynomial section
$P:M\to{\rm Poly}_{d}^{G}(V,W)$ induces another normally polynomial section

$P_{H}:M\times V_{H}\to{\rm Poly}_{d}^{H}(N_{H},W),\ \
P_{H}(p,v^{\prime}):=P(p)(v^{\prime},\cdot).$

One thus needs a good notion of boundary transversality which behaves well
with respect to the _change of groups_ , i.e., $P$ is transverse implies
$P_{H}$ is transverse (or vice versa). Otherwise we will not have the openness
of the transversality condition, which could cause serious difficulties to
construct global perturbations.

The technical core of our construction is the resolution of the above issues
by defining the right notion of boundary transversality using the language of
Whitney stratifications. Indeed, the transversality notion for normally
polynomial sections is related to the complex affine variety

$Z_{d}^{G}(V,W):=\\{(v,P)\in V\times{\rm Poly}_{d}^{G}(V,W)\ |\ P(v)=0\\}$

and how one decomposes it into strata of manifolds. Specifically, in this
paper we define a canonical Whitney stratification on $Z_{d}^{G}(V,W)$ which
is a decomposition into algebraic submanifolds that respects the natural
action stratification of $V$:

$V=\bigsqcup_{H<G}V_{H}^{*},\ {\rm where}\ V_{H}^{*}:=\\{v\in V\ |\ {\rm the\
stabilizer\ of\ }v\ {\rm is\ }H\\}.$

A normally polynomial section $P:M\to{\rm Poly}_{d}^{G}(V,W)$ is said to be
strongly transverse (cf. Definition 5.1) if it is transverse to the image of
all strata of $Z_{d}^{G}(V,W)$ under the map $Z_{d}^{G}(V,W)\to{\rm
Poly}_{d}^{G}(V,W)$. This Whitney stratification is rather robust so that it
transforms naturally with respect to the natural inclusion map

$Z_{d}^{G}(V,W)\rightarrow Z_{d+d^{\prime}}^{G}(V,W)$ (1.6)

related to the change of degree (see Theorem 3.12) and the natural restriction
map

$Z_{d}^{G}(V,W)\to V_{H}^{*}\times Z_{d}^{H}(N_{H},W)$ (1.7)

related to the change of groups (see Theorem 3.18). In this way, the notion of
strong transversality of normally polynomial sections is intrinsic. The
existence of globally defined strongly transverse normal polynomial
perturbations and the relevant cobordism invariance results thus follow by the
technical arguments in Section 5. Therefore, the integral cycles and
invariants expected by Fukaya–Ono [FO97] can be defined. Note that the use of
the canonical Whitney stratification allows us to speak of transversality
along each stratum of the isotropy stratification of the ambient orbifold. As
a result, we actually obtain a system of integral homology classes indexed by
isotropy types.

######  Remark 1.10.

As mentioned in the course of the proof, there are several crucial points
inspired by the unpublished work of Parker [Par13]. Unlike our approach,
Parker tried to achieve transversality using a less canonical notion of _nice
Whitney stratifications_ on the variety $Z_{d}^{G}(V,W)$. Unfortunately, it is
unclear to us how to remove the dependence on the choice of these auxiliary
Whitney stratifications. The canonical Whitney stratification on
$Z_{d}^{G}(V,W)$ we define in this paper is intrinsic to $Z_{d}^{G}(V,W)$ as
it is the “minimal” Whitney stratification respecting the action
stratification, and it has better functorial properties.

######  Remark 1.11.

Parker also introduced the notion of _normally complex sections_ in [Par13]
which generalizes normally polynomial sections as from [FO97]. Although
normally polynomial sections suffice our needs as demonstrated in Lemma 5.5,
working with normally complex sections can provide extra flexibility at
various points. Hence we work with normally complex sections throughout the
paper and call them the Fukaya–Ono–Parker (FOP for short) sections (Definition
2.25).

Note that showing the existence of strongly transverse sections requires us to
work with normally complex effective orbifolds and normally complex
orbibundles. To establish Theorem 1.4 and its corollary Theorem 1.8, we make
use of the flexibility provided by the stabilization equivalences combined
with the “enough vector bundle” theorem proved by Pardon [Par19, Theorem 1.1]
to identify a stable complex derived orbifold chart with an effective derived
orbifold chart such that the ambient orbifold is almost complex and the
orbibundle is complex.

###  1.3. Relations with other work

####  1.3.1. $\mathbb{Z}$-valued invariants in symplectic topology

There are other known $\mathbb{Z}$-valued Gromov–Witten type invariants in
symplectic topology. Firstly, as mentioned above, for semi-positive symplectic
manifolds, Ruan–Tian’s construction ([RT95]) shows that the genus-$0$
Gromov–Witten invariants are indeed $\mathbb{Z}$-valued. Secondly, there are
the celebrated $\mathbb{Z}$-valued Gopakumar–Vafa invariants of Calabi–Yau
$3$-folds [GV98a, GV98b] which govern the $\mathbb{Q}$-valued Gromov–Witten
invariants as demonstrated in full generality by Ionel–Parker [IP18]. It was
speculated by some experts (e.g. [Joy07, Section 6.3]) that the invariants
defined using Fukaya–Ono’s normally polynomial sections are related to the
Gopakumar–Vafa invariants. However, recent developments [DW19], [BS21], which
should be viewed as continuations of the groundbreaking work of Taubes in
dimension $4$ [Tau96], construct $\mathbb{Z}$-valued invariants of Calabi–Yau
$3$-folds by (virtually) counting _embedded_ pseudo-holomorphic curves and
these invariants seems to be better connected with the Gopakumar–Vafa
invariants. Lastly, the $K$-theoretic Gromov–Witten invariants, defined by
Givental and Lee [Lee04] for algebraic objects (which are expected to exist
for general symplectic manifolds in light of [AMS21, Section 6.12]), are also
$\mathbb{Z}$-valued.

It is expected that one can use normally polynomial sections to define Floer
homology with ${\mathbb{Z}}$ or ${\mathbb{Z}}_{2}$-coefficients. A relevant
work along this line is [FOOO13] which uses the notion of “normally conical
perturbations” to define Lagrangian Floer homology for spherically positive
symplectic manifolds.

####  1.3.2. Topological construction and the Floer-theoretic counterpart

Recall that the typical method of defining the (${\mathbb{Q}}$-valued)
traditional Gromov–Witten invariants goes through the inductive construction
of transverse multisections on a Kuranishi atlas (good coordinate system) over
a moduli space. As one only needs transversality in the topological sense,
smoothness of coordinate changes are not necessary for the construction (see
[LT98] and the case of gauged linear sigma model in [TX18, TX21]). Although we
believe that our integral Euler classes are topological invariants, at this
stage we do not know for sure if our construction works without the smoothness
assumption. Working out all the arguments in the topological category would
ease up the technical difficulties in deriving Floer-theoretic applications
which improve the main result from [AB21].

####  1.3.3. $\mathbb{Z}$-valued invariants in algebraic geometry

There are several types of enumerative invariants constructed using algebraic
geometry, most notably the Donaldson–Thomas invariants [Tho00] and the
Pandharipande–Thomas invariants [PT09]. Although these invariants are closely
tied with the Gromov–Witten invariants [MNOP06a, MNOP06b], their constructions
depend on sheaf theory. Fukaya–Ono’s proposal is differential-topological in
nature, as it is still a variant of the general position argument. It would be
interesting to see if the virtual classes
$[\overline{\mathcal{M}}_{0,n}(X,J,A)]^{\text{vir}}_{[\gamma]}$ admit a purely
algebro-geometric interpretation. As remarked before, the classes
$[\overline{\mathcal{M}}_{0,n}(X,J,A)]^{\text{vir}}_{[\gamma]}$ are
refinements of the traditional rational virtual class
$[\overline{\mathcal{M}}_{0,n}(X,J,A)]^{\text{vir}}$, thus an algebro-
geometric construction of them might shed light on refinements other curve-
counting invariants.

####  1.3.4. Wasserman’s theorem and stable homotopy theory

A renowned theorem of Wasserman [Was69], recaptured in [Par20, Theorem 5.6],
provides a sufficient condition for equivariant transversality to hold. Our
result can be interpreted as a variant of Wasserman’s theorem given the
presence of normal complex structures, whose existence allows us to extract
more regularity. Rather than developing a theory of stable _normal_ complex
bordisms, we confine ourselves with stable complex bordisms of derived
orbifolds because the Pontryagin–Thom construction which identifies geometric
bordisms with homotopical cobordisms holds in this case [Par20, Theorem 1.4].
The homotopical cobordism perspective gives far-reaching corollaries of
Wasserman’s theorem, see [Sch18, Theorem 6.2.33]. As mentioned by [Par20,
Remark 5.7], it is an interesting question to understand Fukaya–Ono’s proposal
on the homotopical cobordism side.

###  1.4. Plan of the paper

We introduce basic notions related to orbifolds in Section 2, especially the
notion of _normal complex structure_ (Definition 2.22) and _FOP section_
(Definition 2.27). In Section 3, we study the canonical Whitney stratification
on the universal zero locus $Z_{d}^{G}(V,W)$ and understand its behavior under
the change of degrees and groups. Section 4 proves that any pseudocycle in a
Thom–Mather stratified space defines an integral homology class. The main
theorem is established in Section 5. In Section 6, we discuss how to use FOP
perturbations to study stable complex derived bordisms over orbispaces.
Finally, the necessacry background and results on Whitney stratifications are
discussed in Appendix A.

######  Remark 1.12.

When defining the ${\mathbb{Z}}$-valued genus zero curve counting invariants,
or equivalently the homology class (1.3), as $\overline{\mathcal{M}}_{0,n}$ is
a manifold, we do not need the general result about pseudocycles in
Thom–Mather stratified spaces. We also do not need to the more general
algebraic topology result Theorem 1.4. Hence results in Section 4 and Section
6 can be bypassed.

###  1.5. Acknowledgements

We would like to thank Mohammed Abouzaid, Kenji Fukaya, Helmut Hofer, John
Pardon, Paul Seidel, Mohan Swaminathan, and Dingyu Yang for useful
correspondences and discussions at various stages of this project. The first-
named author would like to thank his advisor John Pardon for constant
encouragement and support. The second-named author would like to thank his
family for their love and support during the Covid-19 pandemic.

##  2\. Normally complex orbifolds and bundles

###  2.1. Orbifolds and orbifold vector bundles

We recall the basic definition of effective orbifolds. We follow the
definition of [ALR07, Section 1.1]. Working with effective orbifolds allows us
to use orbifold charts exclusively without appealing to the language of
groupoids. We remark that for our applications in Section 6, we can drop
effectiveness by choosing an effective presentation of any given derived
orbifold.

Let $X$ be a Hausdorff and second countable topological space. An
$n$-dimensional orbifold chart of $X$ is a triple

$C=(U,{\mathit{\Gamma}},\psi)$

where $U\subset{\mathbb{R}}^{n}$ is a nonempty open subset,
${\mathit{\Gamma}}$ is a finite group acting effectively and smoothly on $U$,
and $\psi:U\to X$ is a ${\mathit{\Gamma}}$-invariant continuous map such that
the induced map

$\underline{\smash{\psi}}:U/{\mathit{\Gamma}}\to X$

is a homeomorphism onto an open subset of $X$. If $x\in\psi(U)$ we also say
that $x$ is contained in the chart $C$.

A chart embedding from another chart
$C^{\prime}=(U^{\prime},{\mathit{\Gamma}}^{\prime},\psi^{\prime})$ to $C$ is a
smooth (open) embedding $\iota:U^{\prime}\hookrightarrow U$ such that

$\psi\circ\iota=\psi^{\prime}.$

It follows that (see [ALR07, Page 3]) given a chart embedding $\iota$ as above
there exists a canonical group injection
${\mathit{\Gamma}}^{\prime}\hookrightarrow{\mathit{\Gamma}}$ such that $\iota$
is equivariant. Therefore we often include the group injection as part of the
data of a chart embedding.

As we are in the smooth category, we can always find “linear” charts around
any point. An orbifold chart $C=(U,{\mathit{\Gamma}},\psi)$ is called linear
if ${\mathit{\Gamma}}$ acts linearly on ${\mathbb{R}}^{n}$ and
$U\subset{\mathbb{R}}^{n}$ is an invariant open subset. We say that a linear
chart is centered at $x\in X$ if $0\in U$ and $x=\psi(0)$.

We say two charts $C_{i}=(U_{i},{\mathit{\Gamma}}_{i},\psi_{i})$, $i=1,2$ are
compatible if for each $p\in\psi_{1}(U_{1})\cap\psi_{2}(U_{2})$, there exists
an orbifold chart $C_{p}=(U_{p},{\mathit{\Gamma}}_{p},\psi_{p})$ and chart
embeddings into both $C_{1}$ and $C_{2}$.

An orbifold atlas ${\mathcal{A}}=\\{C_{i}\ |\ i\in I\\}$ on $X$ is a
collection of mutually compatible charts $C_{i}$ which cover $X$. We say an
atlas ${\mathcal{A}}^{\prime}=\\{C_{j}^{\prime}\ |\ j\in J\\}$ refines
${\mathcal{A}}$, equivalently, ${\mathcal{A}}^{\prime}$ is a refinement of
${\mathcal{A}}$, if for each $C_{j}^{\prime}$ there exists a chart embedding
$C_{j}^{\prime}\hookrightarrow C_{i}$ for some $i\in I$. We say two orbifold
atlases are equivalent if they have a common refinement. A topological space
$X$ together with an equivalence class of orbifold atlases is called a smooth
effective orbifold.

Let $X$ be an effective orbifold. A continuous function $f:X\to{\mathbb{R}}$
is called smooth if its pullback to each chart is a smooth function.

######  Remark 2.1.

One can see that orbifolds are all locally compact. As we also assume they are
Hausdorff and second countable, they are paracompact spaces. Hence for any
open cover by charts, there exists a subordinate continuous partition of
unity. Moreover, as one can approximate continuous functions by smooth
functions on each chart, there always exist a subordinate smooth partition of
unity.

######  Remark 2.2.

We also need the notion of orbifolds with boundary. In that case, the domain
of a chart $C=(U,{\mathit{\Gamma}},\psi)$ is allowed to be a smooth manifold
with boundary. The ${\mathit{\Gamma}}$-action is required to fixed the
boundary $\partial U\subset U$ set-wise; moreover, for each $x\in\partial U$,
the stabilizer of $x$ is required to fix $\partial U$ pointwise.

Similarly we can define the notion of orbifold vector bundles. Let $X$ be an
orbifold, ${\mathcal{E}}$ be a topological space, and $\pi:{\mathcal{E}}\to X$
be a continuous map. A bundle chart of ${\mathcal{E}}\to X$ consists of an
orbifold chart $C=(U,{\mathit{\Gamma}},\psi)$ of $X$, a
${\mathit{\Gamma}}$-equivariant smooth vector bundle $\pi_{C}:E\to U$, and a
${\mathit{\Gamma}}$-invariant continuous map $\hat{\psi}:E\to{\mathcal{E}}$
such that the following diagram commutes:

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\psi}}$$\scriptstyle{\pi_{C}}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{X.}$

In notation we will use a quadruple
$\hat{C}=(U,{\mathit{\Gamma}},E,\hat{\psi})$ to denote the bundle chart where
the map $\psi:U\to X$ is determined by the map $\hat{\psi}:E\to{\mathcal{E}}$.
We can similarly define the notions of chart embeddings, compatibility,
atlases for bundles. Then an orbifold vector bundle structure over
$\pi:\mathcal{E}\rightarrow X$ is defined to be an equivalence class of bundle
atlases as before. We spell out the definition of sections of an orbifold
vector bundle because of their importance in this paper.

######  Definition 2.3 (Sections).

Let ${\mathcal{E}}\to X$ be an orbifold vector bundle.

  1. (1)

Let $\hat{C}_{i}=(U_{i},E_{i},{\mathit{\Gamma}}_{i},\hat{\psi}_{i})$, $i=1,2$
be two bundle charts. We say that a ${\mathit{\Gamma}}_{1}$-equivariant
section $s_{1}:U_{1}\to E_{1}$ and a ${\mathit{\Gamma}}_{2}$-equivariant
section $s_{2}:U_{2}\to E_{2}$ are compatible if for any bundle chart
$\hat{C}_{0}=(U_{0},E_{0},{\mathit{\Gamma}}_{0},\hat{\psi}_{0})$ of $E$ and
chart embeddings $\hat{\iota}_{i}:\hat{C}_{0}\hookrightarrow\hat{C}_{i}$,
$i=1,2$ there holds

$\hat{\iota}_{1}^{-1}\circ s_{1}\circ\iota_{1}=\hat{\iota}_{2}^{-1}\circ
s_{2}\circ\iota_{2}$

as sections of $E_{0}\to U_{0}$.

  2. (2)

Suppose
$\hat{\mathcal{A}}=\\{\hat{C}_{i}=(U_{i},E_{i},{\mathit{\Gamma}}_{i},\hat{\psi}_{i})\
|\ i\in I\\}$ is a bundle atlas on ${\mathcal{E}}$. A section of
$({\mathcal{E}},\hat{\mathcal{A}})$ is a collection $s=(s_{i})_{i\in I}$ where
$s_{i}:U_{i}\to E_{i}$ is a ${\mathit{\Gamma}}_{i}$-equivariant smooth section
such that each pair in this collection are compatible.

  3. (3)

We say two atlases together with sections $(\hat{\mathcal{A}}_{1},s_{1})$ and
$(\hat{\mathcal{A}}_{2},s_{2})$ are equivalent if $\hat{\mathcal{A}}_{1}$ and
$\hat{\mathcal{A}}_{2}$ are equivalent as orbifold atlases and their local
representations are all pairwise compatible.

  4. (4)

A section of ${\mathcal{E}}$ is an equivalence classes of pairs
$(\hat{\mathcal{A}},s)$.

######  Lemma 2.4.

For each section of ${\mathcal{E}}\to X$ and each bundle atlas
$\hat{\mathcal{A}}$, there exists a representative $(\hat{\mathcal{A}},s)$ of
the given section.

###### Proof.

Straightforward. ∎

###  2.2. The isotropy prestratification

One can see from the definition of effective orbifolds that the isomorphism
class of the stabilizer of a point $x$ in an orbifold chart
$C=(U,{\mathit{\Gamma}},\psi)$ only depends on the point $\psi(x)\in X$. One
can use this isomorphism class to decompose the orbifold. Moreover, we can
actually obtain a more refined decomposition by using the information in the
normal direction.

######  Definition 2.5.

Consider triples $(G,V,W)$ where $G$ is a finite group, $V$ and $W$ are
finite-dimensional (real) representations of $G$ such that the decompositions
of $V$ and $W$ into irreducible representations contain no trivial summands.
Two triples $(G,V,W)$ and $(G^{\prime},V^{\prime},W^{\prime})$ are called
_isomorphic_ if there is a group isomorphism $\varphi_{G}:G\to G^{\prime}$ and
equivariant linear isomorphisms $\varphi_{V}:V\to V^{\prime}$,
$\varphi_{W}:W\to W^{\prime}$. An isomorphism class of triples is called an
_isotropy type_ , denoted by $\gamma$.

Now consider an effective orbifold $X$ with an orbifold vector bundle
${\mathcal{E}}\to X$. For each $p\in X$, consider a bundle chart
$\hat{C}=(U,G,E,\hat{\psi})$ centered at $p$. Let $U^{G}\subset U$ be the
$G$-fixed point locus. Then over $U^{G}$ one has the $G$-equivariant
decomposition

$TU|_{U^{G}}\cong TU^{G}\oplus NU^{G}$

where the first resp. the second summand is the direct sum of trivial resp.
nontrivial summands of the splitting of fibers into irreducible
$G$-representations. One also has the decomposition

$E|_{U^{G}}\cong\mathring{E}^{G}\oplus\check{E}^{G}$

where $\mathring{E}^{G}$ is a trivial $G$-representation and $\check{E}^{G}$
is the direct sum of nontrivial irreducible $G$-representations. Define
$\gamma_{p}$ to be the isotropy type represented by the triple $(G,V,W)$ where
$V$ is fiber of $NU^{G}\to U^{G}$ and $W$ is the fiber of
$\check{E}^{G}|_{U^{G}}$. Then $\gamma_{p}$ is well-defined, independent of
the choice of charts.

######  Lemma 2.6.

For each isotropy type $\gamma$ represented by a triple
$(G_{\gamma},V_{\gamma},W_{\gamma})$, define

$X_{\gamma}^{*}=\\{p\in X\ |\ \gamma_{p}=\gamma\\}$

Then $X_{\gamma}^{*}$ is a topological manifold of dimension ${\rm dim}X-{\rm
dim}V_{\gamma}$ equipped with a natural smooth structure.

###### Proof.

By definition, for each $p\in X_{\gamma}^{*}$, there exists a bundle chart
$(U,G,E,\hat{\psi})$ centered at $p$ such that $\gamma$ is represented by
$(G,V,W)$ where $V$ is the fibre of the normal bundle to $U^{G}\subset U$ and
$W$ is the fibre of $\check{E}^{G}$ at $0$. Then $U^{G}$ has dimension equal
to ${\rm dim}X-{\rm dim}V$. Moreover, a neighborhood of $p$ in
$X_{\gamma}^{*}$ is homeomorphic to $\psi(U^{G})$. Hence $X_{\gamma}^{*}$ is a
topological manifold and $\psi^{-1}:X_{\gamma}^{*}\cap\psi(U)\to U^{G}\subset
U$ is a local chart. It is easy to verify that the atlas on $X_{\gamma}^{*}$
obtained in this way is a smooth atlas. ∎

By Lemma 2.6, we obtain a decomposition of the orbifold $X$ as a locally
finite disjoint union of locally closed subsets

$X=\bigsqcup_{\gamma}X_{\gamma}^{*}.$ (2.1)

An argument similar to the proof of Lemma 2.6 shows that the frontier
condition is satisfied. Therefore, (2.1) defines a prestratification (see
Appendix A) called the _isotropy prestratification_.

For each isotropy type $\gamma$ represented by
$(G_{\gamma},V_{\gamma},W_{\gamma})$, define

$n_{\gamma}:={\rm dim}X-{\rm rank}{\mathcal{E}}-{\rm dim}V_{\gamma}+{\rm
dim}W_{\gamma}\in{\mathbb{Z}}.$ (2.2)

This is the expected dimension of the intersection of $X_{\gamma}^{*}$ and the
zero locus of a single-valued section $s:X\to{\mathcal{E}}$ (see Proposition
5.6).

###  2.3. Riemannian metric

We need to specify certain auxiliary data to aid our later construction. These
data include Riemannian metrics on orbifolds and connections on orbifold
vector bundles.

######  Definition 2.7.

A Riemannian metric on an effective orbifold $X$ is a collection of invariant
Riemannian metrics $g_{C}$ on all orbifold charts
$C=(U,{\mathit{\Gamma}},\psi)$ such that every chart embedding

$(U^{\prime},{\mathit{\Gamma}}^{\prime},\psi^{\prime})\hookrightarrow(U,{\mathit{\Gamma}},\psi)$

is isometric.

One can use the standard way to construct Riemannian metrics on orbifolds
using smooth partition of unity on orbifolds (see Remark 2.1).

####  2.3.1. Bundle metrics

We need some technical discussion on metrics on total spaces of vector
bundles. Suppose $(M,g^{TM})$ is a Riemannian manifold. Let $\pi:E\to M$ be a
vector bundle equipped with a metric $h^{E}$ and a metric-preserving
connection $\nabla^{E}$. On the total space $E$ we define a Riemannian metric
$g^{TE}$ as follows. Via the horizontal-vertical decomposition
$TE\cong\pi^{*}TM\oplus\pi^{*}E$ induced by the connection $\nabla^{E}$ we
define $g^{TE}=\pi^{*}g^{TM}\oplus\pi^{*}h^{E}$. We call $g^{TE}$ the _bundle
metric_ induced from $g^{TM}$, $h^{E}$, and $\nabla^{E}$.

######  Lemma 2.8.

Suppose $E$ is endowed with the bundle metric as above. Then

  1. (1)

Each fiber $E_{p}\subset E$ is totally geodesic.

  2. (2)

We identify the normal bundle of the $0$-section $(TM)^{\bot}\to M$ naturally
with $E$. Then the connection on $(TM)^{\bot}$ induced from the Levi–Civita
connection of $g^{TE}$ coincides with $\nabla^{E}$.

###### Proof.

(1) As the restriction of $g^{TE}$ to each fiber is Euclidean, geodesics in
each fiber are straight line segments. To show that a fiber is totally
geodesic, one only needs to show that any (short) straight line segments are
also geodesics in the totally space. Fix $x\in M$. Choose a local orthonormal
frame $e_{1},\ldots,e_{k}$ of $E$ defined over a neighborhood $U_{x}\subset M$
of $x$. For any $p\in E_{x}$, choose $r>0$ sufficiently small such that for
any $q\in B_{r}(p,E_{x})$, the shortest geodesic connecting $p$ and $q$ are
contained in $\pi^{-1}(U_{x})$. We also assume that $U_{x}$ supports a local
coordinate chart $\varphi=(x^{1},\ldots,x^{m}):U_{x}\to{\mathbb{R}}^{m}$. One
only needs to show that the straight line segment between $p$ and any $q\in
B_{r}(p,E_{x})$ is the shortest geodesic. Suppose $\gamma:[0,d]\to E$ is the
shortest geodesic parametrized by arc length. Let $y^{1},\ldots,y^{k}$ be the
bundle coordinates associated to the frame $e_{1},\ldots,e_{k}$ and write

$\gamma(t)=(x^{1}(t),\ldots,x^{m}(t),y^{1}(t),\ldots,y^{k}(t)).$

We can decompose $\dot{\gamma}(t)$ orthogonally as

$\dot{\gamma}(t)=\dot{\gamma}^{H}(t)+\dot{\gamma}^{V}(t)$

where $\dot{\gamma}^{H}(t)$ resp. $\dot{\gamma}^{V}(t)$ is the horizontal
resp. vertical part. Then one has

$1=|\dot{\gamma}(t)|=\sqrt{|\dot{\gamma}^{H}(t)|^{2}+|\dot{\gamma}^{V}(t)|^{2}}\geq|\dot{\gamma}^{V}(t)|=\sqrt{\dot{y}^{1}(t)^{2}+\cdots+\dot{y}^{k}(t)^{2}}.$

Hence

$d\geq\int_{0}^{d}\sqrt{\dot{y}^{1}(t)^{2}+\cdots+\dot{y}^{k}(t)^{2}}dt\geq
d_{E_{x}}(p,q).$

However, as $\gamma$ is the shortest geodesic, the above must be an equality.
Hence $\gamma(t)$ is contained in the fiber $E_{x}$ and $\gamma(t)$ is the
straight line segment. Hence the fiber is totally geodesic.

(2) Choose local coordinates $(x^{1},\ldots,x^{m},y^{1},\ldots,y^{k})$ on the
total space as above where the bundle coordinates $y^{\alpha}$ correspond to
the choice of local orthogonal frame $e_{1},\ldots,e_{k}$ of $E$. Suppose

$\nabla_{\partial_{i}}^{E}e_{\alpha}=\omega_{i,\alpha}^{\beta}e_{\beta}.$

To show that the induced connection on the normal bundle $(TM)^{\bot}\cong E$
coincides with $\nabla^{E}$, one only needs to verify that

$\langle\nabla_{i}^{TE}\frac{\partial}{\partial
y^{\alpha}},\frac{\partial}{\partial
y^{\beta}}\rangle=\omega_{i,\alpha}^{\beta}.$

Indeed, by the formula for the Levi–Civita connection, one has

$\begin{split}\langle\nabla_{i}^{TE}\frac{\partial}{\partial
y^{\alpha}},\frac{\partial}{\partial y^{\beta}}\rangle=&\
\frac{1}{2}\left(\frac{\partial}{\partial
x^{i}}\langle\frac{\partial}{\partial y^{\alpha}},\frac{\partial}{\partial
y^{\beta}}\rangle+\frac{\partial}{\partial
y^{\alpha}}\langle\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
y^{\beta}}\rangle-\frac{\partial}{\partial
y^{\beta}}\langle\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
y^{\alpha}}\rangle\right)\\\ =&\ \frac{1}{2}\left(\frac{\partial}{\partial
y^{\alpha}}\langle\sum_{\gamma,\delta}\omega_{i,\gamma}^{\delta}y^{\gamma}\frac{\partial}{\partial
y^{\delta}},\frac{\partial}{\partial
y^{\beta}}\rangle-\frac{\partial}{\partial
y^{\beta}}\langle\sum_{\gamma,\delta}\omega_{i,\gamma}^{\delta}y^{\gamma}\frac{\partial}{\partial
y^{\delta}},\frac{\partial}{\partial y^{\alpha}}\rangle\right)\\\ =&\
\frac{1}{2}\left(\omega_{i,\alpha}^{\beta}-\omega_{i,\beta}^{\alpha}\right)\\\
=&\ \omega_{i,\alpha}^{\beta}.\end{split}$

Therefore $\nabla^{(TM)^{\bot}}$ coincides with $\nabla^{E}$. ∎

We would like to identify tubular neighborhoods of fixed point locus with disk
bundles of the normal bundle and equip the tubular neighborhoods with bundle
metrics. To globalize such bundle metric construction we need to verify
certain properties of bundle metrics related to orthogonal decompositions of
vector bundles. Let $(M,g^{TM})$ be a Riemannian manifold and
$\pi_{\mathring{E}}:\mathring{E}\to M$, $\pi_{\check{E}}:\check{E}\to M$ be
two vector bundles equipped with metrics $h^{\mathring{E}}$, $h^{\check{E}}$
and metric preserving connections $\nabla^{\mathring{E}}$,
$\nabla^{\check{E}}$. Denote $E=\mathring{E}\oplus\check{E}$ equipped with the
product metric $h^{E}=h^{\mathring{E}}\oplus h^{\check{E}}$ and the product
connection $\nabla^{E}=\nabla^{\mathring{E}}\oplus\nabla^{\check{E}}$. Then
over the total space of $E$ there is the induced bundle metric $g^{TE}$. On
the other hand, the total space of $E$ can also be viewed as the total space
of the bundle

$\pi_{\mathring{E}}^{*}\check{E}\to\mathring{E}.$

There is hence another bundle metric $g^{T\pi_{\mathring{E}}^{*}\check{E}}$
induced from the base metric $g^{T\mathring{E}}$ on $\mathring{E}$, the
fiberwise bundle metric $\pi_{\mathring{E}}^{*}h^{\check{E}}$, and the
connection $\pi_{\mathring{E}}^{*}\nabla^{\check{E}}$. We would like to show
that $g^{TE}$ and $g^{T\pi_{\mathring{E}}^{*}\check{E}}$ coincide.

######  Corollary 2.9.

The following items are true.

  1. (1)

Over $\mathring{E}\subset E$, the direct sum decomposition

$TE|_{\mathring{E}}\cong T\mathring{E}\oplus\pi_{\mathring{E}}^{*}\check{E}$

is orthogonal with respect to the bundle metric $g^{TE}$. Moreover,

$g^{TE}|_{\pi_{\mathring{E}}^{*}\check{E}}=\pi_{\mathring{E}}^{*}h^{\check{E}}.$

  2. (2)

The connection on $\pi_{\mathring{E}}^{*}\check{E}$ (viewed as a subbundle of
$TE|_{\mathring{E}}$) induced from the Levi–Civita connection of $g^{TE}$
coincides with the pullback $\pi_{\mathring{E}}^{*}\nabla^{\check{E}}$.

  3. (3)

For $(x,\mathring{v})\in\mathring{E}|_{x}$ and $\check{v}\in\check{E}|_{x}$,
viewing the latter as a normal vector in
$(T_{(x,\mathring{v})}\mathring{E})^{\bot}$, one has

$\exp_{(x,\mathring{v})}(\check{v})=(x,\mathring{v}+\check{v}).$

  4. (4)

The bundle metric $g^{TE}$ on $E$ (viewed as the total space of $E\to M$)
coincides with the bundle metric $g^{T\pi_{\mathring{E}}^{*}\check{E}}$ on $E$
(viewed as the total space of $\pi_{\mathring{E}}^{*}\check{E}$).

###### Proof.

(1) is straightforward, following from the definition of the bundle metric.
(2) is a consequence of the second item of Lemma 2.8. (3) follows from the
first item of Lemma 2.8. (4) is a straightforward check using (1)—(3). ∎

####  2.3.2. Straightened metrics

Now we consider Riemannian metrics on an orbifold chart. Let $U$ be a smooth
manifold acted on by a finite group ${\mathit{\Gamma}}$. Let $g$ be a
${\mathit{\Gamma}}$-invariant Riemannian metric on $U$. Then for each subgroup
$G\subseteq{\mathit{\Gamma}}$, the metric induces the following objects.

  1. (1)

The orthogonal decomposition splitting $TU|_{U^{G}}\cong TU^{G}\oplus NU^{G}$
which agrees with the decomposition into the direct sum of trivial
$G$-representations and nontrivial $G$-representations.

  2. (2)

A metric on the normal bundle $NU^{G}\to U^{G}$. Let
$N_{\epsilon}(U^{G})\subset NU^{G}$ be the $\epsilon$-disk bundle.

  3. (3)

The restriction of the Levi–Civita connection on $NU^{G}$. This connection
then induces a splitting of $TNU^{G}$ into horizontal and vertical
distributions and hence a Riemannian _bundle_ metric on the total space of
$NU^{G}$.

  4. (4)

The exponential map $N_{\epsilon}(U^{G})\to U$ which pushes forward the metric
on the total space of $NU^{G}$ to a metric on a tubular neighborhood of
$U^{G}$.

######  Definition 2.10.

Let $G\subseteq{\mathit{\Gamma}}$ be a subgroup. A
${\mathit{\Gamma}}$-invariant Riemannian metric $\tilde{g}$ on $U$ is called
_straightened along_ $NU^{G}$ if within a $G$-invariant neighborhood of
$U^{G}\subset U$ the push-forward metric $\tilde{g}^{G}$ coincides with
$\tilde{g}$. A ${\mathit{\Gamma}}$-invariant metric $\tilde{g}$ is called
_straightened_ if it is straightened along $NU^{G}$ for all subgroups
$G\subseteq{\mathit{\Gamma}}$.

######  Lemma 2.11.

Suppose $\tilde{g}$ is straightened along $NU^{G}$ for some subgroup
$G\subseteq{\mathit{\Gamma}}$. Then there is a $G$-invariant neighborhood
$V\subset U$ of $U^{G}$ such that for every subgroup $H\subseteq G$, the
restriction of $\tilde{g}$ to $V$ is straightened along $NV^{H}$.

###### Proof.

It is a direct consequence of Corollary 2.9. ∎

######  Definition 2.12.

A Riemannian metric on an effective orbifold $X$ is called _straightened_
resp. _straightened near a subset $Y\subset X$_ if its pullback to each chart
is straightened resp. if its restriction to an open neighborhood of $Y$ is
straightened.

It is easy to see that for a Riemannian metric on an orbifold, being
straightened is an intrinsic property independent of the choice of an orbifold
chart, therefore the above definition is well-defined. Before we show that
straightened metrics exist, we consider the following good property of them.

######  Definition 2.13.

Suppose $W$ is a finite-dimensional real representation of a finite group $G$.
Then $W$ can be decomposed as the direct sum of irreducible representations.
We call the decomposition

$W=\mathring{W}^{G}\oplus\check{W}^{G}$

where $\mathring{W}^{G}$ is the direct sum of all trivial summands and
$\check{W}^{G}$ is the direct sum of all nontrivial summands the _basic
decomposition_ of $W$ with respect to $G$.

######  Lemma 2.14.

Suppose $X$ is equipped with a straightened metric. Then for each orbifold
chart $C=(U,{\mathit{\Gamma}},\psi)$ with the pullback metric $g_{C}$, for
each pair of subgroups $H<G<{\mathit{\Gamma}}$, for each $x\in U^{G}$ and
$v\in N_{\epsilon}U^{G}\cap N_{x}U^{G}$ for $\epsilon$ sufficiently small, the
following holds. Using the (orthogonal) basic decomposition with respect to
$H$

$NU^{G}\cong(TU^{H}\cap NU^{G})\oplus NU^{H}|_{U^{G}}$

we write

$v=(\mathring{v}_{H},\check{v}_{H}).$

Moreover, we use the exponential map $\exp:N_{\epsilon}U^{G}\to U$ to identify
$\check{v}_{H}\in T_{x}U$ with a vector of
$NU^{H}|_{\exp_{x}\mathring{v}_{H}}$ and still denote it by $\check{v}_{H}$.
Then one has

$\exp_{x}v=\exp_{\exp_{x}\mathring{v}_{H}}(\check{v}_{H}).$

###### Proof.

As the metric is straightened, locally $U$ is isometric to the total space
$NU^{G}$. Then the property follows from Corollary 2.9. ∎

Now we prove the existence of straightened metrics by a prototypical induction
argument.

######  Lemma 2.15.

Let $X$ be an effective orbifold and $Y\subset X$ be a compact subset. Then
there exists a Riemannian metric $g$ on $X$ which is straightened near $Y$.

###### Proof.

We define a filtration

$X_{0}\subset X_{1}\subset\cdots\subset X_{n}=X$

where $n={\rm dim}X$ as follows: $p\in X_{d}$ if for any orbifold chart
$(U,{\mathit{\Gamma}},\psi)$ and any $x\in U$ with $\psi(x)=p$, the dimension
of $U^{{\mathit{\Gamma}}_{x}}$ through $x$ is at most $d$, where
${\mathit{\Gamma}}_{x}\subset{\mathit{\Gamma}}$ is the isotropy group of $x$.
Then $X_{d}$ is closed. Let $g$ be an arbitrary Riemannian metric on $X$. We
modify $g$ inductively such that the modification $g_{d}$ is straightened near
$X_{d}\cap Y$. For $d=0$, $X_{0}$ is discrete. Then for each $p\in X_{0}\cap
Y$, define a Riemannian metric $g_{p}$ as follows. Choose a linear orbifold
chart $C=(U,{\mathit{\Gamma}},\psi)$ centered at $p$. Let $\tilde{g}$ be the
metric on $U$ obtained by pulling back $g$ to the chart $C$. Define a metric
$\tilde{g}_{p}$ on $U$ whose restriction to a neighborhood of $0\in U$ is
equal to the pushforward of the Euclidean metric on
$T_{0}U\cong{\mathbb{R}}^{n}$ (induced from $\tilde{g}$) via the exponential
map on $U$ centered at $0$. Then $\tilde{g}_{p}$ induces a Riemannian metric
in a neighborhood of $p\in X$. As $\tilde{g}_{p}$ is Euclidean near $0$, it is
straightened near $p$. Using cut-off functions one can obtain a metric $g_{0}$
which is straightened near $X_{0}\cap Y$.

Suppose we have obtained a metric $g_{d-1}$ straightened over an open
neighborhood $V_{d-1}\supset X_{d-1}\cap Y$ for some $d\geq 1$. Then by the
compactness of $Y$, one can find finitely many linear orbifold charts
$C_{i}=(U_{i},{\mathit{\Gamma}}_{i},\psi_{i})$ for $i=1,2,\ldots,N$ with
$U_{i}$ being a radius $2r_{i}$-ball of ${\mathbb{R}}^{n}$ such that

  1. (1)

$x_{i}=\psi_{i}(0)\in(X_{d}\setminus X_{d-1})\cap Y$.

  2. (2)

The union of $F_{i}^{\prime}:=\psi_{i}(U_{i}^{\prime})$ with $U_{i}^{\prime}$
being the radius $r_{i}$ ball in $U_{i}$ covers $(Y\cap X_{d})\setminus
V_{d-1}$.

We claim that one can find a metric $g_{d,k}$ on $X$ which is straightened
near $Y\cap X_{d}\cap(V_{d-1}\cup F_{1}^{\prime}\cup\cdots F_{k}^{\prime})$.
Indeed, suppose we have find $g_{d,k-1}$ for some $k\geq 1$. For the chart
$C_{k}$, let $\tilde{g}_{d,k-1}$ be the corresponding
${\mathit{\Gamma}}_{k}$-invariant metric on $U_{k}$. Then we can use the
exponential map centered at the origin of $U_{k}$ to pushforward the induced
bundle metric on the $\epsilon$-disk bundle
$N_{\epsilon}U_{k}^{{\mathit{\Gamma}}_{k}}$ (associated to
$\tilde{g}_{d,k-1}$) to a neighborhood of $U_{k}^{{\mathit{\Gamma}}_{k}}$, and
use a ${\mathit{\Gamma}}_{k}$-invariant cut-off function to extend it to
$U_{k}$. This provides a ${\mathit{\Gamma}}_{k}$-invariant metric
$\tilde{g}_{C_{k}}$ on the manifold $U_{k}$. Then $\tilde{g}_{C_{k}}$ is
straightened near $U_{k}^{{\mathit{\Gamma}}_{k}}$. Let $g_{C_{k}}$ be the
corresponding orbifold metric on the open set $F_{k}=\psi_{k}(U_{k})\subset
X$. Now choose a cut-off function $\rho_{k}:X\to[0,1]$ supported in $F_{k}$
such that $\rho_{k}\equiv 1$ near $\overline{F_{k}^{\prime}}$ and $\rho_{k}$
vanishes on $X\setminus F_{k}$. Then define

$g_{d,k}=(1-\rho_{k})g_{d,k-1}+\rho_{k}g_{C_{k}}$

which is a new Riemannian metric on $X$. We only need to show that this metric
is straightened near any point $p\in Y\cap X_{d}\cap(V_{d-1}\cup
F_{1}^{\prime}\cup\cdots\cup F_{k}^{\prime})$. Indeed, if $p\in
F_{k}^{\prime}$, then locally $g_{d,k}=g_{C_{k}}$ which is straightened near
$p$. If $p\notin F_{k}$, then locally $g_{d,k}=g_{d,k-1}$ which is also
straightened near $p$. If $p\in F_{k}\setminus F_{k}^{\prime}$, then $p\in
Y\cap X_{d}\cap(V_{d-1}\cup F_{1}^{\prime}\cup\cdots\cup F_{k-1}^{\prime})$.
Then near $p$ the metric $g_{d,k-1}$ is already straightened. Let $x\in U_{k}$
be a point such that $\psi_{k}(x)=p$. Then $x\in
U_{k}^{{\mathit{\Gamma}}_{k}}$. Then near $x$ the metrics $\tilde{g}_{d,k-1}$
and $\tilde{g}_{C_{k}}$ are identical. Hence $g_{d,k}=g_{d,k-1}$ near $x$ so
$g_{d,k}$ is straightened near $x$. Then inductively, one can obtain a metric
$g_{d}$ which is straightened near $Y\cap X_{d}$. The induction on $d$
provides a metric with desired property. ∎

From the construction used in the proof one can see the following: for a
closed subset $Q\subset X$, if we start with a Riemannian metric $g$ on $X$
which is already straightened near $Y\cap Q$, then one can obtain a metric
$g^{\prime}$ which is straightened near $Y$ such that it coincides with $g$ in
a neighborhood of $Y\cap Q$. Therefore one can obtain a “relative version” of
Lemma 2.15. In particular, one can connect any two straightened metric via a
one-parameter family.

######  Lemma 2.16.

For any pair of Riemannian metrics $g_{0}$ and $g_{1}$ on $X$ which are
straightened near $Y$, there exists a straightened Riemannian metric on
$X\times[0,1]$ which coincides with $g_{i}+dt^{\otimes 2}$ near
$X\times\\{i\\}$ for $i=0,1$.

###### Proof.

Define $\tilde{X}=X\times[0,1]$ and $\tilde{Y}=Y\times[0,1]$. Let $t$ be the
coordinate on $[0,1]$. Consider an arbitrary Riemannian metric
$\tilde{g}^{\prime}$ on $\tilde{X}$ which coincides with $g_{0}+dt^{\otimes
2}$ on $X\times[0,\frac{1}{3}]$ and coincides with $g_{1}+dt^{\otimes 2}$ on
$X\times[\frac{2}{3},1]$. Then $g$ is straightened near
$Y\times(X\times([0,\frac{1}{4}]\cup[\frac{3}{4},1]))$ Then the inductive
construction of Lemma 2.15 produces a straightened metric $\tilde{g}$ which
coincides with $\tilde{g}$ near
$Y\times(X\times([0,\frac{1}{4}]\cup[\frac{3}{4},1]))$. ∎

###  2.4. Connections

We would also like the orbifold vector bundle to be “straightened” in the
direction normal to fixed point loci in a way analogous to Riemannian metrics.
We first look at connections on a chart. Let $U$ be a smooth manifold acted on
effectively by ${\mathit{\Gamma}}$, and $E\to U$ is a
${\mathit{\Gamma}}$-equivariant vector bundle. Let $\nabla^{E}$ be a
${\mathit{\Gamma}}$-invariant connection on $E$. Suppose also $U$ is equipped
with a straightened metric. Then the connection $\nabla^{E}$ together with the
Riemannian metric induce the following objects. For each subgroup
$H<{\mathit{\Gamma}}$, there is a neighborhood of $U^{H}$ identified with a
disk bundle $N_{r}U^{H}\subset NU^{H}\to U^{H}$. Let $\pi_{H}:NU^{H}\to U^{H}$
be the projection. Then the parallel transport of $E$ along normal geodesics
induces an $H$-equivariant bundle isomorphism

$\rho^{H}:\pi_{H}^{*}(E|_{U^{H}})\cong E|_{N_{r}U^{H}}.$

Therefore the $H$-equivariant splitting
$E|_{U^{H}}\cong\mathring{E}^{H}\oplus\check{E}^{H}$ extends to an
$H$-equivariant splitting

$E|_{N_{r}U^{H}}\cong\mathring{E}^{H}\oplus\check{E}^{H}.$ (2.3)

######  Definition 2.17.

(Straightened connections) Let $U$ be a smooth manifold acted on effectively
by a finite group ${\mathit{\Gamma}}$. Let $E\to U$ be a
${\mathit{\Gamma}}$-equivariant vector bundle. Suppose $U$ is equipped with a
${\mathit{\Gamma}}$-invariant straightened Riemannian metric. For a subgroup
$H<{\mathit{\Gamma}}$, a ${\mathit{\Gamma}}$-equivariant connection
$\nabla^{E}$ on $E$ is said to be _straigtened along_ $NU^{H}$ if in a
neighborhood $N_{r}U^{H}$ of $U^{H}$ there holds

$(\rho^{H})^{*}(\pi^{H})^{*}\nabla^{E|_{U^{H}}}=\nabla^{E}.$

The following statement guarantees that splittings of the form (2.3) induced
by straightened connections behave well with respect to group injections
$H\hookrightarrow G$.

######  Lemma 2.18.

Suppose $\nabla^{E}$ is straightened along $NU^{G}$. Then there exists a
$G$-invariant neighborhood $V\subset U$ of $U^{G}$ such that for each subgroup
$H\subseteq G$, the restriction of $\nabla^{E}$ to $V$ is straightened along
$NV^{H}$.

###### Proof.

Basically, this lemma follows from the functoriality of pullback connections.
Indeed, as a pullback connection has no curvature in each fiber, one has

$(\pi_{H}^{*}(\rho_{G}|_{V^{H}}))\circ\rho_{H}=\rho_{G}:E\to\pi_{G}^{*}E|_{U^{G}}$

near $U^{G}$. Therefore,

$\nabla^{E}=\rho_{G}^{*}(\nabla^{E|_{U^{G}}})=\rho_{H}^{*}(\pi_{H}^{*}(\rho_{G}|_{V^{H}}))^{*}(\nabla^{E|_{U^{G}}})\\\
=\rho_{H}^{*}\big{(}\pi_{H}^{*}(\rho_{G}^{*}(\nabla^{E|_{U^{G}}}))|_{V^{H}})\big{)}=\rho_{H}^{*}(\pi_{H}^{*}\nabla^{E|_{V^{H}}}).$

Hence $\nabla^{E}$ is also straigtened along $NV^{H}$. ∎

We define the notion of straightenedness for connections on effective
orbifolds.

######  Definition 2.19.

Let $X$ be an effective orbifold equipped with a straightened Riemannian
metric. Let ${\mathcal{E}}\to X$ be an orbifold vector bundle. Then a
connection $\nabla^{\mathcal{E}}$ on ${\mathcal{E}}$ is called _straightened_
if for every bundle chart $\hat{C}=(U,E,{\mathit{\Gamma}},\hat{\psi})$ and
every subgroup $G<{\mathit{\Gamma}}$, the ${\mathit{\Gamma}}$-equivariant
connection $\nabla^{E}$ on $E\to U$ obtained by pulling back
$\nabla^{\mathcal{E}}$ is straigtened along $NU^{G}$.

######  Lemma 2.20.

Let $X$ be an effective orbifold with or without boundary equipped with a
straightened Riemannian metric. Let ${\mathcal{E}}\to X$ be a vector bundle.
Let $Y\subset X$ be a compact subset and $Q\subset X$ be a closed set. Suppose
$\nabla^{\mathcal{E}}$ is a connection on ${\mathcal{E}}$ which is straigtened
near $Y\cap Q$. Then there exists a connection on ${\mathcal{E}}$ which is
straightened near $Y$ and which coincides with $\nabla^{\mathcal{E}}$ near
$Y\cap Q$.

###### Proof.

The proof is similar to that of Lemma 2.15. ∎

######  Definition 2.21.

Let $X$ be an effective orbifold and ${\mathcal{E}}\to X$ be an orbifold
vector bundle. A _straightened structure_ on the pair $(X,{\mathcal{E}})$
consists of a straightened Riemannian metric on $X$ and a straightened
connection on ${\mathcal{E}}$ with respect to the straightened Riemannian
metric. If $(X,{\mathcal{E}})$ is equipped with a straightened structures,
then we say that $(X,{\mathcal{E}})$ is straightened.

###  2.5. Normal complex structures

Now we introduce the most important geometric condition which plays the
central role in our construction. In applications, normal complex structures
appear naturally as the Cauchy–Riemann operator has a complex linear principal
symbol.

######  Definition 2.22.

Let $X$ be an effective orbifold. A _normal complex structure_ on $X$ is a
collection of $H$-equivariant complex structures

$I_{H}:NU^{H}\to NU^{H}$

on the normal bundles $NU^{H}\to U^{H}$ for each orbifold chart
$(U,{\mathit{\Gamma}},\psi)$ and each subgroup $H\subset{\mathit{\Gamma}}$.
Moreover, the collection must satisfy the following compatibility conditions.

  1. (1)

For each chart $(U,{\mathit{\Gamma}},\psi)$, each $y\in U$ with stabilizer
${\mathit{\Gamma}}_{y}<{\mathit{\Gamma}}$, and each pair of subgroups
$H<G\subset{\mathit{\Gamma}}_{y}$, notice that we have the $H$-equivariant
decomposition

$N_{y}U^{G}\cong(T_{y}U^{H}\cap N_{y}U^{G})\oplus N_{y}U^{H}$

into trivial and nontrivial summands of $H$-representations. This
decomposition is also $I_{G}$-invariant. Then we require that the restriction
of $I_{G}$ to the second summand $N_{y}U^{H}$ of the above decomposition
agrees with $I_{H}$ at $y$.

  2. (2)

For each chart embedding from $(V,{\mathit{\Pi}},\phi)$ to
$(U,{\mathit{\Gamma}},\psi)$ given by an injection
${\mathit{\Pi}}\hookrightarrow{\mathit{\Gamma}}$ and an equivariant open
embedding $\iota:V\hookrightarrow U$, for any $z\in V$ sent to $y=\iota(z)$,
with ${\mathit{\Gamma}}_{y}\cong{\mathit{\Pi}}_{z}$, one has an equivariant
isomorphism $T_{z}V\cong T_{y}U$ sending $N_{z}V^{{\mathit{\Pi}}_{z}}$
isomorphically to $N_{y}U^{{\mathit{\Gamma}}_{y}}$, we require that this
isomorphism is complex linear.

Similarly we can define normal complex structures on bundles. Let $X$ be an
effective orbifold and ${\mathcal{E}}\to X$ be a vector bundle. For each
bundle chart $\hat{C}=(U,E,{\mathit{\Gamma}},\psi)$ and each subgroup
$G<{\mathit{\Gamma}}$, one has the basic decomposition

$E|_{U^{G}}\cong\mathring{E}^{G}\oplus\check{E}^{G}$

with respect to $G$.

######  Definition 2.23.

A normal complex structure on ${\mathcal{E}}$ is a collection of
$G$-equivariant complex structures

$J_{G}:\check{E}^{G}\to\check{E}^{G}$

on the bundle $\check{E}^{G}\to U^{G}$ for each bundle chart
$\hat{C}=(U,E,{\mathit{\Gamma}},\hat{\psi})$ and each subgroup
$G<{\mathit{\Gamma}}$. Moreover, the collection must satisfy the following
compatibility conditions.

  1. (1)

For each bundle chart $\hat{C}=(U,E,{\mathit{\Gamma}},\hat{\psi})$, each $y\in
U$ with stabilizer ${\mathit{\Gamma}}_{y}<{\mathit{\Gamma}}$, and each pair of
subgroups $H\subseteq G<{\mathit{\Gamma}}_{y}$, notice that we have the
$H$-equivariant decomposition

$\check{E}^{G}|_{y}\cong(\check{E}_{y}^{G})^{H}\oplus\check{E}_{y}^{H}.$

Notice that the decomposition is $J_{G}$-invariant. We require that the
restriction of $J_{G}$ to the summand $\check{E}_{y}^{H}$ coincides with
$J_{H}$.

  2. (2)

For each chart embedding from $(V,F,{\mathit{\Pi}},\hat{\phi})$ to
$(U,E,{\mathit{\Gamma}},\hat{\psi})$ given by an injection
${\mathit{\Pi}}\hookrightarrow{\mathit{\Gamma}}$ and an equivariant bundle
embedding $\hat{\iota}:F\hookrightarrow E$ covering a base embedding
$\iota:V\hookrightarrow U$, for any $z\in V$ (with stabilizer $G_{z}$) sent to
$y=\iota(z)$ (with stabilizer $G_{y}$), one has an equivariant isomorphism
$F|_{z}\cong E|_{y}$. We require that the induced isomorphism between
$\check{F}^{G_{z}}|_{z}\cong\check{E}^{G_{y}}|_{y}$ is complex linear.

######  Remark 2.24.

If $(X,J)$ is an almost complex orbifold, $J$ also endows $X$ with a normal
complex structure. The notion of normal complex structure is convenient for
the discussion of cobordisms between (derived) almost complex orbifolds, and
it also plays an important role in the discussion of certain invariance
result.

###  2.6. Normally complex sections

Now we discuss the notion of normally complex sections originally introduced
by Parker [Par13] which generalizes Fukaya–Ono’s notion of normally polynomial
sections [FO97]. We first discuss the notion of lifts within a single orbifold
chart. Suppose $G$ is a finite group. Let $V$, $W$ be finite-dimensional
complex representations of $G$. Let ${\rm Poly}^{G}(V,W)$ be the space of
$G$-equivariant polynomial maps and for each $d\geq 0$, define

${\rm Poly}_{d}^{G}(V,W)\subset{\rm Poly}^{G}(V,W)$

to be the subspace of polynomial maps of degrees at most $d$. Then there is an
evaluation map

${\rm ev}:V\times{\rm Poly}^{G}(V,W)\to W.$ (2.4)

These notions can also be defined for the parametrized case. Let $M$ be a
smooth manifold, ${\mathcal{V}},{\mathcal{W}}\to M$ be smooth complex vector
bundles equipped with fiberwise complex linear $G$-actions. Then one has the
bundle of fiberwise polynomial maps

${\rm Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})\to M$

and the corresponding evalutation map

${\rm ev}:{\mathcal{V}}\oplus{\rm
Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})\to{\mathcal{W}}.$ (2.5)

Now let $X$ be an effective orbifold and ${\mathcal{E}}\to X$ be a vector
bundle. Suppose $(X,{\mathcal{E}})$ is normally complex and is straightened.
Then for each bundle chart $\hat{C}=(U,G,E,\hat{\psi})$ of ${\mathcal{E}}$ and
any subgroup $H<G$, using the straightened structures, we can identify a
neighborhood of $U^{H}$ with a disk bundle $N_{r}U^{H}$ and we can extend the
basic decomposition

$E|_{U^{H}}\cong\mathring{E}^{H}\oplus\check{E}^{H}$

to a decomposition of $E$ near $U^{H}$. From now on such identifications and
decompositions will be assumed implicitly.

######  Definition 2.25.

Let $\hat{C}=(U,G,E,\hat{\psi})$ be a bundle chart of ${\mathcal{E}}\to X$ and
let $s:U\to E$ be a smooth ${\mathit{\Gamma}}$-equivariant section. A _local
normally complex lift_ of $s$ of degree at most $d$ near $x$ is a
$G_{x}$-equivariant smooth bundle map (over $U^{G_{x}}$)

$\check{\mathfrak{s}}_{x}:N_{\epsilon}U^{G_{x}}\to{\rm
Poly}_{d}^{G_{x}}(NU^{G_{x}},\check{E}^{G_{x}})$ (2.6)

satisfying the following condition. We define the graph of
$\check{\mathfrak{s}}_{x}$ to be the bundle map

${\rm graph}(\check{\mathfrak{s}}_{x}):N_{\epsilon}U^{G_{x}}\to
NU^{G_{x}}\oplus{\rm Poly}_{d}^{G_{x}}(NU^{G_{x}},\check{E}^{G_{x}}),\
v\mapsto(v,\check{\mathfrak{s}}_{x}(v)).$

Using the basic decomposition of $E$ near $U^{G_{x}}$, we can write $s$ as

$s=(\mathring{s}_{x},\check{s}_{x}):N_{\epsilon}U^{G_{x}}\to\mathring{E}^{G_{x}}\oplus\check{E}^{G_{x}}\cong
E|_{N_{\epsilon}U^{G}},$

then near $x$ one has

$\check{s}_{x}={\rm ev}\circ{\rm graph}(\check{\mathfrak{s}}_{x}).$

By abuse of notions, we also call the bundle map

$\tilde{s}_{x}:=(\mathring{s}_{x},\check{\mathfrak{s}}_{x}):N_{\epsilon}U^{G_{x}}\to\mathring{E}^{G_{x}}\oplus{\rm
Poly}_{d}^{G_{x}}(NU^{G_{x}},\check{E}^{G_{x}})$

or the bundle map

$\tilde{S}_{x}:=(\mathring{s}_{x},{\rm
graph}(\check{\mathfrak{s}}_{x})):N_{\epsilon}U^{G_{x}}\to\mathring{E}^{G_{x}}\oplus\big{(}NU^{G_{x}}\oplus{\rm
Poly}_{d}^{G_{x}}(NU^{G_{x}},\check{E}^{G_{x}})\big{)}$

a local normally complex lift of $s$ near $x$.

If $s:U\to E$ has a local normally complex lift (of degree at most $d$) near
each point of $U$, then we say that $s$ is a _normally complex section_ or an
_FOP section_ 111Stands for Fukaya–Ono–Parker. (of degree at most $d$).

######  Remark 2.26.

In equation (2.6), we use the exponential map implicitly to identify
$N_{\epsilon}U^{G_{x}}$ with a neighborhood of the zero section in
$NU^{G_{x}}$. In other words, a local normally complex lift _depends_ on the
straightened structure and the normal complex structure. On the other hand,
one can see that being an FOP section is a condition invariant under chart
embeddings. Hence it is a condition intrinsic for the orbifold bundle, the
normal complex structure, and the straightened structures.

######  Definition 2.27.

A smooth section $s:X\to{\mathcal{E}}$ is called a _FOP section_ or a
_normally complex section_ (of degree at most $d$) if for each $p\in X$ there
is a bundle chart $\hat{C}=(U,{\mathit{\Gamma}},E,\hat{\psi})$ of
${\mathcal{E}}\to X$ around $p$ such that the pullback of $s$ to $U$ is a
normally complex section (of degree at most $d$) in the sense of Definition
2.25.

######  Remark 2.28.

There is a more restricted notion called _normally polynomial sections_
considered by Fukaya–Ono [FO97]. A section is normally polynomial if in each
chart $\hat{C}=(U,{\mathit{\Gamma}},E,\hat{\psi})$, for each group
$G<{\mathit{\Gamma}}$, near $U^{G}\subset U$ the section
$\check{s}:N_{\epsilon}U^{G}\to\check{E}^{G}$ is of the form

$\check{s}:U^{G}\to{\rm Poly}_{d}^{G}(NU^{G},\check{E}^{G}).$

In particular, there is no ambiguity of choosing lifts. In fact it is enough
to use normally complex sections to achieve the required transversality (see
the proof of Proposition 5.4). However, the more flexible notion of normally
complex sections, introduced by Parker [Par13], is very convenient to work
with. For example one can do cut and paste as shown below. More importantly,
this flexible notion allows us to prove that our notion of transversality is
well-behaved even for normally polynomial sections (see Section 3).

######  Lemma 2.29.

The space of normally complex sections of $E\to X$ is a module over
$C^{\infty}(X)$.

###### Proof.

Left to the reader. ∎

######  Remark 2.30.

In general the set of FOP sections is strictly contained in the set of smooth
sections. Consider ${\mathbb{C}}$ acted nontrivially by ${\mathbb{Z}}_{2}$.
The space of equivariant polynomial maps from ${\mathbb{C}}$ to itself is
generated by monomials $z,z^{3},z^{5},\cdots$. Hence a FOP map from
${\mathbb{C}}$ to ${\mathbb{C}}$ is of the form

$z\mapsto f_{1}(z)z+f_{3}(z)z^{3}+\cdots+f_{2k-1}z^{2k-1}$

where $f_{i}(z)$ is an even smooth function. We can also write such a map as
$z\mapsto f(z)z$ with $f$ an even function. However, not all equivariant
smooth maps are of this form, such as the map $z\mapsto\overline{z}$.

Now we show that smooth sections can be approximated by FOP sections. To
measure the distance between sections, choose a metric on ${\mathcal{E}}$. We
remind the reader that $X$ is assumed to be effective throughout this
subsection.

######  Lemma 2.31.

Let $s:X\to{\mathcal{E}}$ be a smooth section with $s^{-1}(0)$ compact. Let
$D\subset X$ be a precompact open neighborhood of $s^{-1}(0)$. Then there
exists $d\geq 0$ such that for any $\epsilon>0$, there exists a smooth section
$s^{\prime}:X\to{\mathcal{E}}$ such that

  1. (1)

$s^{\prime}$ is an FOP section near $\overline{D}$ of degree at most $d$.

  2. (2)

There holds the estimate

$\|s-s^{\prime}\|_{C^{0}(D)}<\epsilon.$ (2.7)

###### Proof.

By the compactness of $\overline{D}$, one can find a finite collection of
bundle charts
$\hat{C}_{i}^{+}=(U_{i}^{+},E_{i},{\mathit{\Gamma}}_{i},\hat{\psi}_{i})$
($i=1,\ldots,N$) centered at $x_{i}\in\overline{D}$ satisfying the following
conditions.

  1. (1)

The bundle chart $\hat{C}_{i}^{+}$ is linear and
$U_{i}^{+}\subset{\mathbb{R}}^{n}$ is a radius $2r_{i}$-ball centered at the
origin.

  2. (2)

The collection $F_{i}:=\psi_{i}(U_{i})$, $i=1,\ldots,N$ cover $\overline{D}$
where $U_{i}\subset U_{i}^{+}$ is the radius $r_{i}$-ball centered at the
origin.

Define $F_{\infty}:=X\setminus\overline{D}$. Then choose a smooth partition of
unity subordinate to the open cover $\\{F_{1},\ldots,F_{N},F_{\infty}\\}$,
denoted by $\rho_{1},\ldots,\rho_{\infty}$.

Let $s_{i}:U_{i}^{+}\to E_{i}$ be the pullback of $s$ to the linear chart
$\hat{C}_{i}^{+}$. We first show that $s_{i}$ can be approximated by normally
complex sections. Indeed we can write

$s_{i}=(\mathring{s}_{i},\check{s}_{i}):U_{i}^{+}\to\mathring{E}_{i}^{{\mathit{\Gamma}}_{i}}\oplus\check{E}_{i}^{{\mathit{\Gamma}}_{i}}.$

We would like to approximate the smooth map $\check{s}_{i}$ by a smooth map
$\check{\mathfrak{s}}_{i}:U_{i}^{+}\to{\rm
Poly}_{d}^{{\mathit{\Gamma}}_{i}}(NU_{i}^{{\mathit{\Gamma}}_{i}},\check{E}_{i}^{{\mathit{\Gamma}}_{i}})$
near $\overline{U_{i}}$. Indeed, for each $x\in\overline{U_{i}}$, suppose its
stabilizer is $G_{x}\subset{\mathit{\Gamma}}_{i}$. Then
$\check{s}_{i}(x)\in(\check{E}_{i}^{{\mathit{\Gamma}}_{i}})^{G_{x}}$. Then for
a sufficiently large $d$ (which only depends on the group
${\mathit{\Gamma}}_{i}$) there exists a degree at most $d$ equivariant
polynomial map $P_{x}\in{\rm
Poly}_{d}^{{\mathit{\Gamma}}_{i}}(NU_{i}^{{\mathit{\Gamma}}_{i}},\check{E}_{i}^{{\mathit{\Gamma}}_{i}})$
such that $P_{x}(x)=\check{s}_{i}(x)$ (see Lemma 3.10). Then one can find
$r_{x}>0$ such that

$\sup_{x^{\prime}\in
B_{r_{x}}(x)}|\check{s}_{i}(x)-\check{s}_{i}(x^{\prime})|+\sup_{x^{\prime}\in
B_{r_{x}}(x)}|P_{x}(x)-P_{x}(x^{\prime})|<\epsilon.$

As $\overline{U_{i}}$ is compact, one can find finitely many points
$x_{j}\in\overline{U_{i}}$ such that

$\overline{U_{i}}\subset\bigcup_{j=1}^{l}B_{r_{x_{j}}}(x_{j}).$

Choose a partition of unity subordinate to $B_{r_{x_{j}}}(x_{j})$ and
$U_{i}^{+}\setminus\overline{U_{i}}$. Define

$\check{\mathfrak{s}}_{i}^{\prime}:U_{i}^{+}\to{\rm
Poly}_{d}^{{\mathit{\Gamma}}_{i}}(NU_{i}^{{\mathit{\Gamma}}_{i}},\check{E}_{i}^{{\mathit{\Gamma}}_{i}}),\
\check{\mathfrak{s}}_{i}(x)=\sum_{j=1}^{l}\rho_{j}(x)P_{x_{j}}.$

We first check that this (not necessarily ${\mathit{\Gamma}}_{i}$-equivariant
map) is close to the original section $\check{s}_{i}$. Indeed, for each
$x\in\overline{U_{i}}$, then one has

$|\check{s}_{i}(x)-{\rm
ev}(x,\check{\mathfrak{s}}_{i}^{\prime}(x))|\leq\sum_{j=1}^{l}\rho_{j}(x)|\check{s}_{i}(x)-P_{x_{j}}(x)|.$

If $x\notin B_{r_{x_{j}}}(x)$, then $\rho_{j}(x)=0$ and the corresponding
summand above vanishes; if $x\in B_{r_{x_{j}}}(x)$, then

$|\check{s}_{i}(x)-P_{x_{j}}(x)|\leq|\check{s}_{i}(x)-\check{s}_{i}(x_{j})|+|P_{x_{j}}(x)-P_{x_{j}}(x_{j})|<\epsilon.$

Hence one has

$\sup_{x\in\overline{U_{i}}}|\check{s}_{i}(x)-{\rm
ev}(x,\check{\mathfrak{s}}_{i}^{\prime}(x))|<\epsilon.$

Now we make $\check{s}_{i}$ invariant by setting

$\check{\mathfrak{s}}_{i}(x)=\frac{1}{|{\mathit{\Gamma}}_{i}|}\sum_{g\in{\mathit{\Gamma}}_{i}}\check{\mathfrak{s}}_{i}^{\prime}(gx).$

Then for each $x\in\overline{U_{i}}$, one has

$|\check{s}_{i}(x)-{\rm
ev}(x,\check{\mathfrak{s}}_{i}(x))|=\frac{1}{|{\mathit{\Gamma}}_{i}|}\left|\sum_{g\in{\mathit{\Gamma}}_{i}}g^{-1}\check{s}_{i}(gx)-\sum_{g\in{\mathit{\Gamma}}_{i}}{\rm
ev}(x,\check{\mathfrak{s}}_{i}^{\prime}(gx))\right|\\\
\leq\frac{1}{|{\mathit{\Gamma}}_{i}|}\sum_{g\in{\mathit{\Gamma}}_{i}}\left|g^{-1}\check{s}_{i}(gx)-g^{-1}{\rm
ev}(gx,\check{\mathfrak{s}}_{i}^{\prime}(gx))\right|<\epsilon.$

Then $\check{\mathfrak{s}}_{i}$ together with $\mathring{s}_{i}$ defines an
FOP section of $E_{i}\to U_{i}^{+}$ and hence an FOP section $s_{i}^{\prime}$
of the orbifold bundle ${\mathcal{E}}\to X$ over $\psi_{i}(U_{i}^{+})$. Then
define

$s^{\prime}:=\sum_{i=1}^{N}\rho_{i}s_{i}^{\prime}+\rho_{\infty}s.$

By Lemma 2.29, this is a smooth section of ${\mathcal{E}}$ which is an FOP
section in a neighborhood of $\overline{D}$. The chartwise estimates imply
that (2.7) holds. ∎

##  3\. Whitney stratifications on the variety $Z$

The purpose of this section is to specify the notion of transversality for
normally complex sections via the language of Whitney stratifications. The
fundamental idea is from Fukaya–Ono [FO97]. On the other hand, it is rather
technical to prove that this transversality notion is intrinsic, i.e., behaves
well with respect to chart embeddings. It is the work of Parker [Par13] from
which we learned how to use normally complex sections to compare chartwise
transversality notions and how to prove another similar property, i.e. the
transversality notion is independent of the choice of the cut-off degree of
polynomial maps.

The notion of transversality for FOP sections is based on the intricate study
of a particular class of complex affine varieties which we generally refer to
as “the variety $Z$.” In this section, let $G$ be a finite group and $V,W$ be
two finite-dimensional complex representations of $G$. We require that the
$G$-action on $V$ is effective. We allow $V$ or $W$ to have trivial
$G$-summands, hence the triple $(G,V,W)$ does not represent an isotropy type
in general (see Definition 2.5). Define

$Z^{G}(V,W)=\\{(v,P)\in{\rm Poly}^{G}(V,W)\ |\ P(v)=0\in W\\}$

the zero locus of the evaluation map from (2.4). Its cut-off at any degree $d$
is

$Z_{d}^{G}(V,W):=Z^{G}(V,W)\cap(V\times{\rm Poly}_{d}^{G}(V,W)).$

Similarly, one can define the family of the $Z$-variety for the parametrized
case. Given a smooth manifold $M$ and ${\mathcal{V}},{\mathcal{W}}\rightarrow
M$ smooth complex vector bundles with fiberwise complex linear $G$-actions,
the zero locus of (2.5) is denoted by

${\mathcal{Z}}^{G}({\mathcal{V}},{\mathcal{W}}):=\\{(v,P)\in{\mathcal{V}}\oplus{\rm
Poly}^{G}({\mathcal{V}},{\mathcal{W}})\ |\ P(v)=0\\}$

and ${\mathcal{Z}}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ is defined similarly
by considering fiberwise polynomial maps of degree at most $d$.

###  3.1. The canonical Whitney stratification

We first recall basic notions about Whitney stratifications. More detailed
discussions can be found in Appendix A.

######  Definition 3.1.

Let $M$ be a smooth manifold and $S\subset M$ be a subset.

  1. (1)

A _prestratification_ of $S$ is a decomposition of $S$

$S=\bigsqcup S_{\lambda}$

into nonempty locally closed sets such that 1) the decomposition is locally
finite and 2) if $S_{\lambda}\cap\overline{S_{\mu}}\neq\emptyset$, then
$S_{\lambda}\subset\overline{S_{\mu}}$. Each member $S_{\lambda}$ of this
decomposition is called a _strata_ of this prestratification.

  2. (2)

A prestratification is called a _Whitney prestratification_ if all strata are
smooth submanifolds and each distinct pair $(S_{\lambda},S_{\mu})$ of strata
satisfy Whitney’s condition (b) (Definition A.3).

  3. (3)

A _stratification_ of $S$ is a rule ${\mathfrak{S}}$ which assigns to each
$x\in S$ a set-germ $S_{x}$ containing $x$ such that for each $x\in S$, there
is an open neighborhood $N$ and a prestratification of $N\cap S$ such that for
all $y\in S\cap N$, the set-germ $S_{y}$ is the germ of the strata in the
prestratification which contains $y$. In particular, any prestratification of
$S$ induces a stratification.

  4. (4)

A stratification of $S$ is called a _Whitney stratification_ if it is induced
from a Whitney prestratification.

  5. (5)

Let $N$ be a smooth manifold and $f:N\to M$ be a smooth map. $f$ is said to be
_lemma412_ to $S$ with respect to a Whitney stratification ${\mathfrak{S}}$ if
$f$ is transverse to each set-germ of ${\mathfrak{S}}$.

  6. (6)

If $M$ is a nonsingular complex algebraic variety and $S\subset M$ is a
constructible set, a Whitney prestratification of $S$ is called _complex
algebraic_ if all of its strata are nonsingular complex algebraic subsets of
$M$. A Whitney stratification is called complex algebraic if it is induced
from a complex algebraic Whitney prestratification.

Now consider the variety $Z$ associated to the triple $(G,V,W)$ and a
nonnegative integer $d$. For each $v\in V$, let $G_{v}\subset G$ be the
isotropy subgroup (or stabilizer) of $v$. The total space $V$ has a
prestratification indexed by subgroups of $G$. More precisely, for each
subgroup $H\subset G$, denote

$\displaystyle\ V_{H}:=\mathring{V}^{H}:=\\{v\in V\ |\ H\subset G_{v}\\},\ $
$\displaystyle\ V_{H}^{*}:=\\{v\in V\ |\ G_{v}=H\\}.$

The top stratum is also called the isotropy-free part of $V$, denoted by

$V^{\rm free}:=\\{v\in V\ |\ G_{v}=\\{e\\}\\}.$

Then the decomposition

$V=\bigsqcup_{H\subseteq G}V_{H}^{*}$

is a Whitney prestratification of $V$ whose strata are all regular complex
algebraic sets. We will call it the _action prestratification_ on $V$. The
induced prestratification on $V\times{\rm Poly}_{d}^{G}(V,W)$, where $G$ acts
trivially on the second factor, is also called the _action prestratification_.

######  Definition 3.2.

A smooth Whitney stratification on $Z:=Z_{d}^{G}(V,W)$ is said to _respect_
the action prestratification if for each $(v,P)\in Z_{d}^{G}(V,W)$, if $v\in
V_{H}^{*}$ for some subgroup $H\subseteq G$, then the set-germ $Z_{x}$ is
contained in $V_{H}^{*}\times{\rm Poly}_{d}^{G}(V,W)$.

This definition is a special case of Definition A.20.

######  Theorem 3.3.

For each $d\geq 0$, there exists a unique minimal Whitney stratification
${\mathfrak{Z}}$ on $Z_{d}^{G}(V,W)$ which respects the action
prestratification. Moreover, it has the following additional properties.

  1. (1)

It is complex algebraic, i.e., it is induced from a Whitney prestratification
whose strata are all nonsingular complex algebraic sets.

  2. (2)

It is $G$-invariant.

  3. (3)

If a self-diffeomorphism $f$ of $V\times{\rm Poly}_{d}^{G}(V,W)$ preserves the
action prestratification and preserves the set $Z_{d}^{G}(V,W)$, then $f$ also
preserves this Whitney stratification.

###### Proof.

The construction of such a Whitney stratification, denoted by
${\mathfrak{Z}}$, is a special case of Theorem A.21. From that theorem one
also knows that ${\mathfrak{Z}}$ is algebraic and is minimal among all smooth
ones which respect the action prestratification. The minimality implies the
uniqueness (see Definition A.6).

To prove that ${\mathfrak{Z}}$ is $G$-invariant, pick any $g\in G$. Then $g$
is a particular diffeomorphism of the vector space $V\times{\rm
Poly}_{d}^{G}(V,W)$ which satisfies assumptions of the map $f$ in Proposition
A.23. Hence $g^{*}{\mathfrak{Z}}={\mathfrak{Z}}$. Similarly, a self-
diffeomorphism $f$ of $V\times{\rm Poly}_{d}^{G}(V,W)$ which preserves the
action prestratification and preserves the set $Z_{d}^{G}(V,W)$ also satisfies
assumptions of Proposition A.23. Hence the last condition is also true. ∎

######  Definition 3.4.

We call the Whitney stratification ${\mathfrak{Z}}$ of Theorem 3.3 the
_canonical Whitney stratification_ on the variety $Z_{d}^{G}(V,W)$.

######  Remark 3.5.

To talk about transversality, one does not need to specify a Whitney
prestratification. However, when we discuss the pseudocycle property in
Section 5, it is convenient to have a distinguished set of strata. Indeed,
here is a canonically associated prestratification which induces the canonical
Whitney stratification (cf. Definition A.17). The _dimension filtration_ of
$Z_{d}^{G}(V,W)$ associated to the canonical Whitney stratification is

$Z_{d}^{G}(V,W)_{k}:=\\{x\in Z_{d}^{G}(V,W)\ |\ {\rm
dim}_{\mathbb{R}}Z_{x}\leq k\\},\ k=0,1,\ldots.$

Here $Z_{x}$ is the set-germ at $x$ given by the Whitney stratification. Then
each $Z_{d}^{G}(V,W)_{k}\setminus Z_{d}^{G}(V,W)_{k-1}$ is a nonsingular
complex algebraic set of real dimension $k$. Each of its connected components
(in the Euclidean topology) is also a nonsingular algebraic set (a connected
component is also an irreducible component in Zariski topology). Hence the
collection of connected components of $Z_{d}^{G}(V,W)_{k}\setminus
Z_{d}^{G}(V,W)_{k-1}$ for all $k$ is a complex algebraic Whitney
prestratification which induces the canonical Whitney stratification. From now
on, a _stratum_ of $Z_{d}^{G}(V,W)$ means a stratum of this canonical Whitney
prestratification.

One can also define the notion of the canonical Whitney stratification of the
parametrized case. Let $M$ be a base manifold acted on trivially by $G$ and
let ${\mathcal{V}},{\mathcal{W}}\to M$ be complex $G$-vector bundles with
fibers isomorphic to $V$ and $W$ respectively. Then the structure group of
${\mathcal{V}}\oplus{\rm Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ is
canonically reduced to $GL(V)^{G}\times GL(W)^{G}$, whose elements are
diffeomorphisms of $V\times{\rm Poly}_{d}^{G}(V,W)$ preserving both the action
prestratification and the set $Z_{d}^{G}(V,W)$. Hence by the last property of
Theorem 3.3, the canonical Whitney stratification on $Z_{d}^{G}(V,W)$ could be
patched into a Whitney stratification of
$\mathcal{Z}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ which is “locally trivial”.
We also call this stratification on
$\mathcal{Z}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ the canonical Whitney
stratification.

######  Remark 3.6.

The canonical Whitney stratification is an example of Parker’s nice Whitney
stratification ([Par13, Definition 4.6]).

Following Parker, we prove two very useful facts concerning the canonical
Whitney stratification.

######  Lemma 3.7.

(cf. [Par13, Lemma 4.10]) Let $G,M,{\mathcal{V}},{\mathcal{W}}$ be as above.
Fix an integer $d>0$. Let ${\mathcal{V}}^{+}\subset{\mathcal{V}}$ be an open
subset and let
$\underline{\smash{f}}_{1},\underline{\smash{f}}_{2}:{\mathcal{V}}^{+}\to{\rm
Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ be two smooth bundle maps such
that

${\rm
graph}(\underline{\smash{f}}_{1}-\underline{\smash{f}}_{2})\subset{\mathcal{Z}}_{d}^{G}({\mathcal{V}},{\mathcal{W}}):={\mathcal{Z}}_{d}^{G}.$

Then the graph of $\underline{\smash{f}}_{1}$ (viewed as a submanifold of
${\mathcal{V}}^{+}\times_{M}{\rm Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$)
is transverse (cf. Definition A.7) to ${\mathcal{Z}}_{d}^{G}$ if and only if
the graph of $\underline{\smash{f}}_{2}$ is transverse to
${\mathcal{Z}}_{d}^{G}$.

###### Proof.

Suppose the graph of $\underline{\smash{f}}_{1}$ is transverse to
${\mathcal{Z}}_{d}^{G}$. For any
$(v_{0},\underline{\smash{f}}_{2}(v_{0}))\in{\mathcal{Z}}_{d}^{G}$, we would
like to show that the graph of $\underline{\smash{f}}_{2}$ is transverse to
${\mathcal{Z}}_{d}^{G}$ at this point
$(v_{0},\underline{\smash{f}}_{2}(v_{0}))$. Choose a compactly supported cut-
off function

$\rho_{0}:{\mathcal{V}}^{+}\to[0,1]$

which is identically $1$ near $v_{0}$. Consider the smooth vertical vector
field on ${\mathcal{V}}^{+}\times_{M}{\rm
Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ defined by

$\xi(v,P)=(0,\rho_{0}(v)(\underline{\smash{f}}_{1}(v)-\underline{\smash{f}}_{2}(v))).$
(3.1)

Then $\xi$ can be regarded as a smooth vector field on
${\mathcal{V}}\oplus{\rm Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$.
Moreover, the flow of $\xi$ is the 1-parameter family of fibre-preserving
diffeomorphisms

$\Phi_{t}(v,P)=(v,P+t\rho_{0}(v)(\underline{\smash{f}}_{1}(v)-\underline{\smash{f}}_{2}(v)))$

of ${\mathcal{V}}\oplus{\rm Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ which
exists for all time $t$. It is also easy to see that $\Phi_{t}$ preserves the
action prestratification on ${\mathcal{V}}\oplus{\rm
Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ and the set
${\mathcal{Z}}_{d}^{G}$. Hence $\Phi_{t}$ pulls back the canonical Whitney
stratification on ${\mathcal{Z}}_{d}^{G}$ to itself. Moreover, $\Phi_{1}$ maps
a neighborhood of $(v_{0},\underline{\smash{f}}_{2}(v_{0}))$ in ${\rm
graph}(\underline{\smash{f}}_{2})$ to a neighborhood of
$(v_{0},\underline{\smash{f}}_{1}(v_{0}))$ in ${\rm
graph}(\underline{\smash{f}}_{1})$. Hence the graph of
$\underline{\smash{f}}_{2}$ is also transverse to
${\mathcal{Z}}_{d}^{G}({\mathcal{E}},{\mathcal{F}})$ at $v_{0}$. ∎

######  Lemma 3.8.

Under the same hypothesis as above. Let
$f_{1},f_{2}:{\mathcal{V}}^{+}\to{\mathcal{V}}\oplus{\rm
Poly}_{d}^{G}({\mathcal{V}},{\mathcal{W}})$ be the maps defined by

$f_{i}(v)=(v,\underline{\smash{f}}_{i}(v)).$

Suppose both the graphs of $\underline{\smash{f}}_{1}$ and
$\underline{\smash{f}}_{2}$ are transverse to ${\mathcal{Z}}_{d}^{G}$ (with
respect to the canonical Whitney stratification). Then the Whitney
stratifications on
$f_{1}^{-1}({\mathcal{Z}}_{d}^{G})=f_{2}^{-1}({\mathcal{Z}}_{d}^{G})$ pulled
back from the canonical Whitney stratifications on ${\mathcal{Z}}_{d}^{G}$ via
$f_{1}$ and $f_{2}$ coincide.

###### Proof.

We just need to check locally around a point $v_{0}\in
f_{1}^{-1}({\mathcal{Z}}_{d}^{G})=f_{2}^{-1}({\mathcal{Z}}_{d}^{G})$. Let
$\Phi_{1}$ be the time-1 map of the flow of the vector field (3.1). Then in a
neighborhood of $v_{0}$ one has

$f_{1}=\Phi_{1}\circ f_{2}.$

On the other hand, as $\Phi_{1}$ preserves the canonical Whitney
stratification ${\mathfrak{Z}}$ on ${\mathcal{Z}}_{d}^{G}$, it follows that
near $v_{0}$,
$f_{1}^{*}{\mathfrak{Z}}=f_{2}^{*}\Phi_{1}^{*}{\mathfrak{Z}}=f_{2}^{*}{\mathfrak{Z}}$.
∎

###  3.2. Regularity of the isotropy-free part

We want the isotropy-free part of the variety $Z$ to be transversely cut-out.
This is true if $d$ is sufficiently large as proved by Fukaya–Ono [FO97]. For
the convenience of the reader we rewrite their proof here.

######  Proposition 3.9.

([FO97, Lemma 5],[FOOO, Proposition 35.3]) For any finite group $G$, there
exists a positive integer $d_{0}$ satisfying the following conditions. Let
$V,W$ be finite-dimensional complex representations. Suppose $G$ acts on $V$
faithfully. Then for any $d\geq d_{0}$ and for each subgroup $H\subset G$, the
set

$Z_{d,H}^{*}:=Z_{d}^{G}(V,W)\cap(V_{H}^{*}\times{\rm Poly}_{d}^{G}(V,W))$

is a nonsingular complex algebraic set of complex dimension

${\rm dim}_{\mathbb{C}}Z_{d,H}^{*}={\rm dim}_{\mathbb{C}}{\rm
Poly}_{d}^{G}(V,W)+{\rm dim}_{\mathbb{C}}\mathring{V}^{H}-{\rm
dim}_{\mathbb{C}}\mathring{W}^{H}.$

We first prove a lemma.

######  Lemma 3.10.

[FO97, Lemma 5] There exists $d_{0}>0$ such that for all
$\mathring{v}_{H}\in\mathring{V}_{H}^{*}$ and
$\mathring{w}_{H}\in\mathring{W}_{H}$, there exists $P\in{\rm
Poly}_{d_{0}}^{G}(V,W)$ such that $P(\mathring{v}_{H})=\mathring{w}_{H}$.

###### Proof.

By decomposing $W$ into irreducible components, we may assume that $W$ is an
irreducible representation of $G$. Define the $G$-vector space

${\bm{W}}:=\bigoplus_{\gamma\in G}{\mathbb{C}}\\{\langle\gamma\rangle\\}$

with $G$-action defined as

$g\left(\sum_{\gamma}c_{\gamma}\cdot\langle\gamma\rangle\right)=\sum_{\gamma}c_{\gamma}\cdot\langle\gamma
g^{-1}\rangle=\sum_{\gamma}c_{\gamma g}\cdot\langle\gamma\rangle.$

Since ${\bm{W}}$ is a regular representation, there is a $G$-equivariant
homomorphism $\Psi:{\bm{W}}\to W$ and an element ${\bm{w}}\in{\bm{W}}$ such
that

$\Psi({\bm{w}})=\Psi\left(\sum_{\gamma}w_{\gamma}\cdot\langle\gamma\rangle\right)=\mathring{w}_{H}.$

Since $\mathring{w}_{H}\in\mathring{W}_{H}$, for all $h\in H$, one has

$\Psi(h{\bm{w}})=h\Psi({\bm{w}})=\mathring{w}_{H}.$

Hence by taking average over $H$, one may assume that

$\gamma^{\prime}H=\gamma^{\prime\prime}H\Longrightarrow
w_{\gamma^{\prime}}=w_{\gamma^{\prime\prime}}.$

Now we claim that for some $d_{0}>0$ which only depends on $G$, one can choose
a polynomial $f:V\to{\mathbb{C}}$ (not necessarily $G$-invariant) of degree at
most $d_{0}$ such that

$\forall\gamma\in G,\ f(\gamma\mathring{v}_{H})=w_{\gamma}.$

Indeed, there are $n:=|G/H|$ distinct elements in the $G$-orbit of
$\mathring{v}_{H}$. One can choose a linear decomposition $V=V_{1}\oplus
V_{2}$ such that $V_{1}$ is one-dimensional and that the projection of these
$n$ distinct elements are still distinct in $V_{1}$. Then by Lagrange’s method
of interpolation, one can find a complex polynomial $f:V_{1}\to{\mathbb{C}}$
of degree at most $|G|$ taking the prescribed values $w_{\gamma}$ at the
corresponding projection image of $\gamma\mathring{v}_{H}$ in $V_{1}$. Extend
$f$ trivially to $V$ one obtains a polynomial $f:V\to{\mathbb{C}}$ satisfying
the required conditions. Now define $P:V\to W$ by

$P(v)=\Psi\left(\sum_{\gamma\in G}f(\gamma
v)\cdot\langle\gamma\rangle\right).$

Then this is a $G$-equivariant polynomial map sending $\mathring{v}_{H}$ to
$\mathring{w}_{H}$. ∎

###### Proof of Proposition 3.9.

For each subgroup $H\subset G$, we can write any polynomial map $P\in{\rm
Poly}_{d}^{G}(V,W)$ as

$P=(\mathring{P}^{H},\check{P}^{H}):V\to\mathring{W}^{H}\oplus\check{W}^{H}.$

Then the equivariance implies that

$\check{P}^{H}|_{\mathring{V}_{H}}\equiv 0.$

Therefore

$Z_{d,H}^{*}:=\\{(v,P)\in\mathring{V}_{H}^{*}\times{\rm Poly}_{d}^{G}(V,W)\ |\
\mathring{P}^{H}(v)=0\\}.$

Then when $d\geq d_{0}$, Lemma 3.10 and the _faithfulness_ of the $G$-action
on $V$ imply that $Z_{d,H}^{*}$ is a nonsingular complex algebraic set of
dimension

${\rm dim}_{\mathbb{C}}{\rm Poly}_{d}^{G}(V,W)+{\rm
dim}_{\mathbb{C}}\mathring{V}^{H}-{\rm dim}_{\mathbb{C}}\mathring{W}^{H}.$

Lastly, by the construction of the canonical Whitney stratification of
$Z_{d}^{G}$ which respects the action prestratification (see Theorem A.21), as
$Z_{d}^{\rm free}$ is smooth of dimension $k$, it is entirely contained in
$Z_{d,k}^{*}$. ∎

######  Remark 3.11.

If $W$ is the trivial representation, then $d_{0}$ can be taken to be $0$. In
fact, all constant maps from $V$ to $W$ is $G$-equivariant and
$Z_{G}^{0}(V,W)=V\times\\{0\\}\subset V\times{\rm Poly}_{0}^{G}(V,W)\cong
V\times W$ so the

$Z_{0,H}^{*}=V_{H}^{*}\times\\{0\\}.$

###  3.3. Change of degrees

Here we prove that the canonical Whitney stratifications from different cut-
off degrees are compatible with each other. For each $d\geq 0$, consider the
inclusion map

$\phi:V\times{\rm Poly}_{d}^{G}(V,W)\to V\times{\rm Poly}_{d+1}^{G}(V,W).$

Then obviously

$\phi^{-1}(Z_{d+1}^{G}(V,W))=Z_{d}^{G}(V,W).$

Our main theorem of this subsection is the following.

######  Theorem 3.12.

$\phi$ is transverse to $Z_{d+1}^{G}(V,W)$ and pulls back the canonical
Whitney stratification on $Z_{d+1}^{G}(V,W)$ to the canonical Whitney
stratification on $Z_{d}^{G}(V,W)$.

The idea of the proof is similar to the discussion in [Par13] using Parker’s
notion of nice Whitney stratifications. First we prove an algebraic result
which was also used in [Par13] without providing a proof or reference.

######  Lemma 3.13.

${\rm Poly}^{G}(V,W)$ is a finitely generated ${\mathbb{C}}[V]^{G}$-module.

###### Proof.

The proof follows from [hk], we present it here for completeness. By Hilbert’s
basis theorem, given any finite-dimensional complex $G$-representation
$V^{\prime}$, the ring of $G$-invariant polynomials
$\mathbb{C}[V^{\prime}]^{G}$ is finitely generated. Now let
$V^{\prime}=V\oplus W^{\vee}$ where $W^{\vee}$ is the dual to $W$ endowed with
the corresponding $G$-action. The ring $\mathbb{C}[V^{\prime}]^{G}$ has a bi-
grading by keeping track of the degree of the $V$-coordinates and
$W^{\vee}$-coordinates respectively. Choose
$h_{1},\dots,h_{r},h_{r+1},\dots,h_{r+m},\dots,h_{n}$ which generate
$\mathbb{C}[V^{\prime}]^{G}$ such that $h_{1},\dots,h_{r}$ have
$W^{\vee}$-degree $0$, $h_{r+1},\dots,h_{r+m}$ have $W^{\vee}$-degree $1$, and
$h_{r+m+1},\dots,h_{n}$ have higher $W^{\vee}$-degrees.

Note that ${\rm Poly}^{G}(V,W)$ can be identified with the subset of
$\mathbb{C}[V^{\prime}]^{G}$ consisting of elements with $W$-degree $1$. Then
any element in ${\rm Poly}^{G}(V,W)$ could be written as a linear combination
of products of $h_{1},\dots,h_{r},h_{r+1},\dots,h_{r+m}$. Observe that ${\rm
Poly}^{G}(V,W)$ is naturally a module over ${\mathbb{C}}[V]^{G}$, therefore it
is generated by $h_{r+1},\dots,h_{r+m}$ as a ${\mathbb{C}}[V]^{G}$-module. ∎

Next we construct a left inverse of the map $\phi$.

######  Lemma 3.14.

(cf. [Par13, Lemma 4.11]) Given $G$, $V$, $W$. There exists an integer
$d_{0}=d_{0}(G,V,W)$ such that for all $d\geq d_{0}$, there is a $G$-invariant
map

$\underline{\smash{\phi}}^{\prime}:V\times{\rm Poly}_{d+1}^{G}(V,W)\to{\rm
Poly}_{d}^{G}(V,W)$

satisfying the following properties. Denote

$\phi^{\prime}:V\times{\rm Poly}_{d+1}^{G}(V,W)\to V\times{\rm
Poly}_{d}^{G}(V,W),\ (v,P)\mapsto(v,\underline{\smash{\phi}}^{\prime}(v,P)).$
(3.2)

Then

  1. (1)

$\phi^{\prime}\circ\phi={\rm Id}_{V\times{\rm Poly}_{d}^{G}(V,W)}$.

  2. (2)

For each $(v,P)\in V\times{\rm Poly}_{d+1}^{G}(V,W)$, ${\rm
ev}(\phi^{\prime}(v,P))={\rm ev}(v,P)$. In particular, $\phi^{\prime}$ maps
$Z_{d+1}^{G}$ surjectively onto $Z_{d}^{G}(V,W)$

  3. (3)

$\phi\circ\phi^{\prime}$ is transverse to $Z_{d+1}^{G}$ and pulls back the
canonical Whitney stratification on $Z_{d+1}^{G}$ to itself.

###### Proof.

As ${\rm Poly}^{G}(V,W)$ is finitely generated over ${\mathbb{C}}[V]^{G}$ (see
Lemma 3.13), one can find a sufficiently large $d_{0}$ such that ${\rm
Poly}_{d_{0}}^{G}(V,W)$ contains a set of generators $Q_{1},\ldots,Q_{m}$.
Then when $d\geq d_{0}$, let ${\rm Homo}_{d+1}^{G}(V,W)\subset{\rm
Poly}_{d+1}^{G}(V,W)$ be the subset of $G$-equivariant homogeneous polynomial
maps of degree $d+1$. Then one has

${\rm Poly}_{d+1}^{G}(V,W)={\rm Poly}_{d}^{G}(V,W)\oplus{\rm
Homo}_{d+1}^{G}(V,W).$

Choose a basis $P_{1},\ldots,P_{k}$ of ${\rm Homo}_{d+1}^{G}(V,W)$. Then there
exist polynomials $h_{ij}\in{\mathbb{C}}[V]^{G}$ for $1\leq i\leq k$, $1\leq
j\leq m$ such that

$P_{i}=\sum_{j=1}^{m}h_{ij}Q_{j}.$

Then for each $P\in{\rm Poly}_{d+1}^{G}(V,W)$, let $P^{\prime}\in{\rm
Homo}_{d+1}^{G}(V,W)$ be its degree $d+1$ part which can be uniquely written
as

$P^{\prime}=\sum_{i=1}^{k}a_{i}P_{i}.$

Then define

$\underline{\smash{\phi}}^{\prime}(v,P)=(P-P^{\prime}+\sum
a_{i}h_{ij}(v)Q_{j}).$ (3.3)

This map (which is not canonical) is linear in the variable $P$. Then it is
easy to see that for the associated map $\phi^{\prime}$ defined by (3.2) there
holds

$\phi^{\prime}\circ\phi={\rm Id}_{V\times{\rm Poly}_{d}^{G}(V,W)}$

and for all $(v,P)\in V\times{\rm Poly}_{d+1}^{G}(V,W)$ one has

${\rm ev}(\phi(\phi^{\prime}(v,P)))={\rm ev}(\phi^{\prime}(v,P))={\rm
ev}(v,P).$ (3.4)

Therefore $\phi^{\prime}$ is surjective and maps $Z_{d+1}^{G}$ to $Z_{d}^{G}$.

Now we prove the last property. Define ${\mathcal{V}}$ and ${\mathcal{W}}$ to
be the trivial bundles over $M_{d+1}:={\rm Poly}_{d+1}^{G}(V,W)$ with fibers
$V$ and $W$ respectively. Then we have two bundle maps
$\underline{\smash{f}}_{1},\underline{\smash{f}}_{2}:{\mathcal{V}}\cong
V\times M_{d+1}\to{\rm Poly}_{d+1}^{G}({\mathcal{V}},{\mathcal{W}})\cong{\rm
Poly}_{d+1}^{G}(V,W)\times M_{d+1}$ defined by

$\displaystyle\ \underline{\smash{f}}_{1}(v,P)=(P,P),\ $ $\displaystyle\
\underline{\smash{f}}_{2}(v,P)=(\underline{\smash{\phi}}(\phi^{\prime}(v,P)),P)$

where $\underline{\smash{\phi}}(v,P)=P$. Then (3.4) implies that

${\rm
graph}(\underline{\smash{f}}_{1}-\underline{\smash{f}}_{2})\subset{\mathcal{Z}}_{d+1}^{G}({\mathcal{V}},{\mathcal{W}}).$

As the identity map of $V\times{\rm Poly}_{d+1}^{G}(V,W)$ is transverse to
$Z_{d+1}$ (which means the graph of $\underline{\smash{f}}_{1}$ is transverse
to ${\mathcal{Z}}_{d+1}$), it follows from Lemma 3.7 that the graph of
$\underline{\smash{f}}_{2}$ is also transverse to ${\mathcal{Z}}_{d+1}$. It is
equivalent to say that the map $\phi\circ\phi^{\prime}$ is transverse to
$Z_{d+1}$. Moreover, as the identity map preserves the canonical Whitney
stratification, by Lemma 3.8, $\phi\circ\phi^{\prime}$ also pulls back the
canonical Whitney stratification on $Z_{d+1}$ to itself. ∎

###### Proof of Theorem 3.12.

Still abbreviate $Z_{d}^{G}(V,W)$ by $Z_{d}$. By Lemma 3.14, we know that
$\phi\circ\phi^{\prime}$ resp. $\phi^{\prime}\circ\phi$ is transverse to
$Z_{d+1}$ resp. $Z_{d}$, it follows that $\phi$ resp. $\phi^{\prime}$ is
transverse to $Z_{d+1}$ resp. $Z_{d}$ along images of $\phi^{\prime}$ resp.
$\phi$. As $\phi^{\prime}$ is surjective, it follows that $\phi$ is transverse
to $Z_{d+1}$. Furthermore, notice that

$\phi^{\prime}=\phi^{\prime}\circ\phi\circ\phi^{\prime}.$

Hence the image of $d\phi^{\prime}$ at any point coincides with the image of
$d\phi^{\prime}$ at some point on the image of $\phi$. As $\phi^{\prime}$ is
transverse to $Z_{d}$ along the image of $\phi$, it follows that
$\phi^{\prime}$ is transverse to $Z_{d}$ everywhere.

Now we show that $\phi$ resp. $\phi^{\prime}$ pulls back the canonical Whitney
stratification to the canonical one. We prove inductively that for each
subgroup $H\subseteq G$, $\phi$ resp. $\phi^{\prime}$ pulls back the canonical
Whitney stratification on $Z_{d+1,H}^{*}$ resp. $Z_{d,H}^{*}$ to the canonical
Whitney stratification on $Z_{d,H}^{*}$ resp. $Z_{d+1,H}^{*}$. For $H$ being
the trivial group, notice that the pair of maps $\phi$ and $\phi^{\prime}$
satisfies the assumptions of the absolute case of Lemma A.11 for $f=\phi$,
$g=\phi^{\prime}$, $M=V^{\rm free}\times{\rm Poly}_{d}^{G}(V,W)$, $N=V^{\rm
free}\times{\rm Poly}_{d+1}^{G}(V,W)$, and $Y=Y^{\circ}=Z_{d}^{\rm free}$,
$Z=Z^{\circ}=Z_{d+1}^{\rm free}$. Hence by Lemma A.11, $\phi$ resp.
$\phi^{\prime}$ pulls back the canonical Whitney stratification on
$Z_{d+1}^{\rm free}$ resp. $Z_{d}^{\rm free}$ to the canonical Whitney
stratification on $Z_{d}^{\rm free}$ resp. $Z_{d+1}^{\rm free}$.222Because of
Proposition 3.9, the base case of the induction does not need Lemma A.11.
Suppose for a subgroup $H\subset G$, one has proved the claim for all proper
subgroups $H^{\prime}\subsetneq H$. Then by the relative case of Lemma A.11,
one can show that $\phi$ pulls back the canonical Whitney stratification on
$Z_{d+1,H}^{*}$ resp. $Z_{d,H}^{*}$ to the canonical Whitney stratification on
$Z_{d,H}^{*}$ resp. $Z_{d+1,H}^{*}$. ∎

######  Remark 3.15.

If $W$ is a trivial $G$-representation, then the statement of Theorem 3.12 is
true for $d=0$. In fact for any $d>0$, the projection map

$\underline{\smash{\phi}}^{\prime}:V\times{\rm Poly}_{d}^{G}(V,W)\to{\rm
Poly}_{0}^{G}(V,W)\cong W$

constructed in the above proof coincides with the evaluation map.

By Theorem 3.12, for any sufficiently large cut-off degree $d$, there is a
canonical Whitney stratification on $Z_{d}^{G}(V,W)$ which is natural with
respect to inclusion maps. We need a slight extension of this result. Suppose
$W_{1}$ and $W_{2}$ are two complex $G$-representations, then for all
sufficiently large $d_{1}$ and $d_{2}$, consider the set

$Z_{d_{1},d_{2}}^{G}(V,W_{1},W_{2})=Z_{d_{1}}^{G}(V,W_{1})\times_{V}Z_{d_{2}}^{G}(V,W_{2})\\\
:=\\{(v,P_{1},P_{2})\in V\times{\rm Poly}_{d_{1}}^{G}(V,W_{1})\times{\rm
Poly}_{d_{2}}^{G}(V,W_{2})\ |\ P_{1}(v)=0,\ P_{2}(v)=0\\}.$

Abbreviate it by $Z_{d_{1},d_{2}}$. Then the construction of Theorem A.21
provides a minimal Whitney stratification on $Z_{d_{1},d_{2}}$ which respects
the action prestratification, which is also a nice Whitney stratification. We
also call it the canonical Whitney stratification. Consider $d\geq
d_{1},d_{2}$ and the natural inclusion map

$\phi:V\times{\rm Poly}_{d_{1}}^{G}(V,W_{1})\times{\rm
Poly}_{d_{2}}^{G}(V,W_{2})\to V\times{\rm Poly}_{d}^{G}(V,W_{1}\oplus W_{2})$

which maps $Z_{d_{1},d_{2}}$ into $Z_{d}^{G}(V,W_{1}\oplus W_{2})$.

######  Proposition 3.16.

When $d_{1},d_{2}$ are sufficiently large, $\phi$ pulls back the canonical
Whitney stratification on $Z_{d}^{G}(V,W_{1}\oplus W_{2})$ to the canonical
one on $Z_{d_{1},d_{2}}$.

###### Proof.

One can construct a similar projection map

$\underline{\smash{\phi}}^{\prime}:V\times{\rm Poly}_{d}^{G}(V,W_{1}\oplus
W_{2})\to{\rm Poly}_{d_{1}}^{G}(V,W_{1})\times{\rm
Poly}_{d_{2}}^{G}(V,W_{2}).$

The rest of the argument is identical to the proof of Theorem 3.12. ∎

######  Corollary 3.17.

Let $\mathring{W}$ be a trivial complex $G$-representation and $\check{W}$ be
another complex $G$-representation. Let $V$ be an effective complex
$G$-representation. Let $M$ be a base manifold. Let ${\mathcal{V}}$,
$\mathring{\mathcal{W}}$, $\check{\mathcal{W}}$ be the trivial bundles over
$M$ with fibers being $V$, $\mathring{W}$, $\check{W}$ respectively. Let
${\mathcal{V}}^{*}\subset{\mathcal{V}}$ be a $G$-invariant open subset. Let
$d$ be sufficiently large and let

$\underline{\smash{f}}=(\underline{\smash{\mathring{f}}},\underline{\smash{\check{f}}}):{\mathcal{V}}^{*}\to{\rm
Poly}_{d}^{G}({\mathcal{V}},\mathring{\mathcal{W}}\oplus\check{\mathcal{W}})$

be a smooth bundle map. Consider the partial evaluation of
$\underline{\smash{f}}$ which is

$\underline{\smash{f}}^{\prime}:{\mathcal{V}}^{*}\to\mathring{W}\times{\rm
Poly}_{d}^{G}({\mathcal{V}},\check{\mathcal{W}})$

defined by

$\underline{\smash{f}}^{\prime}(v)=(\underline{\smash{\mathring{f}}}(v)(v),\underline{\smash{\check{f}}}).$

Then the graph of $\underline{\smash{f}}$ is transverse to
${\mathcal{Z}}_{d}^{G}({\mathcal{V}}^{*},\mathring{\mathcal{W}}\oplus\check{\mathcal{W}})$
if and only if the graph of $\underline{\smash{f}}^{\prime}$ is transverse to
$\\{0\\}\times{\mathcal{Z}}_{d}^{G}({\mathcal{V}}^{*},\check{\mathcal{W}})$.

###  3.4. Change of groups

Let $V$, $W$ be complex $G$-spaces. In this subsection, fix a proper subgroup
$H\subsetneq G$. Then we have the basic decomposition

$V=\mathring{V}_{H}\oplus\check{V}_{H}$

where $\check{V}_{H}$ is the direct sum of nontrivial irreducible $H$-spaces.
There is a _restriction map_

$\underline{\smash{\theta}}:V\times{\rm Poly}^{G}(V,W)\to{\rm
Poly}^{H}(\check{V}_{H},W)$

defined by (denoting
$v=(\mathring{v}_{H},\check{v}_{H})\in\mathring{V}_{H}\oplus\check{V}_{H}$)

$\underline{\smash{\theta}}(v,P)=P|_{\\{\mathring{v}_{H}\\}\times\check{V}_{H}}.$

Define

$\begin{split}\theta:V\times{\rm Poly}^{G}(V,W)&\ \to V\times{\rm
Poly}^{H}(\check{V}_{H},W)\\\ (v,P)&\
\mapsto(v,\underline{\smash{\theta}}(v,P)).\end{split}$ (3.5)

If we view $\mathring{V}_{H}$ as a base manifold with the trivial $H$-action,
then over $\mathring{V}_{H}$ one has the product vector bundles
$\check{\mathcal{V}}_{H}=\mathring{V}_{H}\times\check{V}_{H}$ and
${\mathcal{W}}=\mathring{V}_{H}\times W$. Then the target of $\theta$ can be
identified with the fiber product
$\check{\mathcal{V}}_{H}\times_{\mathring{V}_{H}}{\rm
Poly}^{H}(\check{\mathcal{V}}_{H},{\mathcal{W}})$. Then for each degree $d$
one has

$\theta(Z_{d}^{G}(V,W))\subset{\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H},{\mathcal{W}})\subset\check{\mathcal{V}}_{H}\times_{\mathring{V}_{H}}{\rm
Poly}_{d}^{H}(\check{\mathcal{V}}_{H},{\mathcal{W}}).$ (3.6)

We will compare the canonical Whitney stratification on $Z_{d}^{G}(V,W)$ and
the canonical Whitney stratification on
${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H},{\mathcal{W}})$, where the
latter is the parametrized version induced from the canonical Whitney
stratification on $Z_{d}^{H}(\check{V}_{H},W)$.

Denote

$V_{H}^{+}:=\\{v\in V\ |\ G_{v}\subset H\\}$

which is the complement of the union of finitely many subspaces. Let
$\check{\mathcal{V}}_{H}^{+}\subset\check{\mathcal{V}}_{H}$ be the
corresponding open subset of $\check{\mathcal{V}}_{H}$. Denote

$Z_{d}^{G}(V_{H}^{+},W):=Z_{d}^{G}(V,W)\cap(V_{H}^{+}\times{\rm
Poly}_{d}^{G}(V,W))$

and

${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}}):={\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}},{\mathcal{W}})\cap(\check{\mathcal{V}}_{H}^{+}\times_{\mathring{V}_{H}}{\rm
Poly}_{d}^{H}(\check{\mathcal{V}}_{H},{\mathcal{W}})),$

which are Zariski open subsets of $Z_{d}^{G}(V,W)$ and
${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H},{\mathcal{W}})$ respectively.
Hence they have the restricted canonical Whitney stratifications. Our main
theorem of this subsection is the following.

######  Theorem 3.18.

$\theta$ is transverse to
${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}})$ and pulls
back the canonical Whitney stratification on
${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}})$ to the
canonical one on $Z_{d}^{G}(V_{H}^{+},W)$.

The main idea of the proof also comes from [Par13] and is analogous to the
proof of Theorem 3.12. We first define a smooth map

$\underline{\smash{\theta}}^{\prime}:V_{H}^{+}\times{\rm
Poly}^{H}(\check{V}_{H},W)\to{\rm Poly}^{G}(V,W)$

which is roughly an “inverse” of $\theta$. Using the Lagrange interpolation
method, one can find an integer $d_{0}=d(G,V)>0$ such that for each $v\in
V_{H}^{+}$, there is a polynomial $L_{v}:V\to{\mathbb{C}}$ of degree at most
$d_{0}$ satisfying

$L_{v}(gv)=\left\\{\begin{array}[]{cc}1,&\ {\rm if}\ g\in H,\\\ 0,&\ {\rm if}\
g\in G\setminus H.\end{array}\right.$ (3.7)

$L_{v}$ can be made smoothly dependent on $v\in V_{H}^{+}$. After averaging
over $H$ (which is a linear transformation) we can require that $L_{v}$ is
$H$-invariant. Also, using the decomposition
$V=\mathring{V}_{H}\oplus\check{V}_{H}$, regard any $P\in{\rm
Poly}^{H}(\check{V}_{H},W)$ an $H$-equivariant polynomial map from $V$ to $W$
which is constant in $\mathring{V}_{H}$-direction. Then define

$\underline{\smash{\theta}}^{\prime\prime}(v,P)=\frac{1}{|H|}\sum_{g\in
G}g^{-1}\circ(L_{v}\cdot P)\circ g.$

This map is clearly $G$-equivariant. Moreover, (3.7) implies that

$\underline{\smash{\theta}}^{\prime\prime}(v,P)(v)=P(\check{v}_{H})\ \ \
\forall v=(\mathring{v}_{H},\check{v}_{H})\in V_{H}^{+}\subset V_{H}\times
N_{H}.$

Moreover, for each $d>0$, one has

$\underline{\smash{\theta}}^{\prime\prime}(V_{H}^{+}\times{\rm
Poly}_{d}^{H}(\check{V}_{H},W))\subseteq{\rm Poly}_{d+d_{0}}^{G}(V,W).$

Now, we use iterations of the map $\underline{\smash{\phi}}^{\prime}$ defined
by (3.3) to reduce the degree. More precisely, by the construction of the map
$\underline{\smash{\phi}}^{\prime}$, if $d>0$ is large enough, there exists a
$G$-equivariant map

$\underline{\smash{\phi}}^{\prime\prime}:V\times{\rm
Poly}_{d+d_{0}}^{G}(V,W)\to{\rm Poly}_{d}^{G}(V,W)$

satisfying the following conditions.

  1. (1)

$\underline{\smash{\phi}}^{\prime\prime}$ is linear in the second variable
such that the associated map

  2. (2)

The associated map

$\phi^{\prime\prime}:V\times{\rm Poly}_{d+d_{0}}^{G}(V,W)\to V\times{\rm
Poly}_{d}^{G}(V,W),\
(v,P)\mapsto(v,\underline{\smash{\phi}}^{\prime\prime}(v,P))$

preserves the evaluation map.

  3. (3)

The restriction of $\phi^{\prime\prime}$ to $V\times{\rm Poly}_{d}^{G}(V,W)$
is the identity map.

Then define

$\underline{\smash{\theta}}^{\prime}=\underline{\smash{\phi}}^{\prime\prime}\circ\underline{\smash{\theta}}^{\prime\prime}:V_{H}^{+}\times{\rm
Poly}_{d}^{H}(\check{V}_{H},W)\to{\rm Poly}_{d}^{G}(V,W).$

We denote

$\theta^{\prime}(v,P)=(v,\underline{\smash{\theta}}^{\prime}(v,P)).$

Note that $\theta^{\prime}$ is not canonically defined, but we will just use
the existence of such maps to deduce propositions independent of the choice of
$\theta^{\prime}$.

Before proving the next result, we do some preparations. Consider the
compositions

$\underline{\smash{\theta}}\circ\theta^{\prime}:V_{H}^{+}\times{\rm
Poly}_{d}^{H}(\check{V}_{H},W)\to{\rm Poly}_{d}^{H}(\check{V}_{H},W)$

and

$\underline{\smash{\theta}}^{\prime}\circ\theta:V_{H}^{+}\times{\rm
Poly}_{d}^{G}(V,W)\to{\rm Poly}_{d}^{G}(V,W).$

For any $Q\in{\rm Poly}_{d}^{H}(\check{V}_{H},W)$, define a vector field on
$V_{H}^{+}\times{\rm Poly}_{d}^{H}(\check{V}_{H},W)$ by

$v_{Q}(x,P)=(0,Q-\underline{\smash{\theta}}(\theta^{\prime}(x,Q)));$

for any $Q\in{\rm Poly}_{d}^{G}(V,W)$, define a vector field on
$V_{H}^{+}\times{\rm Poly}_{d}^{G}(V,W)$ by

$w_{Q}(x,P)=(0,Q-\underline{\smash{\theta}}^{\prime}(\theta(x,Q))).$

Their flows are

$\begin{split}&\
\Phi_{Q,t}(x,P)=(x,P+t(Q-\underline{\smash{\theta}}(\theta^{\prime}(x,Q)))),\\\
&\
\Psi_{Q,t}(x,P)=(x,P+t(Q-\underline{\smash{\theta}}^{\prime}(\theta(x,Q))))\end{split}$

By the construction of $\theta^{\prime}$, one can see that the flows preserve
the corresponding $Z$-varieties.

######  Lemma 3.19.

For any $Q\in{\rm Poly}_{d}^{H}(\check{V}_{H},W)$ resp. $Q\in{\rm
Poly}_{d}^{G}(V,W)$ and $t\in{\mathbb{R}}$, consider the map

$F_{Q,t}:=\theta^{\prime}\circ\Phi_{Q,t}\circ\theta:V_{H}^{+}\times{\rm
Poly}_{d}^{G}(V,W)\to V_{H}^{+}\times{\rm Poly}_{d}^{G}(V,W)$

resp.

$G_{Q,t}:=\theta\circ\Psi_{Q,t}\circ\theta^{\prime}:V_{H}^{+}\times{\rm
Poly}_{d}^{H}(\check{V}_{H},W)\to V_{H}^{+}\times{\rm
Poly}_{d}^{H}(\check{V}_{H},W).$

Then the following is true.

  1. (1)

$F_{Q,t}^{-1}(Z_{d}^{G}(V_{H}^{+},W))=Z_{d}^{G}(V_{H}^{+},W)$ and
$G_{Q,t}^{-1}({\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}}))={\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}})$.

  2. (2)

$F_{Q,t}$ resp. $G_{Q,t}$ is transverse to $Z_{d}^{G}(V_{H}^{+},W)$ resp.
${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}})$ and pulls
back the canonical Whitney stratification to itself.

###### Proof.

Notice that $\theta$, $\theta^{\prime}$, $\Phi_{Q,t}$, and $\Psi_{Q,t}$ all
preserve the evaluation map. Hence one has

${\rm ev}(F_{Q,t}(v,P))={\rm ev}(v,P)=P(v),\ {\rm ev}(G_{Q,t}(v,P))={\rm
ev}(v,P)=P(v).$

Therefore, the first item is true.

To prove (2), regard ${\rm Poly}_{d}^{G}(V,W)$ as a base manifold $M_{d}^{G}$
with trivial $G$-action, over which we have the trivial vector bundles

$\displaystyle\ {\mathcal{E}}:=M_{d}^{G}\times V,\ $ $\displaystyle\
{\mathcal{F}}:=M_{d}^{G}\times W.$

Denote

${\mathcal{E}}_{H}^{+}:=V_{H}^{+}\times M_{d}^{G}\subset{\mathcal{E}}.$

Then $F_{Q,t}$ can be viewed as a bundle map

$\underline{\smash{f}}_{Q,t}:{\mathcal{E}}_{H}^{+}\to{\rm
Poly}_{d}^{G}({\mathcal{E}},{\mathcal{F}})\cong M_{d}^{G}\times{\rm
Poly}_{d}^{G}(V,W).$

On the other hand, there is the natural inclusion map

$\underline{\smash{\iota}}:{\mathcal{E}}_{H}^{+}\to{\rm
Poly}_{d}^{G}({\mathcal{E}},{\mathcal{F}}),\
\underline{\smash{\iota}}(P,v):=(P,P).$

Then the construction implies that

${\rm
graph}(\underline{\smash{f}}_{Q,t}-\underline{\smash{\iota}})\subset{\mathcal{Z}}_{d}^{G}({\mathcal{E}},{\mathcal{F}}).$

As the graph of $\underline{\smash{\iota}}$ is transverse to the canonical
Whitney stratification on
${\mathcal{Z}}_{d}^{G}({\mathcal{E}},{\mathcal{F}})$, by Lemma 3.7 the graph
of $\underline{\smash{f}}_{Q,t}$ is also transverse to
${\mathcal{Z}}_{d}^{G}({\mathcal{E}},{\mathcal{F}})$. Equivalently, it means
that the composition $F_{Q,t}$ is transverse to $Z_{d}^{G}(V_{H}^{+},W)$. The
case of $G_{Q,t}$ is the same. ∎

###### Proof of Theorem 3.18.

We first prove that $\theta^{\prime}$ resp. $\theta$ is transverse to
$Z_{d}^{G}(V_{H}^{+},W)$ resp.
${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}})$. Consider
$\theta^{\prime}$. The above lemma implies that $\theta^{\prime}$ is
transverse to $Z_{d}^{G}(V_{H}^{+},W)$ along the image of
$\Phi_{Q,t}\circ\theta$. The global transversality of $\theta^{\prime}$
follows if we can show that any point in $Z_{d}^{H}(V_{H}^{+},W)$ lies in the
image of $\Phi_{Q,t}\circ\theta$ for some $Q$ and some $t$. Indeed, for any
$(x,Q)\in Z_{d}^{H}(V_{H}^{+},W)$, one has

$(x,Q)=\Phi_{Q,1}(x,\underline{\smash{\theta}}(\theta^{\prime}(x,Q)))=\Phi_{Q,1}(\theta(x,\underline{\smash{\theta}}^{\prime}(x,Q)))\in{\rm
im}(\Phi_{Q,1}\circ\theta).$

Hence $\theta^{\prime}$ is transverse to $Z_{d}^{G}(V,W)$ at $(x,Q)$. As
$(x,Q)$ is an arbitrary point of $Z_{d}^{H}(V_{H}^{+},W)$, it follows that
$\theta^{\prime}$ is transverse to $Z_{d}^{G}(V,W)$. The transversality
statement for $\theta$ is similar.

It is then similar to Theorem 3.12 to show that $\theta$ resp.
$\theta^{\prime}$ pulls back the canonical Whitney stratification to the
canonical one, using Lemma A.11 and an induction on strata of the action
prestratification. Here are the details. Abbreviate ${\rm Poly}_{d}^{G}(V,W)$
by $P_{d}^{G}$ and ${\rm Poly}_{d}^{H}(\check{V}_{H},W)$ by $P_{d}^{H}$;
abbreviate $Z_{d}^{G}(V_{H}^{+},W)$ by $Z_{d}^{G}$ and
${\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}})$ by
${\mathcal{Z}}_{d}^{H}$. First notice that

$\displaystyle\ Z_{d}^{G}(V_{H}^{+},W)\subset V_{H}^{+}\times P_{d}^{G},\ $
$\displaystyle\
{\mathcal{Z}}_{d}^{H}(\check{\mathcal{V}}_{H}^{+},{\mathcal{W}})\subset
V_{H}^{+}\times P_{d}^{H}.$

The the strata of the action prestratifications on $V_{H}^{+}\times P_{d}^{G}$
and $V_{H}^{+}\times P_{d}^{H}$ are both indexed by the set

$B=\\{I<G\ |\ I\subseteq H\\}.$

Then by Remark A.22, the canonical Whitney stratifications on $Z_{d}^{G}$ and
${\mathcal{Z}}_{d}^{H}$ are still the minimal smooth Whitney stratifications
which respect the action prestratifications. For each $I\in B$, denote

$\displaystyle\ Z_{d,I}^{G}:=Z_{d}^{G}\cap(V_{I}^{*}\times P_{d}^{G}),\ $
$\displaystyle\
{\mathcal{Z}}_{d,I}^{H}:={\mathcal{Z}}_{d}^{H}\cap(V_{I}^{*}\times
P_{d}^{H}).$

Then

$\displaystyle\ \theta^{-1}({\mathcal{Z}}_{d,I}^{H})=Z_{d,I}^{G},\ $
$\displaystyle\ (\theta^{\prime})^{-1}(Z_{d,I}^{G})={\mathcal{Z}}_{d,I}^{H}.$

Then we prove inductively the following statement.

$\bullet$ Suppose $I\in B$. Then $\theta$ resp. $\theta^{\prime}$ pulls back
the restriction of the canonical Whitney stratifications on
${\mathcal{Z}}_{d,I}^{H}$ resp. $Z_{d,I}^{G}$ to the canonical one on
$Z_{d,I}^{G}$ resp. ${\mathcal{Z}}_{d,I}^{H}$.

Notice that we just verified condition (1) of Lemma A.11. On the other hand,
by the $Q=0$, $t=0$ case of Lemma 3.19, conditions (3) and (4) of Lemma A.11
are checked. Hence by the absolute case of Lemma A.11, the base case of the
induction is verified. Suppose we have proved that for some $I_{0}\in B$, the
above statement is true for all $I\in B$ with $I\subsetneq I_{0}$. Then the
above statement follows from the relative case of Lemma A.11 where

$\displaystyle\ Y^{\bullet}=\bigsqcup_{I\subsetneq I_{0}}Z_{d,I}^{G},\ $
$\displaystyle\ Z^{\bullet}=\bigsqcup_{I\subsetneq
I_{0}}{\mathcal{Z}}_{d,I}^{H}$

as the induction hypothesis verifies condition (5) of Lemma A.11.

∎

##  4\. Pseudocycles in Thom–Mather stratified spaces

In this section we define the notion of pseudocycles in Thom–Mather stratified
spaces and extend half of the main results of Zinger [Zin08] about
pseudocycles in manifolds. This discussion is necessary as we would like to
define the homology classes supported on the perturbed zero loci in effective
orbifolds, which are naturally Thom–Mather stratified spaces. However, we
remark that we do not need the result of this section to do intersection
theory such as to define Gromov–Witten type invariants or Floer homology.

######  Definition 4.1.

[Tho64, Tho69][Mat12] A _Thom–Mather stratified space_ is a triple
$(X,{\mathcal{S}},{\mathfrak{J}})$ satisfying the following axioms (A1)–(A7).

  1. (A1)

$X$ is a locally compact, Hausdorff, and second countable topological space.

  2. (A2)

${\mathcal{S}}$ is a prestratification of $X$ with strata $X_{\alpha}\subset
X$.

  3. (A3)

Each member of ${\mathcal{S}}$ is a smooth manifold.

  4. (A4)

${\mathfrak{J}}$ (called the _collection of control data_) is a triple
$\\{(N_{\alpha}),(\pi_{\alpha}),(\rho_{\alpha})\\}$ where for each
$X_{\alpha}\in{\mathcal{S}}$, $N_{\alpha}$ is an open neighborhood of
$X_{\alpha}$, $\pi_{\alpha}:N_{\alpha}\to X_{\alpha}$ is a continuous
retraction onto $X_{\alpha}$, and $\rho_{\alpha}:N_{\alpha}\to[0,+\infty)$ is
a continuous function.

  5. (A5)

$X_{\alpha}=\\{v\in N_{\alpha}\ |\ \rho_{\alpha}(v)=0\\}$.

  6. (A6)

For each pair of strata $X_{\alpha},X_{\beta}\in{\mathcal{S}}$, define
$N_{\beta\alpha}=N_{\alpha}\cap X_{\beta}$ (which is an open subset of the
manifold $X_{\beta}$), $\pi_{\beta\alpha}=\pi_{\alpha}|_{N_{\beta\alpha}}$,
and $\rho_{\beta\alpha}=\rho_{\alpha}|_{N_{\beta\alpha}}$. We require that the
map

$(\pi_{\beta\alpha},\rho_{\beta\alpha}):N_{\beta\alpha}\to
X_{\alpha}\times(0,+\infty)$

is a smooth submersion.

  7. (A7)

For any three strata $X_{\alpha},X_{\beta},X_{\gamma}$ one has

$\begin{split}\pi_{\beta\alpha}\circ\pi_{\gamma\beta}=&\pi_{\gamma\alpha},\\\
\rho_{\beta\alpha}\circ\pi_{\gamma\beta}=&\rho_{\gamma\alpha}\end{split}$

whenever both sides of the equations are defined.

For each $\epsilon>0$ sufficiently small, denote

$\displaystyle\ N_{\alpha}(\epsilon)=\\{v\in N_{\alpha}\ |\
\rho_{\alpha}(v)<\epsilon\\},\ $ $\displaystyle\ S_{\alpha}(\epsilon)=\\{v\in
N_{\alpha}\ |\ \rho_{\alpha}(v)=\epsilon\\},$

and

$X_{\beta}^{0}(\epsilon):=X_{\beta}\setminus\bigcup_{\alpha\leq\beta}N_{\alpha}(\epsilon).$

######  Definition 4.2.

(Family of lines)[Gor76, Gor78] Given a Thom–Mather stratified space
$(X,{\mathcal{S}},{\mathfrak{J}})$, a _family of lines_ consists of a positive
number $\delta>0$ and a system of retractions

$r_{\alpha}(\epsilon):N_{\alpha}\setminus X_{\alpha}\to S_{\alpha}(\epsilon),\
\epsilon\in(0,\delta),\ X_{\alpha}\in{\mathcal{S}}$

satisfying the following conditions. Let $X_{\alpha}<X_{\beta}$.

  1. (1)

$r_{\alpha}(\epsilon)\circ
r_{\beta}(\epsilon^{\prime})=r_{\beta}(\epsilon^{\prime})\circ
r_{\alpha}(\epsilon)$.

  2. (2)

$\rho_{\alpha}\circ r_{\beta}(\epsilon)=\rho_{\alpha}$.

  3. (3)

$\rho_{\beta}\circ r_{\alpha}(\epsilon)=\rho_{\beta}$.

  4. (4)

$\pi_{\alpha}\circ r_{\beta}(\epsilon)=\pi_{\alpha}$.

  5. (5)

If $0<\epsilon<\epsilon^{\prime}<\delta$ then
$r_{\alpha}(\epsilon^{\prime})\circ
r_{\alpha}(\epsilon)=r_{\alpha}(\epsilon^{\prime})$.

  6. (6)

$\pi_{\alpha}\circ r_{\alpha}(\epsilon)=\pi_{\alpha}$.

  7. (7)

$r_{\alpha}(\epsilon)|_{N_{\alpha}(\epsilon)\cap
X_{\beta}}:N_{\alpha}(\epsilon)\cap Y\to S_{\alpha}(\epsilon)\cap Y$ is
smooth.

######  Example 4.3.

  1. (1)

Mather [Mat12] proved that a Whitney stratified set $S$ in a smooth manifold
$M$ is a Thom–Mather stratified space.

  2. (2)

Let $G$ be a compact Lie group acting smoothly on a smooth manifold $M$. Then
Lellmann [Lel75] proved that the orbit space $M/G$ is a Thom–Mather stratified
space.333The authors cannot access the original paper of Lellmann. A complete
proof can be found in [DS06, Section II.4]. As all effective orbifolds can be
expressed as a global quotient by a compact Lie group, an effective orbifold
admits the structure of Thom–Mather stratified space whose strata are labelled
by orbit types. Given an orbifold with a uniform bound the orders of the
isotropy groups, it can also be presented as a global quotient of a compact
Lie group [Par19, Corollary 1.3] so it is also a Thom–Mather stratified space
as above.

######  Theorem 4.4.

[Gor76][Gor78] Any Thom–Mather stratified space admits a family of lines.

The family of lines induces a system of maps

$\begin{split}h_{\alpha}:N_{\alpha}\setminus X_{\alpha}\to&\
S_{\alpha}(\epsilon)\times{\mathbb{R}}_{+}\\\ v\mapsto&\
(r_{\alpha}(\epsilon)(v),\rho_{\alpha}(v))\end{split}$

which is a homeomorphism onto its image and which is a diffeomorphism on each
stratum.

###  4.1. Pseudocycles

We generalize the notion of pseudocycles in smooth manifolds to the case of
Thom–Mather stratified spaces.

######  Definition 4.5.

Let $X$ be a Thom–Mather stratified space with strata $X_{\alpha}$. Let $W$ be
a smooth manifold. A continuous map $F:W\to X$ is called _smooth_ if for each
$\alpha$, $W_{\alpha}:=F^{-1}(X_{\alpha})$ is a smooth submanifold of $W$ and
the restriction $F_{\alpha}:=F|_{W_{\alpha}}:W_{\alpha}\to X_{\alpha}$ is a
smooth map.

######  Definition 4.6.

A subset $A\subset X$ is said to _have dimension $\leq k$_ if there exists a
smooth manifold $W$ and a smooth map $F:W\to X$ such that $A\subseteq F(W)$
such that for each strata $X_{\alpha}$, ${\rm dim}W_{\alpha}\leq k$.

Moreover, for any continuous map $f:A\to B$ between topological spaces. The
_boundary_ of $f$ is defined to be

${\rm Bd}f:=\bigcap_{K\subset A\ {\rm compact}}\overline{f(A\setminus K)}.$
(4.1)

######  Definition 4.7.

(Pseudocycles and cobordism)

  1. (1)

A smooth map $F:W\to X$ is called a _$k$ -dimensional pseudocycle_ in $X$ if
$W$ is oriented, $F(W)$ is precompact in $X$, ${\rm dim}W=k$, and ${\rm Bd}F$
has dimension at most $k-2$.

  2. (2)

Two $k$-dimensional pseudocycles $F_{i}:W_{i}\to X$ ($i=0,1$) is called
_cobordant_ if there exist an oriented $k+1$-dimensional manifold $\tilde{W}$
with boundary such that $\partial\tilde{W}=W_{0}\sqcup W_{1}$ compatible with
the orientations, and there exists a smooth map $\tilde{F}:\tilde{W}\to X$
such that $\tilde{F}(\tilde{W})$ is precompact in $X$ and such that

$\tilde{F}|_{W_{0}}=F_{0},\ \tilde{F}|_{W_{1}}=F_{1},\ {\rm Bd}\tilde{F}\leq
k-1.$

One can define an abelian group ${\mathcal{H}}_{k}(X)$ whose elements are
cobordism classes of oriented pseudocycles in $X$ and whose addition is
induced from disjoint unions of pseudocycles. Our main conclusion is

######  Theorem 4.8.

There exists a homomorphism

$\Psi_{X}:{\mathcal{H}}_{k}(X)\to H_{*}(X;{\mathbb{Z}})$

such that if an element $A\in{\mathcal{H}}_{k}(X)$ is represented by a smooth
map $F:W\to X$ where $X$ is compact and oriented, then

$\Psi_{X}(A)=F_{*}([W]).$

When $X$ is a smooth manifold, this theorem is rigorously proved by Zinger
[Zin08]. In fact Zinger showed that $\Psi$ is an isomorphism.

###  4.2. A technical lemma

The proof of Theorem 4.8 follows the same line as in [Zin08]. The key step is
a generalization of [Zin08, Proposition 2.2].

######  Lemma 4.9.

Suppose ${\rm dim}W=k$ and $F:W\to X$ is a smooth map. Then there exists an
open neighborhood $U\subset X$ of $F(W)$ such that

$H_{s}(U;{\mathbb{Z}})=0,\ \forall s>k.$

We first introduce a few new notations. Given a pair of strata $X_{\alpha}\leq
X_{\beta}$, define $S_{\beta\alpha}(\epsilon):=S_{\alpha}(\epsilon)\cap
X_{\beta}$. Choose a sufficiently small $d_{0}>0$ and consider the open set

$W_{\beta\alpha}(d_{0}):=F^{-1}(N_{\alpha}(d_{0}))\cap
W_{\beta}=F^{-1}(N_{\alpha}(d_{0})\cap X_{\beta})\subset W_{\beta}.$

Then the composition

$\pi_{\beta\alpha}\circ F_{\beta}:W_{\beta\alpha}(d_{0})\to X_{\alpha}$

is smooth.

######  Lemma 4.10.

There exists a collection of real numbers $\\{d_{\alpha}\in(0,d_{0})\ |\
\alpha\in{\mathcal{S}}\\}$ satisfying the following conditions.

  1. (1)

For each stratum $\beta$ and $\alpha_{1},\ldots,\alpha_{s}\leq\beta$, the
intersection

$S_{\beta\alpha_{1}}(d_{\alpha_{1}})\cap\cdots\cap
S_{\beta\alpha_{s}}(d_{\alpha_{s}})$

is transverse (in $X_{\beta}$).

  2. (2)

For each pair of strata $\beta\leq\beta^{\prime}$ and for
$\alpha_{1},\ldots,\alpha_{s}\leq\beta$, the map

$\pi_{\beta^{\prime}\beta}\circ
F_{\beta^{\prime}}:W_{\beta^{\prime}\beta}(d_{0})\to X_{\beta}$

is transverse to $S_{\beta\alpha_{1}}(d_{\alpha_{1}})\cap\cdots\cap
S_{\beta\alpha_{s}}(d_{\alpha_{s}})$.

###### Proof.

Sard’s theorem. ∎

Now by rescaling the control data by positive constants, we may assume that
all $d_{\alpha}$ are equal to some $d\in(0,d_{0})$. We can also assume that
$d$ is sufficiently small so that

$N_{\alpha}(d)\cap N_{\beta}(d)\neq\emptyset\Longrightarrow\alpha\leq\beta\
{\rm or}\ \beta\leq\alpha.$

Now following Goresky [Gor78], we would like to construct an “interior
triangulation.”

######  Definition 4.11.

Let $(X,{\mathcal{S}},{\mathfrak{J}})$ be a Thom–Mather stratified space. For
$d>0$ sufficiently small, a (smooth) _interior $d$-triangulation_ consists of
a simplicial complex $K$, a continuous map $\eta:|K|\to X$ satisfying the