# On fillings of contact links of quotient singularities

Zhengyi Zhou

###### Abstract.

We study several aspects of fillings for links of general isolated quotient
singularities using Floer theory, including co-fillings, Weinstein fillings,
strong fillings, exact fillings and exact orbifold fillings, focusing on the
non-existence of exact fillings of contact links of isolated terminal quotient
singularities. We provide an extensive list of isolated terminal quotient
singularities whose contact links are not exactly fillable, including
${\mathbb{C}}^{n}/({\mathbb{Z}}/2)$ for $n\geq 3$, which settles a conjecture
of Eliashberg, quotient singularities from general cyclic group actions and
finite subgroups of $SU(2)$, and all terminal quotient singularities in
complex dimension $3$. We also obtain uniqueness of the _orbifold_
diffeomorphism type of _exact orbifold fillings_ of contact links of some
isolated terminal quotient singularities.

###### Contents

  1. 1 Introduction
  2. 2 Basics of the contact link $(S^{2n-1}/G,\xi_{\rm std})$
  3. 3 Symplectic cohomology of fillings of $(S^{2n-1}/G,\xi_{\rm std})$
  4. 4 The secondary coproduct on symplectic cohomology
  5. 5 Fundamental groups
  6. 6 Non-existence of exact fillings for real projective spaces
  7. 7 More non-existence results
  8. A Bernoulli numbers

##  1\. Introduction

Contact structures arise naturally from symplectic manifolds with a convex
boundary. The study of contact manifolds as convex boundaries of symplectic
manifolds and their symplectic fillings was pioneered by Gromov [40] and
Eliashberg [29]. In particular, they showed that any contact 3-fold with a
(weak) symplectic filling is necessarily tight. This result was generalized to
all dimensions by Massot, Niederkrüger and Wendl,[45], see also [61]. As a
consequence, the different types of fillability give a coarse understanding of
the landscape of contact structures beyond the fundamental dichotomy between
overtwistedness and tightness [9, 28]. Namely, we have the following
inclusions among classes of contact structures:

$\\{\text{Weinstein fillable}\\}\subset\\{\text{exactly
fillable}\\}\subset\\{\text{strongly fillable}\\}\subset\\{\text{weakly
fillable}\\}\subset\\{\text{tight}\\}.$

All of the inclusions are proper: in dimension three, they were shown by
Bowden [13], Ghiggini [36], Eliashberg [32], Etnyre and Honda [34], and in
higher dimensions by Bowden, Crowley, and Stipsicz [14], the author [72],
Bowden, Gironella, and Moreno [15], Massot, Niederkrüger, and Wendl [45].

The guiding question in understanding the second inclusion is the following
conjecture by Eliashberg:

######  Conjecture A ([25, §1.9]).

$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ is not exact fillable if $n\geq
3$.

Fortunately, the real projective space is natural and simple enough so that
one can understand both pseudo-holomorphic curves as well as the topology
sufficiently well. We showed in [72] that Conjecture A holds if $n\neq 2^{k}$.
The odd $n$ cases were also proved by Ghiggini and Niederkrüger [37] using
different methods. In this paper, we fill the gap, hence settle the
conjecture.

######  Theorem B.

For $n\geq 3$, $(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ is not exactly
fillable.

When $n=2^{k}$, the total Chern class of $\xi_{\rm std}$ becomes trivial,
which sabotages the topological obstructions in [72]. Hence to prove B, we
need to enhance both the pseudo-holomorphic curve argument and the topological
argument. On the pseudo-holomorphic curve side, we use symplectic cohomology
for exact orbifolds developed by Gironella and the author [38] and study the
secondary coproduct on positive symplectic cohomology, which was proposed by
Seidel [63], defined by Ekholm and Oancea [27], and recently studied
extensively by Cieliebak, Hingston and Oancea [20, 21, 23]. Then by a result
of Eliashberg, Ganatra, and Lazarev [30], we can get information on the
intersection form of the filling. On the topological side, the contradiction
is a non-integer Chern number derived from the Hirzebruch signature theorem.
The argument essentially boils down to studying the $2$ factors of the
numerators of Bernoulli numbers, whose proof is motivated by a proof of the
Clausen-Staudt Theorem dating back to 1883 [71]. When $n=4$, the arguments
above do not yield a contradiction. But we can build a spin manifold from a
hypothetical exact filling. Finally, we can arrive at a contradiction by
showing its $\hat{A}$-genus is not integer, violating the celebrated
Atiyah–Singer index theorem for the Dirac operator.

###  1.1. Fillings of Contact links of isolated quotient singularities

Real projective spaces with the standard contact structure fall in a much
larger class of contact manifolds: contact links of isolated singularities.
Links of isolated singularities are natural and interesting examples of
contact manifolds, which are always strongly fillable by Hironaka’s resolution
of singularities. In the case of smoothable singularities, the smoothing
provides natural exact/Weinstein fillings to the contact link. In complex
dimension $2$, contact properties of links, especially filling properties, in
particular Weinstein fillings, were extensively studied, see [55] for an
introduction and references therein for details. In such a low dimension,
tools from low dimensional topology and intersection theory of pseudo-
holomorphic curves allow us to obtain classification results. For example, the
diffeomorphism type of symplectic fillings of lens spaces is classified by
Lisca [44]. Moreover, such classification can be strengthened to a
classification up to symplectic deformation by a powerful result of Wendl
[69]. Moreover, there are surprising connections between the contact
properties of the link and algebro-geometric properties of the singularity,
e.g. the beautiful theorem of McLean [48] relating the terminality of the
singularity with the SFT degrees.

In this paper, we study, by means of Floer theories, symplectic fillings of
contact links of isolated quotient singularities in any dimension, where
$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ should be understood as the
contact link of the simplest quotient singularity
${\mathbb{C}}^{n}/({\mathbb{Z}}/2)$. More generally, let $G$ be a finite
subgroup of $U(n)$, such that every nontrivial element in $G$ does not have
one as an eigenvalue. Since $G$ preserves the standard contact structure on
the unit sphere $S^{2n-1}\subset{\mathbb{C}}^{n}$, we get a natural contact
manifold $(S^{2n-1}/G,\xi_{\rm std})$, which is the contact link of the
isolated quotient singularity ${\mathbb{C}}^{n}/G$. Our discussion will cover
co-fillings (a.k.a. semi-fillings), Weinstein fillings, and strong fillings,
but with an emphasis on exact fillings and exact orbifold fillings, recalled
in the following.

######  Definition 1.1.

Let $(Y,\xi)$ be a co-oriented contact manifold, then a symplectic manifold
$(W,\omega)$ is called

  1. (1)

a strong filling of $Y$, if $\partial W=Y$ and there exits a outward-pointing
Liouville vector field $X$ near $\partial W$ (i.e. $L_{X}\omega=\omega$), such
that $\iota_{X}\omega$ restricts to a contact form for $\xi$;

  2. (2)

a co-filling (semi-filling) of $Y$, if $W$ is a connected strong filling of
$Y\sqcup Y^{\prime}$, where $\bm{Y^{\prime}\neq\emptyset}$ is another contact
manifold;

  3. (3)

an exact filling of $Y$, if $W$ is a strong filling and the Liouville vector
field $X$ is globally defined;

  4. (4)

an exact orbifold filling of $Y$, if $W$ is an orbifold and exact;

  5. (5)

a Weinstein filling of $Y$, if $W$ is exact and the Liouville vector field $X$
is gradient-like for a Morse function $\phi$ with $\partial W$ a regular level
set of $\phi$.

An almost complex manifold $(W,J)$ is an almost Weinstein filling of $Y$, if
$\partial W=Y$, $JTY\cap TY=\xi$ such that $J|_{\xi}$ is homotopic to the
almost complex structure compatible with the contact structure and $J$ sends
the outward normal vector to the co-orientation along $Y$, and there exists a
Morse function $\phi$ with $\partial W$ a regular level set of $\phi$, whose
critical points have indices at most $\dim W/2$.

######  Remark 1.2.

The general definition of a weak filling was given in [45, Definition 4].
Since $H^{2}(S^{2n-1}/G;\mathbb{R})=0$, there is no difference between weak
fillings and strong fillings for contact links in this paper up to deformation
by [45, Lemma 2.10].

###  1.2. Weinstein fillings, strong fillings, and co-fillings.

Unlike the situation in complex dimension $2$, it is a classical fact that
isolated quotient singularities of complex dimension at least $3$ are not
smoothable, which was proved using algebraic methods by showing those
singularities are rigid [60]. In particular, the contact link of an isolated
quotient singularity of complex dimension at least $3$ can not be filled by a
Milnor fiber. On the symplectic side, it is an easy corollary of the
Eliashberg-McDuff-Floer theorem [46] that the contact link of an isolated
quotient singularity of complex dimension at least $3$ has no Weinstein
filling, see Proposition 2.5. We first state the following result on
fundamental groups of symplectic fillings, which also yields the non-existence
of Weinstein fillings in complex dimensions at least $3$.

######  Theorem C.

Let ${\mathbb{C}}^{n}/G$ be an isolated singularity and $n\geq 2$.

  1. (1)

If $W$ is an _exact filling_ of the contact link $(S^{2n-1}/G,\xi_{\rm std})$,
then $G=\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$ is not injective.

  2. (2)

If ${\mathbb{C}}^{n}/G$ is a terminal quotient singularity (Definition 2.10)
and $W$ is a _strong filling_ of the contact link, then
$\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$ is surjective. If moreover, $W$ is an
_exact filling_ , then $W$ is simply connected.

  3. (3)

If $G={\mathbb{Z}}/m$, where $m$ is not divisible by a non-trivial square, and
${\mathbb{C}}^{n}/G$ is terminal, then any _strong filling_ of the contact
link is simply connected.

######  Remark 1.3.

(1) of C also implies that the contact link of an isolated quotient
singularity of complex dimension at least $3$ has no Weinstein filling. In
fact, by the topological argument in [31, §6.2], those contact links have no
almost Weinstein filling whenever $n\geq 4$, see Proposition 2.6.

Terminal singularities are important players in the minimal model program in
higher dimensions introduced by Reid [56]. A deep theorem due to McLean [48]
shows that the terminal property is in fact a contact property of the link. In
the context of isolated quotient singularities, such relation is contained in
the terminal criterion due to Reid, Shepherd-Barron [56] and Tai [66]. By
arguments in the same spirit of the proof of Theorem C, we can obtain the
following result on co-fillings.

######  Proposition D.

Let ${\mathbb{C}}^{n}/G$ be an isolated terminal singularity, then
$(S^{2n-1}/G,\xi_{std})$ is not co-fillable.

Gromov [40], Floer, Eliashberg and McDuff [46] showed that $(S^{2n-1},\xi_{\rm
std})$ is not co-fillable when $n\geq 2$. Note that $(S^{2n-1},\xi_{\rm std})$
can be viewed as the link of a smooth point, which is terminal if and only if
$n\geq 2$. Hence Proposition D can be viewed as a generalization of the above
result. Obstructions to co-fillings were used to obstruct strong fillings,
e.g. [32]

###  1.3. Non-existence of exact fillings

The first example of a strongly but not exactly fillable contact manifold was
found in dimension $3$ by Ghiggini [36], and many more examples in dimension
$3$ were found by Min [51]. In higher dimensions, such examples were predicted
by Conjecture A. It is worth noting that the threshold $n\geq 3$ in Conjecture
A is precisely the threshold for the singularity
${\mathbb{C}}^{n}/({\mathbb{Z}}/2)$ to be terminal. Therefore B leads to the
following conjecture in [72].

######  Conjecture E.

Let ${\mathbb{C}}^{n}/G$ be an isolated terminal singularity, then the contact
link $(S^{2n-1}/G,\xi_{\rm std})$ is not exactly fillable.

In view of the conjecture above and Hironaka’s resolution of singularities,
isolated terminal quotient singularities shall provide tons of natural contact
manifolds with strong fillings that are not exactly fillable. In this paper,
we will prove this conjecture for a list of cases.

For general group actions, we have the following result with a simple
assumption. Let $\operatorname{Conj}(G)$ be the set of conjugacy classes of
$G$.

######  Theorem F.

Assume ${\mathbb{C}}^{n}/G$ is an isolated terminal singularity with $n$ odd
and $|\operatorname{Conj}(G)|$ even, then the contact link
$(S^{2n-1}/G,\xi_{\rm std})$ has no exact filling.

In the smallest non-trivial dimension, namely complex dimension $3$, terminal
quotient singularities are completely classified in [52], where the group can
be any finite cyclic group. Conjecture E indeed holds in those cases.

######  Theorem G.

The contact link of any isolated terminal quotient singularity in complex
dimension $3$ has no exact filling.

We also consider the case of ${\mathbb{Z}}/3,{\mathbb{Z}}/4$ actions, more
general cyclic group actions as well as finite subgroups of $SU(2)$ acting on
${\mathbb{C}}^{2n}$ diagonally. The proof of the above results is based on the
study of symplectic cohomology of a hypothetical exact filling. The starting
point is that the rational first Chern class
$c_{1}^{{\mathbb{Q}}}(S^{2n-1}/G,\xi_{\rm std})$ vanishes, and all Reeb orbits
are of torsion homotopy classes, therefore we have a well-defined rational
Conley-Zehnder index assigned to each Reeb orbit. It is important to note
that, as a hypothetical filling may not have vanishing first Chern class,
those “boundary” Conley-Zehnder indices may not agree with actual Conley-
Zehnder indices used in the construction of symplectic cohomology. However,
when the singularity is terminal, those boundary Conley-Zehnder indices induce
a filtration on the cochain complex of the positive symplectic cohomology,
which is compatible with the differential. Moreover, certain pages of the
spectral sequence are independent of the filling/augmentation. Therefore, we
can compare it with the computation we did in [38, Theorem B] for the standard
exact _orbifold_ filling ${\mathbb{C}}^{n}/G$ and conclude that the spectral
sequence degenerates at some point due to parity reasons. Moreover, we can
show that the total dimension $\dim H^{*}(W;{\mathbb{Q}})$ is
$|\operatorname{Conj}(G)|$, the size of the set of conjugacy classes, for a
hypothetical exact filling $W$ whenever ${\mathbb{C}}^{n}/G$ is terminal. Many
topological properties (e.g. intersection form, fundamental groups, cup
product length) of the hypothetical filling $W$ can be derived this way
through the Floer theory, which in many cases lead to contradictions to the
existence of $W$.

###  1.4. Uniqueness of the diffeomorphism type of exact orbifold fillings

The celebrated Eliashberg-Floer-McDuff theorem [46] states that any exact
filling of $(S^{2n-1},\xi_{\rm std})$ is diffeomorphic to a ball whenever
$n\geq 2$. This theorem has been generalized in many directions [6, 53, 73,
75]. From the singularity perspective, $(S^{2n-1},\xi_{\rm std})$ is the link
of a smooth point and a smooth point viewed as a singularity is terminal if
and only if $n\geq 2$. Hence it naturally leads to the following conjecture,
which in particular will imply Conjecture E.

######  Conjecture H.

Let ${\mathbb{C}}^{n}/G$ be an isolated terminal singularity, then exact
_orbifold_ fillings of the contact link $(S^{2n-1}/G,\xi_{\rm std})$ have a
unique diffeomorphism type111More aggressively, we can conjecture that exact
orbifold fillings are unique..

######  Remark 1.4.

By [38, Theorem C], any exact orbifold filling of $(S^{2n-1},\xi_{\rm std})$
for $n\geq 2$ must be a manifold. Therefore Conjecture H holds for a smooth
point by the Eliashberg-Floer-McDuff theorem.

In this paper, we prove this conjecture in some simple cases.

######  Theorem I.

Let $n\geq 3$ be an odd number. Then any exact orbifold filling of
$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ is diffeomorphic to
${\mathbb{D}}^{2n}/({\mathbb{Z}}/2)$ as an orbifold with boundary.

### Organization of the paper

We discuss basic topological, contact, and algebro-geometric properties of
quotient singularities in §2. We study symplectic cohomology of hypothetical
exact fillings in §3 and secondary coproduct in §4. §5 focuses on fundamental
groups of strong fillings of links. We prove B and I in §6 and other non-
existence of exact fillings in §7. Appendix A is devoted to proving some
properties of the Bernoulli numbers.

### Acknowledgments

The author would like to thank Ruizhi Huang for the helpful discussion on the
modified signature and the anonymous referee for many helpful suggestions that
improved this paper. The author is supported by the National Key R&D Program
of China under Grant No. 2023YFA1010500, the National Natural Science
Foundation of China under Grant No. 12288201 and 12231010.

##  2\. Basics of the contact link $(S^{2n-1}/G,\xi_{\rm std})$

In this section, we recall and prove some basics of the contact topology of
$(S^{2n-1}/G,\xi_{\rm std})$. Unless specified otherwise, $G\subset U(n)$ acts
on ${\mathbb{C}}^{n}$ with an isolated singularity throughout this paper. We
write $\mathbf{i}:=\sqrt{-1}$.

###  2.1. Topology of $S^{2n-1}/G$

Since $G$ acts on ${\mathbb{C}}^{n}$ with an isolated singularity, the same is
true for the diagonal action of $G$ on $({\mathbb{C}}^{n})^{k}$. In
particular, this gives rise to a model of $K(G,1)=BG$ as
$S^{\infty}/G:=\varinjlim\partial({\mathbb{D}}^{2nk})/G$, where
${\mathbb{D}}^{2nk}$ is the unit ball in ${\mathbb{C}}^{nk}.$ As a
consequence, we have the following.

######  Proposition 2.1.

The inclusion $S^{2n-1}/G\hookrightarrow
BG=\varinjlim\partial({\mathbb{D}}^{2nk})/G$ induces an isomorphism on
cohomology over any coefficient for degrees smaller than $2n-1$.

###### Proof.

On the unit sphere $\partial({\mathbb{D}}^{2nk})$, we have a Morse-Bott
function

$f=\sum_{s=0}^{k-1}s\sum_{j=1}^{n}|z_{sn+j}|^{2}$ (2.1)

which is $G$ invariant. The critical submanifolds of the induced function on
$S^{2nk-1}/G$ are $k$ copies of $S^{2n-1}/G$ with Morse-Bott indices
$0,2n,\ldots,2(k-1)n$. By the standard Morse-Bott theory, for any $k\geq 1$,
we have that $S^{2n-1}/G\hookrightarrow\partial({\mathbb{D}}^{2nk})/G$ induces
an isomorphism on cohomology over any coefficient for degrees smaller than
$2n-1$. Hence the claim follows. ∎

######  Remark 2.2.

A few remarks on the cohomology of $BG$ are in order.

  1. (1)

Let $\bm{z}_{s}$ denote $(z_{sn+1},\ldots,z_{(s+1)n})$ in ${\mathbb{C}}^{nk}$
for $0\leq s\leq k-1$. Then

$\phi_{t}:(\bm{z}_{0},\ldots,\bm{z}_{k-1})\mapsto\sum_{s=0}^{k-1}e^{st}\bm{z}_{s}\left/|\sum_{s=0}^{k-1}e^{st}\bm{z}_{s}|\right.$

defines the flow of a $G$-equivariant gradient like vector field for (2.1).
Then the moduli space of Morse trajectories on
$\partial({\mathbb{D}}^{2nk})/G$ connecting the critical submanifolds of
Morse-Bott indices $2jn$ and $2(j+1)n$ is diffeomorphic to $(S^{2n-1}\times
S^{2n-1})/G$, with the evaluation maps to the two critical submanifolds given
by the two projections. To see this, note that the moduli space of Morse
trajectories on $\partial({\mathbb{D}}^{2nk})$ connecting the critical
submanifolds of Morse-Bott indices $2jn$ and $2(j+1)n$ is diffeomorphic to
$S^{2n-1}\times S^{2n-1}$ by the evaluation maps at the two ends (as it is
direct to check that given two endpoints, there is a unique Morse trajectory).
Then the claim follows from the fact that the evaluation maps are obviously
$G$-equivariant. Then the Morse-Bott cochain complex of $BG$ is
$\prod_{j=1}^{\infty}H^{*}(S^{2n-1}/G;{\mathbb{Z}})[-2jn]$, i.e. the direct
product of the cohomology of critical manifolds with degree shifted by the
Morse-Bott indices. The differential $\delta$ is defined by the push-pull
formula

$\delta:H^{*}(S^{2n-1}/G;{\mathbb{Z}})[-2jn]\to
H^{*}(S^{2n-1}/G;{\mathbb{Z}})[-2(j+1)n],\quad
x\mapsto(ev_{+})_{*}\circ(ev_{-})^{*}(x)$ (2.2)

where $ev_{\pm}$ are the evaluation maps on the moduli spaces connecting the
critical manifolds of Morse-indices $2jn$ and $2(j+1)n$, i.e. the natural
projections $(S^{2n-1}\times S^{2n-1})/G\to S^{2n-1}/G$. Then by degree
reasons, the only non-trivial term of $\delta$ is $\delta(\mathrm{vol}[-2jn])$
for the generator $\mathrm{vol}\in H^{2n-1}(S^{2n-1}/G;{\mathbb{Z}})$, which
is a multiple of $1\in H^{0}(S^{2n-1}/H;{\mathbb{Z}})$. The pushforward
$(ev_{+})_{*}$ is the pushforward of homology composed with Poincaré
dualities, hence we can compute that $\delta(\mathrm{vol}[-2jn])=|G|\cdot
1[-2(j+1)n]$, $H^{2n-1+2jn}(BG;{\mathbb{Z}})=0$,
$H^{2(j+1)n}(BG;{\mathbb{Z}})={\mathbb{Z}}/|G|$. As a consequence, we have
that $H^{*}(BG;{\mathbb{Z}})=H^{*+2n}(BG;{\mathbb{Z}})$ for $*>0$. This puts
strong restrictions on which finite group $G$ is allowed to have an isolated
singularity ${\mathbb{C}}^{n}/G$.

The push-pull formula (2.2) is a special case of a more general statement,
where the push-pull on cohomology is the differential between critical
manifolds with critical values next to each other. In particular, the moduli
space of Morse trajectories between them is a closed manifold, hence we have
Poincaré duality and the pushforward map. In general, there are more
components to the differential between critical manifolds with critical values
not next to each other, which can be roughly interpreted as the correction
terms from the homological perturbation lemma. However, in the special case
here, we do not need to consider those by degree reasons. The real coefficient
version of such theory was worked out in [77].

Instead of working out the integer version of [77], we can prove (2.2)
directly. We start by considering the case on $S^{4n-1}/G=:X_{2}$ with two
critical manifolds of Morse-Bott indices $0$ and $2n$. Let $X_{1}$ denote the
sub-level set $f^{-1}((-\infty,1/2])$. Then $\partial
X_{1}\simeq(S^{2n-1}\times S^{2n-1})/G$ can be identified with the moduli
space of Morse trajectories. In view of the long exact sequence from the pair
$(X_{2},X_{1})$, we have
$H^{j}(X_{2};{\mathbb{Z}})=H^{j}(X_{1};{\mathbb{Z}})=H^{j}(S^{2n-1}/G;{\mathbb{Z}})$
when $j<2n-1$ and
$H^{j}(X_{2};{\mathbb{Z}})=H^{j}(X_{2},X_{1};{\mathbb{Z}})=H^{j-2n}(S^{2n-1}/G;{\mathbb{Z}})$
when $j>2n$, where the last identification is from excision and the Thom
isomorphism. The remaining term is determined by the following exact sequence

$0\to H^{2n-1}(X_{2};{\mathbb{Z}})\to H^{2n-1}(X_{1};{\mathbb{Z}})\to
H^{2n}(X_{2},X_{1};{\mathbb{Z}})\mapsto H^{2n}(X_{2})\to 0.$

Note that the middle map factors as

$H^{2n-1}(X_{1};{\mathbb{Z}})\to H^{2n-1}(\partial X_{1};{\mathbb{Z}})\to
H^{2n}(\overline{X_{2}\backslash X_{1}},\partial
X_{1};{\mathbb{Z}})\stackrel{{\scriptstyle\text{excision}}}{{=}}H^{2n}(X_{2},X_{1};{\mathbb{Z}})$

Since $H^{2n-1}(S^{2n-1}/G;{\mathbb{Z}})=H^{2n-1}(X_{1};{\mathbb{Z}})$ and the
pullback $H^{2n-1}(X_{1};{\mathbb{Z}})\mapsto H^{2n-1}(\partial
X_{1};{\mathbb{Z}})$ can be understood as $(ev_{-})^{*}$ in (2.2), it suffices
to show that $H^{2n-1}(\partial X_{1};{\mathbb{Z}})\to
H^{2n}(\overline{X_{2}\backslash X_{1}},\partial
X_{1};{\mathbb{Z}})\stackrel{{\scriptstyle\text{Thom
iso}}}{{=}}H^{0}(S^{2n-1}/G;{\mathbb{Z}})$ is the pushforward $(ev_{+})_{*}$.
To see this, note that we have a commutative diagram

$\textstyle{H^{2n-1}(\partial
X_{1};{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{PD}}$$\textstyle{H^{2n}(\overline{X_{2}\backslash
X_{1}},\partial
X_{1};{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{LD}}$Thom
iso$\textstyle{H^{0}(S^{2n-1}/G;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{PD}}$$\textstyle{H_{2n-1}(\partial
X_{1};{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{2n-1}(\overline{X_{2}\backslash
X_{1}};{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{H_{2n-1}(S^{2n-1}/G;{\mathbb{Z}})}$

where $\mathrm{PD},\mathrm{LD}$ are Poincaré duality and Lefschetz duality,
and $\pi_{*}$ is the pushforward by the projection of the normal bundle of
$f^{-1}(1)$ and all of them are isomorphisms. Since the composition
$H_{2n-1}(\partial X_{1};{\mathbb{Z}})\to H_{2n-1}(\overline{X_{2}\backslash
X_{1}};{\mathbb{Z}})\stackrel{{\scriptstyle\pi_{*}}}{{\to}}H_{2n-1}(S^{2n-1}/G;{\mathbb{Z}})$
is the pushforward of $ev_{+}$ on homology, the claim follows. The general
case follows from looking at the pair of sub-level sets
$f^{-1}((-\infty,j+\frac{1}{2}])$ and $f^{-1}((-\infty,j+\frac{3}{2}])$, which
up to homotopy is $(X_{2},X_{1})$ after $2jn$ suspensions.

  2. (2)

The above discussion can be generalized to the topological setting. Assume $G$
acts $S^{2n-1}$ freely. Then the diagonal $G$ action on the join
$S^{4n-1}\simeq S^{2n-1}\star S^{2n-1}:=S^{2n-1}\times
S^{2n-1}\times[0,1]/\sim$ is again free. We can inductively apply the argument
above to obtain a model for $BG$. One can chop up $S^{4n-1}/G$ along the
middle slice $(S^{2n-1}\times S^{2n-1}\times\\{1/2\\})/G$ (the moduli space of
“Morse trajectories”) to compute the cohomology using the long exact sequence
of pairs as before. Similarly, we also have the periodicity on
$H^{*}(BG;{\mathbb{Z}})$.

  3. (3)

The above discussion is an interpretation of a classical result on free
actions on spheres using Morse theory. Smith [64] showed that if a finite
group $G$ acts on $S^{n}$ freely222If $n$ is even, it is easy to show that $G$
has to be ${\mathbb{Z}}/2$ and reverses the orientation., then any abelian
subgroup of $G$ must be cyclic. This turns out to be equivalent to the
periodicity of the cohomology of $G$ [17, Theorem 11.6] due to Artin and Tate.

  4. (4)

Groups that admit complex representations with isolated singularities are
completely classified, c.f. [70, Theorem 6.1.11,6.3.1].

######  Proposition 2.3.

Let $G$ be a finite abelian group acting on ${\mathbb{C}}^{n}$ with an
isolated singularity, then $G={\mathbb{Z}}/m$ and the action up to conjugation
is given by sending the generator to multiplication of
$\operatorname{diag}(e^{\frac{2\pi a_{1}\mathbf{i}}{m}},\ldots,e^{\frac{2\pi
a_{n}\mathbf{i}}{m}})$ for $1\leq a_{i}<m$ coprime to $m$.

###### Proof.

In our linear setting333Even if we only assume $G$ is a continuous free action
on $S^{2n-1}$, $G$ must be cyclic by Remark 2.2 as $H^{*}(BG;{\mathbb{Z}})$ is
not periodic if $G$ is a finite non-cyclic abelian group., $G$ being abelian
implies that they can be simultaneously diagonalized. Since the fixed subspace
of every nontrivial element of $G$ is $\\{0\\}$, $G$ must be cyclic with the
action in the statement. ∎

We denote such lens space quotient $S^{2n-1}/G$ in Proposition 2.3 by
$L(m;a_{1},\ldots,a_{n})$.

######  Proposition 2.4.

Under the ring isomorphism
$H^{*}(L(m;a_{1},\ldots,a_{n});{\mathbb{Z}})=H^{*}(B{\mathbb{Z}}/m;{\mathbb{Z}})$
for $*<2n-1$, the total Chern class of $(L(m;a_{1},\ldots,a_{n}),\xi_{\rm
std})$ is given by $\prod_{i=1}^{n}(1+a_{i}u)$ where $u$ is a generator of
$H^{2}(B{\mathbb{Z}}/m;{\mathbb{Z}})={\mathbb{Z}}/m$.

Although the choice of the generator will be made clear in the proof, we do
not need that level of precision as properties such as whether $c_{k}(\xi_{\rm
std})=0$ and whether $c_{k}(\xi_{\rm std})$ generates
$H^{2k}(L(m;a_{1},\ldots,a_{n});{\mathbb{Z}})$ are independent of the choice
of the generator.

###### Proof of Proposition 2.4.

We use $\underline{{\mathbb{C}}}$ to denote the trivial complex bundle. Note
that $\xi_{\rm std}\oplus\underline{{\mathbb{C}}}$ is isomorphic
$\underline{{\mathbb{C}}}^{n}/({\mathbb{Z}}/m)$, where ${\mathbb{Z}}/m$ acts
diagonally as in Proposition 2.3. Since
$H^{*}(L(m;a_{1},\ldots,a_{n});{\mathbb{Z}})=H^{*}(B{\mathbb{Z}}/m;{\mathbb{Z}})$
for $*<2n-1$. We can compute the total Chern class of
$\underline{{\mathbb{C}}}^{n}/({\mathbb{Z}}/m)$ on
$B{\mathbb{Z}}/m=\varinjlim\partial({\mathbb{D}}^{2nk})/({\mathbb{Z}}/m)$.
Then the claim follows from the fact that
$c(\underline{{\mathbb{C}}}/({\mathbb{Z}}/m))=1+au$ for the ${\mathbb{Z}}/m$
action on ${\mathbb{C}}$ given by $z\mapsto e^{\frac{2\pi a\mathbf{i}}{m}}z$.
Here $u$ is the first Chern class of
$\underline{{\mathbb{C}}}/({\mathbb{Z}}/m)$ for the ${\mathbb{Z}}/m$ action on
${\mathbb{C}}$ given by $z\mapsto e^{\frac{2\pi\mathbf{i}}{m}}z$, which is a
generator of
$H^{2}(B{\mathbb{Z}}/m;{\mathbb{Z}})=H^{2}(\varinjlim\partial({\mathbb{D}}^{2nk})/({\mathbb{Z}}/m);{\mathbb{Z}})$.
∎

######  Proposition 2.5.

When $n\geq 3$, $(S^{2n-1}/G,\xi_{\rm std})$ is not Weinstein fillable.

###### Proof.

If $W$ is a Weinstein filling of $(S^{2n-1}/G,\xi_{\rm std})$, since $n\geq
3$, we have $\pi_{1}(W)=\pi_{1}(S^{2n-1}/G)=G$. Let $\widetilde{W}$ be the
universal cover of $W$. Then $\widetilde{W}$ is a Weinstein filling of
$(S^{2n-1},\xi_{\rm std})$, which by the Eliashberg-Floer-McDuff theorem [46]
is diffeomorphic to a ball in ${\mathbb{C}}^{n}$. Now a non-trivial deck
transformation acts freely, contradicting the Brouwer fixed-point theorem. ∎

In fact, most of them are not even almost Weinstein fillable.

######  Proposition 2.6.

$(S^{2n-1}/G,J_{\rm std})$ is not almost Weinstein fillable when $n\geq 4$.

###### Proof.

Assume for contradiction that there is an almost Weinstein filling $W$. Let
$g\in G$ a non-trivial element, we consider the covering space $\widetilde{W}$
of $W$ corresponding to the subgroup
$\langle\,g\,\rangle\subset\pi_{1}(W)=\pi_{1}(S^{2n-1}/G)=G$. As we assume
$n\geq 3$, $\widetilde{W}$ is also constructed using only handles up to
dimension $n$ with $\partial\widetilde{W}=S^{2n-1}/\langle\,g\,\rangle$. Since
$H^{*}(\partial\widetilde{W};{\mathbb{Z}})=H^{*}(B{\mathbb{Z}}/\operatorname{ord}(g);{\mathbb{Z}})$
as a ring in degrees less than $2n-1$, we have a class $\alpha$ in
$H^{2}(\partial\widetilde{W};{\mathbb{Z}})$ such that $\alpha^{n-1}\neq 0$.
Since $\widetilde{W}$ is almost Weinstein and $n\geq 4$, we know that
$H^{2}(\widetilde{W};{\mathbb{Z}})\to
H^{2}(\partial\widetilde{W};{\mathbb{Z}})$ is an isomorphism. Therefore
$\alpha=\beta|_{\partial\widetilde{W}}$ for $\beta\in
H^{2}(\widetilde{W};{\mathbb{Z}})$ and $\beta^{n-1}\neq 0$ as
$\beta^{n-1}|_{\partial\widetilde{W}}=\alpha^{n-1}\neq 0$. This contradicts
with the fact that $H^{2n-2}(\widetilde{W};{\mathbb{Z}})=0$ when $n\geq 4$,
since $\widetilde{W}$ is built from handles of indexes at most $n$. ∎

######  Example 2.7.

Another intersecting class of examples of isolated quotient singularities
comes from finite subgroups of $SU(2)$, namely the $A,D,E$ types of subgroups.
The associated quotient singularity ${\mathbb{C}}^{2}/G$ is canonical and has
a minimal (also crepant) resolution whose exceptional locus has an
intersection pattern described by the associated Dynkin diagram. Moreover, the
contact link $(S^{3}/G,\xi_{\rm std})$ has a Weinstein filling from plumbings
of $T^{*}S^{2}$ according to the associated Dynkin diagram. The $A_{m-1}$ type
subgroup is ${\mathbb{Z}}/m$, which acts on ${\mathbb{C}}^{2}$ by sending the
generator to
$\operatorname{diag}(e^{\frac{2\pi\mathbf{i}}{m}},e^{\frac{2(m-1)\pi\mathbf{i}}{m}})$.
Details of $D,E$ types of subgroups can be found in [24]. Given any finite
$G\subset SU(2)$, it acts diagonally on $({\mathbb{C}}^{2})^{n}$ giving rise
to an isolated terminal quotient singularity if $n>1$, see Proposition 7.15.
The group cohomology ring $H^{*}(BG;{\mathbb{Z}})$ is computed, e.g. in [67].
In particular, we have a ring isomorphism
$\oplus_{i=0}^{\infty}H^{4i}(BG;{\mathbb{Z}})={\mathbb{Z}}[v]/\langle\,|G|\cdot
v\,\rangle$ with $\deg(v)=4$.

######  Proposition 2.8.

Let $G\subset SU(2)$ be a finite subgroup, then $G$ acts on
$({\mathbb{C}}^{2})^{n}$ by the natural diagonal action. The total Chern class
of $(S^{4n-1}/G,\xi_{\rm std})$ is given by

$(1-v)^{n}\mod\langle\,|G|,v^{n}\,\rangle,$

where $v\in
H^{4}(S^{4n-1}/G;{\mathbb{Z}})=H^{4}(BG,{\mathbb{Z}})={\mathbb{Z}}/|G|$ is a
generator.

###### Proof.

We view ${\mathbb{C}}^{2}$ as the space of quaternions ${\mathbb{H}}$, then
the $SU(2)\simeq S^{3}$ action on ${\mathbb{C}}^{2}$ is identified with the
quaternion product by unit vectors in ${\mathbb{H}}$. Let $V$ denote the
tautological quaternion line bundle over $\mathbb{H}\mathbb{P}^{n-1}$, then
$\underline{{\mathbb{H}}}^{n}/{\mathbb{H}}^{\times}=T\mathbb{H}\mathbb{P}^{n-1}\oplus\underline{{\mathbb{H}}}=\oplus^{n}\operatorname{Hom}_{{\mathbb{H}}}(V,\underline{{\mathbb{H}}})$,
where the first $\underline{{\mathbb{H}}}^{n}$ is the tangent bundle over
${\mathbb{H}}^{n}\backslash\\{0\\}$. Note that the underlying complex bundle
$\left(\operatorname{Hom}_{{\mathbb{H}}}(V,\underline{{\mathbb{H}}})\right)_{{\mathbb{C}}}$
using $\mathbf{i}\in{\mathbb{H}}^{\times}$ is isomorphic to
$\operatorname{Hom}_{{\mathbb{C}}}(V_{{\mathbb{C}}},\underline{{\mathbb{C}}})\simeq\overline{V_{{\mathbb{C}}}}$.
We use $\pi$ to denote the map $S^{4n-1}/G\to\mathbb{H}\mathbb{P}^{n-1}$
induced from the Hopf fibration $S^{4n-1}\to\mathbb{H}\mathbb{P}^{n-1}$. Then
the complex bundle $\xi_{\rm std}\oplus\underline{{\mathbb{C}}}$ is isomorphic
to $\pi^{*}(\underline{{\mathbb{H}}}^{n}/{\mathbb{H}}^{\times})$ as a complex
bundle, where $\underline{{\mathbb{H}}}^{n}/{\mathbb{H}}^{\times}$ is equipped
with complex structure from $\mathbf{i}\in{\mathbb{H}}^{\times}$. Since
$c(V_{{\mathbb{C}}})=c(\overline{V_{{\mathbb{C}}}})=1-v$ for $v\in
H^{4}(\mathbb{H}\mathbb{P}^{n-1};{\mathbb{Z}})$ is a generator and $\pi^{*}$
is reduction modulo $|G|$ on $H^{4i}(\mathbb{H}\mathbb{P}^{n-1};{\mathbb{Z}})$
for $1\leq i\leq n-1$, the claim follows. ∎

######  Example 2.9.

Assume $G\subset O(n)$, such that the only fixed point of the $G$-action on
$\mathbb{R}^{n}$ is $0$. If we view $G$ as in $O(n)\subset SU(n)$, the
quotient singularity ${\mathbb{C}}^{n}/G$ is isolated. Moreover, since each
$g\in G$ is represented by a real matrix, the eigenvalues must be conjugated
complex numbers in pairs. Therefore $\operatorname{age}(g)=\frac{n}{2}$ (see
(2.3)) for any $g\neq\operatorname{id}$ and ${\mathbb{C}}^{n}/G$ is terminal
if $n>2$ by Proposition 2.12. If $M$ is a closed orbifold with isolated
singularities, then $T^{*}M$ is a symplectic orbifold with singularities of
such type.

###  2.2. Reeb dynamics on $S^{2n-1}/G$

Since $G\subset U(n)$, we have that $\alpha=\frac{1}{2}\sum(x_{i}{\rm
d}y_{i}-y_{i}{\rm d}x_{i})=-\frac{1}{2}r{\rm d}r\circ I$ is $G$-invariant,
where $I$ is the standard complex structure on ${\mathbb{C}}^{n}$. Hence
$\alpha$ descends to a contact form $\alpha_{\rm std}$ on
$(S^{2n-1}/G,\xi_{\rm std})$. We now recall the description of the Reeb
dynamics of $\alpha_{\rm std}$ from [49, §2.1]. Since
$c^{{\mathbb{Q}}}_{1}(\xi_{\rm std})=0$ and $\pi_{1}(S^{2n-1}/G)$ is finite,
we can assign a well-defined rational generalized Conley-Zehnder index to
every Reeb orbit following [48, §4.1], see also [38, §3.3].

For $g\in U(n)$, we can diagonalize $g$ as $\operatorname{diag}(e^{\frac{2\pi
a_{1}\mathbf{i}}{\mathrm{o}(g)}},\ldots,e^{\frac{2\pi
a_{n}\mathbf{i}}{\mathrm{o}(g)}})$, where $\mathrm{o}(g)$ is the order of $g$
and $1\leq a_{i}<\mathrm{o}(g)$. Then we define

$\operatorname{age}(g):=\sum_{i=1}^{n}\frac{a_{i}}{\mathrm{o}(g)}\in{\mathbb{Q}}.$
(2.3)

$\operatorname{age}(g)$ only depends on the conjugacy class $(g)$.

For $l>0\in\mathbb{R}$ and $g\in V$, we define

$V_{g,l}=\left\\{v\in{\mathbb{C}}^{n}\left|g(v)=e^{l\mathbf{i}}v\right.\right\\}.$

It is easy to check that $h(V_{g,l})=V_{hgh^{-1},l}$ for $h\in G$. Since the
Reeb flow of $\alpha$ on $S^{2n-1}$ is given by
$\phi_{t}(z)=e^{t\mathbf{i}}z$, we have the following properties of the Reeb
dynamics of $\alpha_{\rm std}$ on $S^{2n-1}/G$.

  1. (1)

Let $B_{g,l}=\pi_{G}(V_{g,l}\cap S^{2n-1})$, where $\pi_{G}$ is the quotient
map $S^{2n-1}\to S^{2n-1}/G$. Then, $B_{g,l}=(V_{g,l}\cap S^{2n-1})/C(g)$ is a
submanifold of $S^{2n-1}/G$ of dimension $2\dim_{{\mathbb{C}}}V_{g,l}-1$,
where $C(g)$ is centralizer of $g$. Moreover, $B_{g,l}=B_{hgh^{-1},l}$, hence,
for $(g)\in\operatorname{Conj}(G)$, one can define $B_{(g),l}$ to just be
$B_{g,l}$.

  2. (2)

$\displaystyle\sqcup_{(g),l}B_{(g),l}$ is the space of all _parameterized_
Reeb orbits of $\alpha_{\rm std}$. For $z\in B_{(g),l}$, the time $l$ Reeb
flow from $z$ is an orbit with free homotopy class
$(g)\in\operatorname{Conj}(G)=\operatorname{Conj}(\pi_{1}(S^{2n-1}/G))$.

  3. (3)

The generalized Conley-Zehnder index of the family $B_{(g),l}$ is given in
[49, Theorem 2.6] by

$\mu_{\rm
CZ}(B_{(g),l})=n-2\operatorname{age}(g)+2\sum_{l^{\prime}<l}\dim_{{\mathbb{C}}}V_{g,l^{\prime}}+\dim_{{\mathbb{C}}}V_{g,l}.$

Then we have $\mu_{\rm CZ}(B_{(g),l+2k\pi})=\mu_{\rm CZ}(B_{(g),l})+2kn$.

  4. (4)

The space of _unparameterized_ Reeb orbits is the orbifold $B_{(g),l}/S^{1}$,
which is Morse-Bott non-degenerate in the sense of [10, Definition 1.7]. By
[10, Lemma 2.3], we can find a small perturbation of $\alpha_{\rm std}$, such
that all Reeb orbits with period smaller than any given $R\gg 0$ are non-
degenerate and correspond to critical points of auxiliary Morse functions on
$B_{(g),l}/S^{1}$. The minimal rational Conley-Zehnder index is given by

$\min_{(g)\in\operatorname{Conj}(G)}\left\\{n-2\operatorname{age}(g)+1\right\\}.$

After such a perturbation, the ${\mathbb{Z}}/2$ Conley-Zehnder index of the
orbit in each conjugacy class with the minimal rational Conley-Zehnder index
(or period) is $n+1$. One way to see this is through direct computation. There
is a roundabout as follows: in the computation of the symplectic cohomology of
${\mathbb{C}}^{n}/G$ in [38, Theorem B], there are differentials from those
orbits with minimal rational Conley-Zehnder index to constant twisted sectors
at the origin. Now by [38, Example 3.7], those constant twisted sectors have
${\mathbb{Z}}/2$ Conley-Zehnder index $n$, which implies the claim.

In §3, we will consider Hamiltonians on hypothetical fillings of
$(S^{2n-1}/G,\xi_{\rm std})$ that are in the form of $H(r)$, where
$r\in[1,\infty)$ is the radical coordinate determined by the contact form
$\alpha_{\rm std}$. We will assume that with $H^{\prime\prime}(r)\geq 0$ for
$r\geq 1$ and $H^{\prime}(r)=a\notin\frac{2\pi{\mathbb{N}}}{|G|}$ for
$x>1+\epsilon$ for some $\epsilon>0$. Then non-constant Hamiltonian orbits of
$H$ are parameterized by $B_{(g),l}$ for $l<a$. In particular, the symplectic
cohomology can be equivalently defined using the cascades construction, c.f.
[11, 26], i.e. we need to choose a Morse function $f_{(g),l}$ for each
$B_{(g),l}$ and the generators from non-constant orbits of $H$ are replaced by
critical points of $f_{(g),l}$. The Conley-Zehnder index of a critical point
$x$ of $f_{(g),l}$ is given by

$\mu_{CZ}(x)=n-2\operatorname{age}(g)+2\sum_{l^{\prime}<l}\dim_{{\mathbb{C}}}V_{g,l^{\prime}}+1+\operatorname{ind}(x),$
(2.4)

where $\operatorname{ind}(x)$ is the Morse index of $x$. With such
conventions, the cascade flow lines in $B_{(g),l}$ are _negative_ gradient
flow lines of $f_{(g),l}$ in the cascade construction. On the other hand, if
we use $f_{(g),l}$ to perturb the Hamiltonian following [11], then each
critical point of $x$ gives rise to a non-degenerate Hamiltonian orbit
$\gamma_{x}$, whose Conley-Zehnder index is given by (2.4). The symplectic
action of $\gamma_{x}$ is greater than that of $\gamma_{y}$ iff
$f_{(g),l}(x)<f_{(g),l}(y)$. Moreover, we may choose $f_{(g),l}$ to have a
unique minimum. Then for each conjugacy class $(g)$, there is a unique
generator $x_{(g)}$ with the minimal rational Conley-Zehnder index
$n-2\operatorname{age}(g)+1$, or equivalently minimal period, or maximal
symplectic action. Those generators will have critical roles in §3, 4.

In general, given a $2n-1$ dimensional contact manifold $(Y,\xi)$ with
$H_{1}(Y;{\mathbb{Z}})$ is torsion and $c^{\mathbb{Q}}_{1}(Y)=0$, then every
Reeb orbit can be assigned with a well-defined rational Conley-Zehnder index
$\mu_{CZ}(\gamma)$444The Reeb orbit is not necessarily non-degenerate, the
Conley-Zehnder index is defined in [58].. Then we define the lower SFT index
by

$\mathrm{lSFT}(\gamma):=\mu_{CZ}(\gamma)+(n-3)-\frac{1}{2}\dim\ker({\rm
D}_{\gamma(0)}\phi_{L}|_{\xi}-\operatorname{id}),$ (2.5)

where $\phi_{L}$ is the time $L$ flow of the Reeb vector field and $L$ is the
period of $\gamma$. When $\gamma$ is non-degenerate, $\mathrm{lSFT}(\gamma)$
is the standard SFT grading. When $\gamma$ comes from a Morse-Bott family,
then $\mathrm{lSFT}$ is the minimal SFT grading of the non-degenerate orbits
after a small perturbation. Following [48], the highest minimal SFT index is
defined as

$\mathrm{hmi}(Y,\xi):=\sup_{\alpha}\inf_{\gamma}\\{\mathrm{lSFT}(\gamma)\\}.$

###  2.3. Chen-Ruan cohomology of ${\mathbb{C}}^{n}/G$

From a string theory point of view, the correct cohomology theory of an almost
complex orbifold is the Chen-Ruan cohomology introduced in [19], which can be
viewed as the constant part of the orbifold quantum product introduced in
[18]. The Chen-Ruan cohomology is the replacement of the ordinary cohomology
in the definition of orbifold symplectic cohomology [38]. We refer readers to
[4] for the definition and properties of the Chen-Ruan cohomology.

For orbifold ${\mathbb{C}}^{n}/G$, with $G\subset U(n)$ having only isolated
singularity at $0$, we have

$H^{*}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;R)=\oplus_{(g)\in\operatorname{Conj}(G)}R[-2\operatorname{age}(g)].$
(2.6)

####  2.3.1. Product on Chen-Ruan cohomology

To describe the product structure on
$H^{*}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;R)$, we denote by $[(g)]$ the
(positive) generator in cohomology for the conjugacy class $(g)$. For
$(g_{1}),(g_{2})\neq(\operatorname{id})$, one can define

$I_{g_{1},g_{2}}=\\{(h_{1},h_{2})|h_{i}\in(g_{i}),\operatorname{age}(h_{1})+\operatorname{age}(h_{2})=\operatorname{age}(h_{1}h_{2})\\}/(h_{1},h_{2})\sim(gh_{1}{g^{-1}},gh_{2}g^{-1}).$

Then by [4, Examples 4.26 and 4.27], we have

$[(g_{1})]\cup[(g_{2})]=\sum_{(h_{1},h_{2})\in
I_{g_{1},g_{2}}}\frac{|C(h_{1}h_{2})|}{|C(h_{1})\cap
C(h_{2})|}[(h_{1}h_{2})],$ (2.7)

for $(g_{1}),(g_{2})\neq(\operatorname{id})$, while $[(\operatorname{id})]$ is
the ring unit in $H^{*}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;R)$.

The formula (2.7) can be explained through holomorphic curves as follows. Just
like the manifold case, the cup product can be defined by counting moduli
spaces of punctured holomorphic curves of zero energy with $2$ inputs
(positive punctures) and $1$ output (negative puncture). But in the orbifold
case at hand, there are constant holomorphic curves which, near punctures, are
asymptotic to twisted sectors at the singularity. In fact, each
$(h_{1},h_{2})\in I_{g_{1},g_{2}}$ corresponds to a constant holomorphic curve
with two positive punctures asymptotic to $(g_{1}),(g_{2})$ and one negative
puncture asymptotic to $(h_{1}h_{2})$, which is cut out transversely (by the
conditions on the $\operatorname{age}$ invariants). Moreover, such a
holomorphic curve is an orbifold point in the space of curves in an orbifold
with isotropy group $C(h_{1})\cap C(h_{2})\cap C(h_{1}h_{2})=C(h_{1})\cap
C(h_{2})$. In view of the pull-push definition of the structural constants in
[38, §2.1.3], the coefficient is given by ${|C(h_{1}h_{2})|}/{|C(h_{1})\cap
C(h_{2})|}$.

####  2.3.2. Coproduct on Chen-Ruan cohomology

By the same philosophy, we can define the Chen-Ruan coproduct on the Chen-Ruan
cohomology, which is dual to the Chen-Ruan cohomology through the
Poincaré/Lefschetz duality on Chen-Ruan cohomology. In the case of
${\mathbb{C}}^{n}/G$, we define

$I_{g}=\\{(h_{1},h_{2})|h\in(g),h_{1}h_{2}=h,\operatorname{age}(h_{1})+\operatorname{age}(h_{2})=\operatorname{age}(h)+n\\}/(h_{1},h_{2})\sim(ph_{1}{p^{-1}},ph_{2}p^{-1}).$

Then we define the Chen-Ruan coproduct to be

$\Delta[(g)]=\sum_{(h_{1},h_{2})\in
I_{g}}\frac{|C(h_{1})|\cdot|C(h_{2})|}{|C(h_{1})\cap
C(h_{2})|}[(h_{1})]\otimes[(h_{2})].$ (2.8)

Here, as before, each element in $I_{g}$ corresponds to a constant holomorphic
curve with two negative punctures asymptotic to $(h_{1}),(h_{2})$ and one
positive puncture asymptotic to $(g)$, which is cut out transversely with
isotropy group $C(h_{1})\cap C(h_{2})$. The coefficient is again a consequence
of the pull-push definition of the structural constants on orbifolds. It is
indeed dual to the Chen-Ruan product on the _compactly supported_ Chern Ruan
cohomology considering the Lefschetz duality [4, §4.2] on the inertia orbifold
of ${\mathbb{C}}^{n}/G$ has coefficients from the isotropy groups.

As a special case, we have

$\Delta[(\operatorname{id})]=\sum_{(g)\neq(\operatorname{id})\in\operatorname{Conj}(G)}|C(g)|[(g)]\otimes[(g^{-1})].$

In other words, the “orbifold intersection form” is non-degenerate.

###  2.4. Minimal discrepancy number of ${\mathbb{C}}^{n}/G$

Let $A$ be a singularity, that is isolated at $0$. Let $\pi:\widetilde{A}\to
A$ be a resolution (if $0$ is smooth, we need to blow up once), such that
$\pi^{-1}(0)$ a union of smooth normal crossing divisors $E_{i}$. We view the
canonical bundle $K_{\widetilde{A}}$ of $\widetilde{A}$ as a
${\mathbb{Q}}$-Cartier divisor. We say $A$ is numerically
${\mathbb{Q}}$-Gorenstein if there exists a ${\mathbb{Q}}$-Cartier divisor
$K^{\rm{num}}_{\widetilde{A}}=\sum a_{i}E_{i}$ with the property that
$C\cdot(K^{\rm{num}}_{\widetilde{A}}-K_{\widetilde{A}})=0$ for any projective
algebraic curve $C$ in $\pi^{-1}(0)$. Here $a_{i}\in{\mathbb{Q}}$ is called
the discrepancy of $E_{i}$. Following [48, Lemma 3.3], $A$ being numerically
${\mathbb{Q}}$-Gorenstein is equivalent to that
$c_{1}^{{\mathbb{Q}}}(L_{A},\xi)=0$, where $L_{A}$ is the link of the
singularity and $\xi$ is induced contact structure [68]. Moreover,
$c^{{\mathbb{Q}}}_{1}(\widetilde{A})$ viewed as a class in
$H^{2}_{c}(\widetilde{A};{\mathbb{Q}})=H^{2}(\widetilde{A},L_{A};{\mathbb{Q}})$
is the Lefschetz dual to $\sum a_{i}[E_{i}]$, where $a_{i}$ is the discrepancy
of $E_{i}$ above. Since $H^{2}(S^{2n-1}/G;{\mathbb{Q}})=0$, isolated quotient
singularities are numerically ${\mathbb{Q}}$-Gorenstein.

######  Definition 2.10 ([48, Definition 3.4]).

The minimal discrepancy number $\mathrm{md}(A,0)$ is the given by infimum of
$a_{i}$ over all resolutions. A singularity is called terminal if the minimal
discrepancy number is positive. A singularity is called canonical if the
minimal discrepancy number is non-negative.

A beautiful theorem of McLean relates the algebraic geometry of a singularity
with the contact geometry of the link, which applies to quotient
singularities.

######  Theorem 2.11 (McLean [48, Theorem 1.1]).

Let $A$ have a normal isolated singularity at $0$ that is numerically
${\mathbb{Q}}$-Gorenstein with $H^{1}(L_{A};{\mathbb{Q}})=0$. Then:

  * •

If $\mathrm{md}(A,0)\geq 0$ then
$\mathrm{hmi}(L_{A},\xi_{A})=2\mathrm{md}(A,0)$;

  * •

If $\mathrm{md}(A,0)<0$ then $\mathrm{hmi}(L_{A},\xi_{A})<0$.

As an application of McLean’s theorem, we obtain a symplectic proof of the
Reid–Shepherd-Barron–Tai criterion [56, (3.1)], [66, Proposition 3.2] for
isolated quotient singularities.

######  Proposition 2.12.

The isolated singularity ${\mathbb{C}}^{n}/G$ is terminal (canonical) if and
only if $\min_{g\neq\operatorname{id}\in G}\\{\operatorname{age}(g)\\}>1$
($\geq 1$). More precisely, we have
$\operatorname{md}({\mathbb{C}}^{n}/G)=\min_{g\neq\operatorname{id}\in
G}\\{\operatorname{age}(g)\\}-1$, if the singularity is canonical.

###### Proof.

By the standard Reeb dynamics on $S^{2n-1}/G$ in §2.2, we have
$\mathrm{hmi}(S^{2n-1}/G,\xi_{\rm std})\geq\min_{g\in
G}\\{2n-2\operatorname{age}(g)-2\\}$. On the other hand, by [38, Theorem B],
$SH^{*}({\mathbb{C}}^{n}/G;{\mathbb{Q}})=0$, which implies that
$SH^{*}_{S^{1}}(C^{n}/G;{\mathbb{Q}})=0$ by the Gysin exact sequence [12,
Theorem 1.1] (this holds in the exact orbifold setting from the discussion in
[38, §3.6, 3.7 ]). As a consequence, we have
$SH^{*-1}_{+,S^{1}}({\mathbb{C}}^{n}/G;{\mathbb{Q}})=H^{*}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;{\mathbb{Q}})\otimes({\mathbb{Q}}[u,u^{-1}]/\langle\,u\,\rangle)$.
In particular, $SH^{*}_{+,S^{1}}({\mathbb{C}}^{n}/G;{\mathbb{Q}})$ is
nontrivial in degree $2\operatorname{age}(g)-1$ for any $g\in G$. If
$\mathrm{hmi}(S^{2n-1}/G,\xi_{\rm std})>\min_{g\in
G}\\{2n-2\operatorname{age}(g)-2\\}$, then there exists a contact form
$\alpha$, with $\inf_{\gamma}\\{\mathrm{lSFT}(\gamma)\\}>\min_{g\in
G}\\{2n-2\operatorname{age}(g)-2\\}$. By [48, §4.3], for arbitrarily large
period threshold $K$, any small perturbation of $\alpha$ yields a contact form
such that all Reeb orbits with period smaller than $R\gg 0$ is non-degenerate
and the SFT degree is at least $\inf_{\gamma}\\{\mathrm{lSFT}(\gamma)\\}$.
Then by [41], we have that $SH^{*}_{+,S^{1}}({\mathbb{C}}^{n}/G;{\mathbb{Q}})$
is supported in degrees at most
$2n-3-\inf_{\gamma}\\{\mathrm{lSFT}(\gamma)\\}<2n-3-\min_{g\in
G}\\{2n-2\operatorname{age}(g)-2\\}=2\max_{g\in
G}\\{\operatorname{age}(g)\\}-1$, contradiction. Therefore we have
$\mathrm{hmi}(S^{2n-1}/G,\xi_{\rm std})=\min_{g\in
G}\\{2n-2\operatorname{age}(g)-2\\}$. Finally note that
$2n-2\operatorname{age}(\operatorname{id})-2=2n-2\geq 2$ and
$\operatorname{age}(g)+\operatorname{age}(g^{-1})=2n$ for
$g\neq\operatorname{id}$. We have $\min_{g\in
G}\\{2n-2\operatorname{age}(g)-2\\}>0(\geq 0)$ is equivalent to that
$\min_{g\neq\operatorname{id}\in G}\\{\operatorname{age}(g)\\}>1(\geq 1)$.
Then the claim follows from Theorem 2.11. ∎

##  3\. Symplectic cohomology of fillings of $(S^{2n-1}/G,\xi_{\rm std})$

###  3.1. Symplectic cohomology

We first give a very brief account of the definition and properties of
symplectic cohomology to set up notations for later reference. More detailed
discussions on various aspects of symplectic cohomology can be found in [22,
63].

Given an exact filling $(W,\lambda)$, roughly speaking, symplectic cohomology
is the “Morse cohomology” of the free loop space w.r.t. the symplectic action
functional

${\mathcal{A}}(x):=-\int x^{*}\widehat{\lambda}+\int_{S^{1}}(x^{*}{H}){\rm
d}t$ (3.1)

where $H:S^{1}\times\widehat{W}\to\mathbb{R}$ is a Hamiltonian on the
completion $(\widehat{W},\widehat{\lambda}):=(W\cup\partial
W\times(1,\infty)_{r},\lambda\cup r\lambda|_{\partial W})$ such that “the
slope $\frac{{\rm d}}{{\rm d}r}H$” goes to infinity as $r$ goes to infinity.
Let $R$ be any commutative ring, the cochain complex $C^{*}(H)$ is a free
$R$-module generated by $1$-periodic orbits of $X_{H}$ and the differential is
defined by counting rigid solutions to the Floer equation. Let $x,y$ be two
generators in $C^{*}(H)$ represented by periodic orbits, we use
${\mathcal{M}}_{x,y}$ to denote the compactified moduli space of Floer
cylinders from $x$ to $y$, whose specific meaning depends on the construction
of the symplectic cohomology as follows.

  1. (1)

If $H$ is a non-degenerate Hamiltonian, i.e. all $1$-periodic orbits are non-
degenerate, then we have

${\mathcal{M}}_{x,y}=\overline{\left\\{u:\mathbb{R}_{s}\times
S^{1}_{t}\to\widehat{W}\left|\partial_{s}u+J(\partial_{u}t-X_{H})=0,\displaystyle\lim_{s\to\infty}u=x,\lim_{s\to-\infty}u=y\right.\right\\}/\mathbb{R}}.$

  2. (2)

If $H$ is Morse-Bott non-degenerate, including the case of $H=0$ on $W$
considered in [26, 73], then $x,y$ are critical points of the auxiliary Morse
functions on the Morse-Bott families of periodic orbits (which can be viewed
as a submanifold in $\widehat{W}$ using the starting point of the orbit). In
this case, ${\mathcal{M}}_{x,y}$ is the compactified moduli space of cascades
from $x$ to $y$, which can be pictorially described as

$u_{1}$$u_{2}$$x$$y$

$x$$y$$u_{1}$$u_{2}$

Figure 1. $2$ level cascades

Here the horizontal lines are _negative gradient flow_ of the auxiliary Morse
function used in (2.4) except for the bottom line of the right of the figure
above, which is the _gradient flow_ of an admissible Morse function [73,
Definition 2.1] on $W$ when $H=0$ on $W$. For the more formal description of
the moduli spaces see [26, Definition 4.7] or [72, §2] for details.

In both cases, for a generic compatible (which is also cylindrically convex to
guarantee the integrated maximum principle [2, 22]) almost complex structure
$J:S^{1}\to\operatorname{End}(T^{*}\widehat{W})$, ${\mathcal{M}}_{x,y}$ is cut
out transversely as a manifold with boundary for those with expected dimension
$\operatorname{virdim}{\mathcal{M}}_{x,y}\leq 1$. Moreover,
${\mathcal{M}}_{x,y}$ can be oriented in a coherent way such that

$\delta(x)=\sum_{y,\operatorname{virdim}{\mathcal{M}}_{x,y}=0}(\\#{\mathcal{M}}_{x,y})y$

defines a differential on $C^{*}(H)$.

######  Remark 3.1.

The symplectic cochain complex is graded by $n-\mu_{\rm CZ}(x)$. In general,
since $\mu_{\rm CZ}$ is only well-defined in ${\mathbb{Z}}/2$, symplectic
cohomology always has a ${\mathbb{Z}}/2$ grading. If
$c^{{\mathbb{Z}}}_{1}(W)=0$, then $\mu_{\rm CZ}$ is well-defined in
${\mathbb{Z}}$ for any periodic orbits with a finite order homology class.
Moreover, if $c^{{\mathbb{Q}}}_{1}(W)=0$, then $\mu_{\rm CZ}$ is well-defined
in ${\mathbb{Q}}$ (as in §2.2) for any periodic orbits with finite order
homology class and symplectic cohomology can be graded by ${\mathbb{Q}}$. For
our situation, although $c^{{\mathbb{Q}}}_{1}(S^{2n-1}/G,\xi_{\rm std})=0$ and
$\pi_{1}(S^{2n-1}/G)$ is of finite order, the Conley-Zehnder index in §2.2 is
only well-defined using a trivialization of
$\det_{{\mathbb{C}}}(\oplus^{N}\xi_{\rm std})$, which can not extend to the
filling $W$ if $c^{{\mathbb{Q}}}_{1}(W)\neq 0$. Therefore typically, the
symplectic cohomology of $W$ is not graded by ${\mathbb{Q}}$. However, the
existence of the boundary grading indeed plays an important role in the proof.

Symplectic cohomology $SH^{*}(W;R)$ has the following properties:

  1. (1)

If we choose $H$ to be $C^{2}$ small on $W$ and to be $h(r)$ on $\partial
W\times(1,\infty)_{r}$ with $h^{\prime\prime}(r)>0$, then the periodic orbits
of $X_{H}$ are either constant orbits on $W$ or non-constant orbits on
$\partial W\times(1,\infty)$, which, in pairs after a small perturbation,
correspond to Reeb orbits on $(\partial W,\lambda|_{\partial W})$. The
constant orbits generate a subcomplex corresponding to the cohomology of $W$
(i.e. the space of constant loops), and the non-constant orbits generate a
quotient complex $C^{*}_{+}(H)$, whose cohomology is called the positive
symplectic cohomology $SH^{*}_{+}(W;R)$. Then we have a tautological long
exact sequence,

$\ldots\to H^{*}(W;R)\to SH^{*}(W;R)\to SH^{*}_{+}(W;R)\to
H^{*+1}(W;R)\to\ldots$ (3.2)

  2. (2)

$SH^{*}(W;R)$ is a unital ring and $H^{*}(W;R)\to SH^{*}(W;R)$ is a unital
ring map.

Most of the above discussion (except for the ring structure) can be
generalized to the orbifold setting using classical techniques, see [38].
Since ${\mathbb{C}}^{n}/G$ is the natural exact orbifold filling of
$(S^{2n-1}/G,\xi_{\rm std})$, the following result provides the natural
reference to compare our hypothetical exact fillings to.

######  Theorem 3.2 ([38, Theorem B]).

Let $R$ be a commutative ring such that $|G|$ is invertible. Then we have
$SH^{*}({\mathbb{C}}^{n}/G;R)=0$ for the isolated quotient singularity
${\mathbb{C}}^{n}/G$. In particular, we have
$SH^{*}_{+}({\mathbb{C}}^{n}/G;R)=H^{*+1}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;R)$
(using the ${\mathbb{Q}}$-grading), which is generated by $x_{(g)}$ for
$(g)\in\operatorname{Conj}(G)$. Here $x_{(g)}$ are those generators
(Hamiltonian orbits) with minimal rational Conley-Zehnder index for conjugacy
class $(g)$ defined after (2.4).

###  3.2. Symplectic cohomology of exact fillings of $(S^{2n-1}/G,\xi_{\rm
std})$

Using the Hamiltonian in §2.2, we define the boundary grading of a non-
constant periodic orbit (or a critical point of the auxiliary Morse function
on a Morse-Bott family of non-constant period orbits) as

$|x|^{\partial}:=n-\mu_{\rm CZ}(x),$

where $\mu_{\rm CZ}$ is the rational Conley-Zehnder index in §2.2. With this
rational boundary grading, we have a filtration $F^{k}C_{+}(H)\supset
F^{k+1}C_{+}(H)$ for $k\in{\mathbb{Z}}$ defined by

$F^{k}C_{+}(H):=\left\langle
x\left||x|^{\partial}\geq\frac{k}{|G|}\right.\right\rangle.$ (3.3)

######  Proposition 3.3.

When ${\mathbb{C}}^{n}/G$ is terminal, the differential is compatible with the
filtration and the first $|G|+1$ pages of the associated spectral sequence are
independent of exact fillings/augmentations. When $|G|$ is invertible in the
coefficient ring, the $(|G|+1)$th page of the spectral sequence is spanned by
$\\{x_{(g)}\\}_{(g)\in\operatorname{Conj}(G)}$ and degenerates.

###### Proof.

Let $x,y$ be two generators of the Hamiltonian-Floer cohomology, we can apply
neck-stretching to the moduli space ${\mathcal{M}}_{x,y}$ along the contact
boundary $S^{2n-1}/G$, see [72, §2.2] for more details on neck-stretching in a
similar setting. In the fully-stretched picture, the top level consists of
Hamiltonian-Floer cylinders possibly with negative punctures asymptotic to
Reeb orbits in $B_{(g_{1}),l_{1}},\ldots,B_{(g_{m}),l_{m}}$. If there is at
least one negative puncture, the virtual dimension of the corresponding moduli
space is

$|y|^{\partial}-|x|^{\partial}-\sum_{i=1}^{m}\operatorname{lSFT}(B_{(g_{i}),l_{i}})-1\leq|y|^{\partial}-|x|^{\partial}-2\operatorname{md}({\mathbb{C}}^{n}/G)-1.$
(3.4)

We point out here even though
$|y|^{\partial},|x|^{\partial},\operatorname{lSFT}(B_{(g_{i}),l_{i}})$ are
only rational, the virtual dimension, i.e. the left hand side of (3.4) is an
integer. Since we use $S^{1}$-dependent almost complex structures, we can
assume the transversality for the top level. Note that
$\operatorname{md}({\mathbb{C}}^{n}/G)>0$ by the terminality of the
singularity, to have a top-level curve after the neck-stretching, we must have
$|y|^{\partial}\geq|x|^{\partial}+1$, hence the differential is compatible
with filtration. Moreover, the filtration is clearly Hausdorff, hence induces
a convergent spectral sequence.

Since the differential on the first $|G|-1$ pages is counting
${\mathcal{M}}_{x,y}$ with $|y|^{\partial}-|x|^{\partial}<1$. In view of
(3.4), the expected dimension of the top level with at least one negative
puncture is $<-2\operatorname{md}({\mathbb{C}}^{n}/G)<0$. Therefore those
differentials are from curves contained in the symplectization. However,
$|y|^{\partial}-|x|^{\partial}<1$, those moduli spaces in the symplectization
have negative virtual dimensions and hence are empty. The differential on the
$|G|$th page counts ${\mathcal{M}}_{x,y}$ with
$|y|^{\partial}-|x|^{\partial}\leq 1$ (note that $|G||x|^{\partial}$ is always
an integer), which by the same argument, is from curves contained in the
symplectization and is independent of fillings. Therefore the first $|G|+1$
pages of the induced spectral sequence are independent of
augmentations/fillings (not necessary for the differential on the $(|G|+1)$th
page).

From the above discussion, we see that moduli spaces ${\mathcal{M}}_{x,y}$
with $|y|^{\partial}-|x|^{\partial}=1$ are the only non-empty moduli spaces
appearing in the computation of the first $|G|+1$ pages of the spectral
sequence. Now, we can consider the orbifold filling ${\mathbb{C}}^{n}/G$ of
$(S^{2n-1}/G,\xi_{\rm std})$, we have
$c_{1}^{{\mathbb{Q}}}({\mathbb{C}}^{n}/G)=0$. Therefore the filtration is just
the natural filtration from the ${\mathbb{Q}}$-grading on the symplectic
cohomology and the differential is solely contributed by those
${\mathcal{M}}_{x,y}$ with $|y|^{\partial}-|x|^{\partial}=1$. By the
independence of fillings, the $(|G|+1)$th page of the spectral sequence is
spanned by $x_{(g)}$, as $SH_{+}^{*}({\mathbb{C}}^{n}/G)$ is spanned by
$x_{(g)}$ by Theorem 3.2. Finally, the degeneracy follows tautologically as
the ${\mathbb{Z}}/2$ grading of $x_{(g)}$ is odd by the discussion after (2.4)
and all differentials in the spectral sequence respect this ${\mathbb{Z}}/2$
grading. ∎

The following is a generalization of [72, Proposition 2.12, 2.14], which is
another place where the terminality is crucial.

######  Proposition 3.4.

If ${\mathbb{C}}^{n}/G$ is an isolated terminal singularity. Let $W$ be an
exact filling of $(S^{2n-1}/G,\xi_{\rm std})$ and $R$ be a ring where $|G|$ is
invertible, then $SH^{*}(W;R)=0$, $H^{*}(W;\mathbb{R})$ is a free $R$-module
with $\operatorname{rank}H^{*}(W;R)=\\#\operatorname{Conj}(G)$ and
$H^{*}(W;R)$ is supported in even degrees.

###### Proof.

Let $a$ represent the cohomology class in $SH^{*}_{+}(W)$ represented by
$x_{(\operatorname{id})}$ in the spectral sequence in Proposition 3.3. This
$x_{(\operatorname{id})}$, when lifted to $S^{2n-1}$, is the Hamiltonian orbit
with the minimal Conley-Zehnder index ($=n+1$) and kills the unit in
$SH^{*}({\mathbb{C}}^{n})$. In the following, we will show that
$x_{(\operatorname{id})}$ kills the unit for hypothetical exact fillings of
$S^{2n-1}/G$ as well. Since $a=x_{(\operatorname{id})}$ in the spectral
sequence, we can only assume
$a=x_{(\operatorname{id})}+\sum_{i=1}^{n}c_{i}x_{i}$ for $c_{i}\in R$ and
$|x_{i}|^{\partial}\geq|x_{(\operatorname{id})}|^{\partial}+\frac{1}{|G|}=-1+\frac{1}{|G|}$.
Now we consider differentials from a non-constant orbit $x$ to $1$, or
equivalently, the moduli space ${\mathcal{M}}_{x,q}$ in [72, Proposition 2.12]
with $q\in S^{2n-1}/G$ a point constraint on the contact boundary555The point
constraint can be chosen freely, typically from the interior of the filling
(as the unique minimum of an auxiliary Morse function on the filling). Here we
choose it to be on the boundary for the convenience of neck-stretching.. We
can apply neck-stretching for ${\mathcal{M}}_{x_{(\operatorname{id})},q}$ as
in [72, Proposition 2.12 step 1]. The virtual dimension of the top SFT level
is

$2n-|x_{(\operatorname{id})}|^{\partial}-\sum_{i=1}^{l}\operatorname{lSFT}(B_{(g_{i}),l_{i}})-2n-1\leq
0,$

where $B_{(g_{i}),l_{i}}$ are where the asymptotics of the negative punctures
land. Since $\operatorname{lSFT}(B_{(g_{i}),l_{i}})>0$ by the terminality
assumption, the equality holds iff $l=0$. Then by the same covering argument
in [72, Proposition 2.12 step 3], we have
$\\#{\mathcal{M}}_{x_{(\operatorname{id}),q}}=|G|$. Next for
${\mathcal{M}}_{x_{i},q}$, the virtual dimension of the top SFT level after
neck-stretching is

$2n-|x_{i}|^{\partial}-\sum_{i=1}^{l}\operatorname{lSFT}(B_{(g_{i}),l_{i}})-2n-1<0.$

Hence we may assume ${\mathcal{M}}_{x_{i},q}=\emptyset$. As a consequence, we
have $SH^{*}_{+}(W;R)\to H^{*+1}(W;R)\to H^{0}(W;R)$ is surjective by $a$,
i.e. $a$ is mapped to $|G|+A$ through $SH^{*}_{+}(W;R)\to H^{*+1}(W;R)$ for
$A\in\oplus_{j>0}H^{j}(W;R)$. Note that $|G|+A$ is a unit in $H^{*}(W;R)$, the
long exact sequence (3.2) implies that $SH^{*}(W;R)=0$ and
$SH_{+}^{*}(W;R)=H^{*+1}(W;R)$ in the ${\mathbb{Z}}/2$-grading sense. Since
the $(|G|+1)$th page of the spectral sequence in Proposition 3.3 consists of
free finitely generated $R$-modules, there is no non-trivial extension, i.e.
$SH_{+}^{*}(W;R)$ is a free $R$-module of rank $\\#\operatorname{Conj}(G)$.
Hence the claim follows as the ${\mathbb{Z}}/2$ grading of $x_{(g)}$ is odd. ∎

######  Remark 3.5.

Under a non-canonical isomorphism
$\oplus_{(g)\in\operatorname{Conj}(G)}\langle\,x_{(g)}\,\rangle\simeq
SH_{+}^{*}(W;R)\simeq H^{*+1}(W;R)$ in Proposition 3.4, $x_{(g)}$ for
$g\neq\operatorname{id}$ corresponds to an element in
$\oplus_{i>0}H^{2i}(W;R)$. For otherwise, $x_{(g)}$ will be mapped to the
generator of $H^{0}(S^{2n-1}/G;R)$ in the projection $H^{*}(W;R)\to
H^{0}(S^{2n-1}/G;R)$, which shall not happen by the argument in in Proposition
3.4. We use $y_{(g)}$ to denote the image of $x_{(g)}$ in
$H^{*}(W;\mathbb{R})$ under the identification above.

######  Remark 3.6.

Assume the theory of symplectic cohomology for exact orbifolds is established,
i.e. in addition to the definition of the group level in [38], we also need
various structures like the ring structure, which will be constructed using
polyfolds in a future work. Then Proposition 3.4 holds for exact orbifold
fillings and $R={\mathbb{Q}}$ (this is due to the nature of polyfolds).

##  4\. The secondary coproduct on symplectic cohomology

###  4.1. Secondary coproduct and intersection form

The ordinary coproduct on symplectic cohomology, i.e. counting solutions to
perturbed Cauchy-Riemann equations on a pair of pants with one positive end
and two negative ends, as explained by Seidel in [63], is highly degenerate
due to degeneration of the Riemann surface (pinching a circle near a negative
end). In particular, the coproduct on the positive symplectic cohomology is
necessarily trivial. From such vanishing result, considering solutions to
perturbed Cauchy-Riemann equations on the family of Riemann surfaces
connecting the two pinchings yields the following secondary coproduct on the
positive symplectic cohomology, which was introduced by Ekholm and Oancea in
[27],

$\bm{\lambda}:SH^{*}_{+}(W;\bm{k})\to\left(SH^{*}_{+}(W;\bm{k})\otimes
SH^{*}_{+}(W;\bm{k})\right)[2n-1].$

Here $\bm{k}$ is some field coefficient666The requirement of field coefficient
is for the sake of the Künneth formula..

  1. (1)

A secondary coproduct on $SH^{*}_{+}(T^{*}M)$ was defined by Abbondandolo and
Schwarz [1]. The corresponding coproduct in string topology, through the
Viterbo isomorphism, was defined by Goresky and Hingston [39] in its dual
form, also see [65].

  2. (2)

Another (extended) secondary coproduct appears in the recent work of Cieliebak
and Oancea [23]. For the case of cotangent bundles, the role of the secondary
coproduct and the equivalence between those secondary coproducts were studied
in recent works of Cieliebak, Hingston and Oancea [20].

The secondary coproduct can be defined pictorially by counting perturbed Floer
equations on the following family of surfaces

$1$$0$$1$$1$c$1$$1$$1$$1$$0$$1$$1$c$\tau$ Figure 2. Secondary coproduct
$\bm{\lambda}$, numbers are weights at the puncture

In a more formal way, the secondary coproduct can be defined as follows [20,
§4.1]. Let $\Sigma$ be a $3$-punctured Riemann sphere with one positive
puncture and two negative punctures. A $1$-form $\beta$ has weight $A$ near a
puncture if $\beta=A{\rm d}t$ in the cylindrical coordinate
$[0,\infty)_{s}\times S^{1}_{t}$ (or $(-\infty,0]_{s}\times S^{1}_{t}$). Let
$H$ be a Hamiltonian as in §3. We choose a $1$-parameter family of $1$-forms
$\beta_{\tau}$ for $\tau\in(0,1)$ with the following properties:

  1. (1)

${\rm d}\beta_{\tau}\leq 0$ for all $\tau$;

  2. (2)

$\beta_{\tau}$ equals ${\rm d}t$ near all three punctures;

  3. (3)

as $\tau\to 0$, $\beta_{\tau}$ equals $0{\rm d}t$ near the first negative
puncture whose length goes to infinity, so that $\beta_{0}$ consists of a
$1$-form with weights $1,0,1$ on $\Sigma$ and $0,1$ on a cylinder attached to
the first negative puncture; hence “c” in Figure 2 stands for the continuation
map corresponding to the cylinder with weights $0,1$.

  4. (4)

as $\tau\to 0$, $\beta_{\tau}$ equals $0{\rm d}t$ near the second negative
puncture whose length goes to infinity, so that $\beta_{1}$ consists of a
$1$-form with weights $1,1,0$ on $\Sigma$ and $0,1$ on a cylinder attached to
the second negative puncture.

The algebraic count of rigid solutions to

$({\rm d}u-X_{H}\otimes\beta_{\tau})^{0,1}=0,\quad\tau\in(0,1)$

defines a secondary coproduct $C^{*}_{+}(H)\to(C^{*}_{+}(H)\otimes
C^{*}_{+}(H))[2n-1]$. It is a cochain map as the extra boundary corresponding
to $\tau=0,1$ will not appear since we only consider the positive cochain
complex. Passing to the direct limit, we get the secondary coproduct
$\bm{\lambda}$ on cohomology.

The significance of the secondary coproduct for our purpose is the following
result due to Eliashberg, Ganatra and Lazarev [30].

######  Theorem 4.1 ([30]).

The following diagram is commutative

$\textstyle{SH^{*}_{+}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bm{\lambda}\hskip
42.67912pt}$$\scriptstyle{\delta}$$\textstyle{(SH^{*}_{+}(W)\otimes
SH^{*}(W)[2n-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\otimes\delta}$$\textstyle{H^{*}(W)[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta\hskip
42.67912pt}$$\textstyle{(H^{*}(W)\otimes H^{*}(W))[2n+1]}$

where $\delta:SH^{*}(W)\to H^{*+1}(W)$ is the connecting map in the
tautological exact sequence and the coproduct
$\Delta:H^{*}(W)\to(H^{*}(W)\otimes H^{*}(W))[2n]$ is the dual of the cup
product $H^{i}_{c}(W)\otimes H^{j}_{c}(W)\to H^{i+j}_{c}(W)$.

The proof can be explained schematically by the following figures. In the
following, we recall a sketch of Eliashberg, Ganatra and Lazarev’s proof.

$\partial$$1$$0$$1$$1$$\operatorname{id}$$=$$\cup$$1$$0$$1$$1$$\operatorname{id}$$\Delta$$\delta$$1$$0$$1$$1$$\operatorname{id}$$\delta$

$1$$0$$1$$1$$\operatorname{id}$$\cup$$\tau$$\bm{\lambda}$$1$$1$$1$$1$$1$$\delta$$\tau$$1$$1$$1$$=$$\partial$

Figure 3. Pictorial proof of Theorem 4.1

The numbers stand for weights as before. By $\bullet$, we mean that the end is
asymptotic to a constant orbit. By a circle, we mean the end is asymptotic to
a non-constant orbit, i.e. a generator of the positive cochain complex. The
configuration equipped with $\tau$ is the moduli space in Figure 2. The
configuration marked with $\Delta$ defines the coproduct on $H^{*}(W)$. The
boundary components correspond to the differential on the positive cochain
complex and zero cochain complex (Morse cochain complex) are omitted, which
will correspond to the homotopy in the commutative diagram in Theorem 4.1.
More precisely, we insert $x$ that is closed in the positive cochain complex
(i.e. a representative of a class in $SH^{*}_{+}(W)$) as the input in the top
row of Figure 3, then we can get

$(\mathrm{c}\otimes\operatorname{id})\circ\Delta\circ\delta\pm(\mathrm{c}\otimes\delta)\circ\Theta=0\in
H^{*}(W)\otimes H^{*}(W)$ (4.1)

Here $\Theta:C_{+}(1\cdot H)\to C(0\cdot H)\otimes C_{+}(1\cdot H)$ is defined
by counting the top level of the top right curve in Figure 3, $\mathrm{c}$ is
the continuation map $C(0\cdot H)\to C(1\cdot H)$, i.e. identity onto the
constant orbits part of $C(H)$. (4.1) only holds on cohomology, as we omit the
Morse breakings in a cascades model or breakings appearing below a constant
orbit in a $C^{2}$-small perturbed model. Now we look at the second row of
Figure 3, the first configuration in the union is the left end of Figure 2. A
priori, there is another boundary from pinching the second negative end in
$\tau$-family, but it will not appear since the asymptotic of the second
negative puncture is a non-constant orbit in Figure 3. Therefore, the second
row of Figure 3 reads

$(\mathrm{c}\otimes\operatorname{id})\circ\Theta\pm(\delta\otimes\operatorname{id})\circ\bm{\lambda}=0\in
H^{*}(W)\otimes SH_{+}^{*}(W).$ (4.2)

As before, this relation holds on cohomology only as we omitted other more
standard boundaries. Combining (4.1) and (4.2), we see that
$\Delta\circ\delta=(\delta\otimes\delta)\circ\bm{\lambda}$ after choosing
appropriate signs and orientation conventions.

###  4.2. The secondary coproduct for ${\mathbb{C}}^{n}/G$

We have the following special case of the orbifold version of Theorem 4.1.

######  Proposition 4.2.

The secondary coproduct is well-defined on
$SH^{*}_{+}({\mathbb{C}}^{n}/G;\bm{k})$. When $|G|$ is invertible in $\bm{k}$,
the following is commutative,

$\textstyle{SH^{*}_{+}({\mathbb{C}}^{n}/G;\bm{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bm{\lambda}\hskip
42.67912pt}$$\scriptstyle{\delta}$$\textstyle{(SH^{*}_{+}({\mathbb{C}}^{n}/G;\bm{k})\otimes
SH^{*}({\mathbb{C}}^{n}/G;\bm{k}))[2n-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\otimes\delta}$$\textstyle{H^{*}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;\bm{k})[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta\hskip
42.67912pt}$$\textstyle{(H^{*}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;\bm{k})\otimes
H^{*}_{\mathrm{C}R}({\mathbb{C}}^{n}/G;\bm{k}))[2n+1]}$

where the bottom row is the Chen-Ruan coproduct in §2.3.1 and the grading is
the rational grading.

###### Proof.

In view of [38, §3.7.2], the difficulty of generalizing Theorem 4.1 to the
orbifold setting is the appearance of constant holomorphic curves with more
than $3$ punctures of negative virtual dimension. Those curves are not
somewhere injective and can not be perturbed away by the enforcement of the
symmetry, hence we need the machinery of virtual perturbations to deal with
them. However, in the special case of ${\mathbb{C}}^{n}/G$, such a problem
will not appear and the proof sketched above for Theorem 4.1 goes through.

First note that the secondary coproduct is defined on $SH^{*}_{+}(W;\bm{k})$
for a general exact orbifold filling $W$, as the relevant moduli spaces do not
involve constant curves in the compactification. Now by Theorem 3.2 ([38,
Theorem B]), $SH^{*}_{+}({\mathbb{C}}^{n}/G;\bm{k})$ is spanned by classes
represented by $x_{(g)}$ and is equipped with a global ${\mathbb{Q}}$-grading.
Moreover, $x_{(g)}$ is sent to the constant twisted sector corresponding to
the conjugacy class $(g)$ in the connecting map
$SH^{*}_{+}({\mathbb{C}}^{n}/G;\bm{k})\to
H_{\mathrm{C}R}^{*+1}({\mathbb{C}}^{n}/G;\bm{k})$. With a slight abuse of
notation, we use $(g)$ to denote the constant twisted sector. Therefore it
suffices to prove the diagram on those generators (instead of the whole
cochain complex). In the moduli spaces in Figure 3, the only place to see
constant holomorphic curves with more than $2$ punctures is from the right
side of the first row. Note that $x_{(g)}$ has rational grading
$2\operatorname{age}(g)-1$, hence $\bm{\lambda}(x_{(g)})$ has rational grading
$2\operatorname{age}(g)+2n-2$ and
$(\delta\otimes\delta)\circ\bm{\lambda}(x_{(g)})$ has rational grading
$2\operatorname{age}(g)+2n$. Therefore to prove the commutative diagram, it is
sufficient to plug in $x_{(g)}$ into the input end and constant twisted
sectors $(h_{1}),(h_{2})$ into the output ends of the top row of Figure 3,
subjected to the following two constraints:

  1. (1)

$(h_{1}h_{2})=(g)\in Conj(G)$, this is the constraint from the homotopy
classes.

  2. (2)

$\operatorname{age}(h_{1})+\operatorname{age}(h_{2})=\operatorname{age}(g)+n$,
this is the constraint from the rational grading above.

With such asymptotics, in the top row of Figure 3, the breaking from the input
end must end in $(g)$ by the homotopy class. Therefore the constant thrice-
punctured curve must have asymptotics $(g),(h_{1}),(h_{2})$, which, due to the
second constraint on $\operatorname{age}$, is cut out transversely777The
kernel of the linearized operator is always zero-dimensional for any constant
curve. Since
$2n+2\operatorname{age}(g)-2\operatorname{age}(h_{1})-2\operatorname{age}(h_{2})$
is the Fredholm index,
$\operatorname{age}(h_{1})+\operatorname{age}(h_{2})=\operatorname{age}(g)+n$
implies transversality.. Therefore the proof of Theorem 4.1 goes through for
${\mathbb{C}}^{n}/G$. ∎

######  Remark 4.3.

To prove Theorem 4.1 for general exact orbifolds with contact manifold
boundary, we must check the diagram on general generators instead of the
special collection $\\{x_{(g)}\\}$ in the special case of Proposition 4.2.
Then in the top row of Figure 3, we may run into a situation where the top
curve for the operator $\delta$ has a moduli space of positive dimension, and
the constant curve for $\Delta$ has a negative virtual dimension, with a
correct total virtual dimension. Such a configuration can not resolved by
classical perturbations.

We endow the tensor product $C_{+}(H)\otimes C_{+}(H)$ with a filtration

$F^{k}C_{+}(H)\otimes C_{+}(H)=\left\langle x\otimes
y\left||x|^{\partial}+|y|^{\partial}-2n+1\geq\frac{k}{|G|}\right.\right\rangle.$

By the same argument of Proposition 3.3, the induced spectral sequence also
degenerates at $(|G|+1)$th page, which is spanned by $\\{x_{(g)}\otimes
x_{(h)}\\}_{(g),(h)\in\operatorname{Conj}(G)}$, whenever ${\mathbb{C}}^{n}/G$
is terminal and $|G|$ is invertible in $\bm{k}$. The extra degree shift by
$-2n+1$ is for $\bm{\lambda}$ capturing the non-trivial leading terms (i.e. on
the associate graded) in the proposition below.

######  Proposition 4.4.

Assume ${\mathbb{C}}^{n}/G$ is terminal. Let $W$ be an exact filling of
$(S^{2n-1}/G;\xi_{\rm std})$. The secondary coproduct on
$SH^{*}_{+}(W;\bm{k})$ is compatible with the boundary grading filtration. In
particular, it induces a map on the associated graded, which is isomorphic to
the secondary coproduct on $SH^{*}_{+}({\mathbb{C}}^{n}/G;\bm{k})$.

###### Proof.

Since $\operatorname{md}({\mathbb{C}}^{n}/G)>0$, the neck-stretching argument
shows that $\langle\,\bm{\lambda}(x),y\otimes z\,\rangle=0$ whenever
$|x|^{\partial}+2n-1-|y|^{\partial}-|z|^{\partial}<0$. Note that the boundary
rational gradings are multiplies of $\frac{1}{|G|}$, this implies that
$\bm{\lambda}$ is compatible with filtration. The induced map of
$\bm{\lambda}$ on the first page of the spectral sequence only depends on
moduli spaces defining $\bm{\lambda}$ with input asymptotic $x$ and output
asymptotic $y,z$ such that
$|x|^{\partial}+2n-1-|y|^{\partial}-|z|^{\partial}=0$. Neck-stretching and
$\operatorname{md}({\mathbb{C}}^{n}/G)>0$ imply that this map is independent
of fillings. Note that the induced map of $\bm{\lambda}$ on the associated
graded is induced from the induced map on the first page. Since for
${\mathbb{C}}^{n}/G$, the filtration is the rational grading and the induced
map on the associated graded is identified with the secondary coproduct
$\bm{\lambda}$ on ${\mathbb{C}}^{n}/G$ (determined by Proposition 4.2 and
(2.8)). The claim for a general filling follows from the independence of
filling and a comparison with ${\mathbb{C}}^{n}/G$. ∎

Recall that the cup product length of a topological space $W$ is defined to be

$\mathrm{cl}(W):=\max\left\\{n\left|\exists\alpha_{i}\in\oplus_{*>0}H^{*}(W;{\mathbb{Q}}),i\leq
n,\text{ such that }\cup\alpha_{i}\neq 0\right.\right\\}.$

A consequence of Proposition 4.4 is the following upper bound.

######  Corollary 4.5.

Assume ${\mathbb{C}}^{n}/G$ is terminal. Let $W$ be an exact filling of
$(S^{2n-1}/G;\xi_{\rm std})$. Then we have

$\mathrm{cl}(W)\leq\left\lfloor\frac{n-1-\operatorname{md}({\mathbb{C}}^{n}/G)}{1+\operatorname{md}({\mathbb{C}}^{n}/G)}\right\rfloor.$

In particular, we always have $\mathrm{cl}(W)\leq n-2$.

###### Proof.

By Proposition 3.4 and Remark 3.5, we can write
$H^{*}(W;{\mathbb{Q}})\simeq\oplus_{(g)\in\operatorname{Conj}(G)}\langle\,y_{(g)}\,\rangle$,
where $y_{(g)}$ is the image of $x_{(g)}$ under a non-canonical isomorphism
$\oplus_{(g)\in\operatorname{Conj}(G)}\langle\,x_{(g)}\,\rangle\simeq
SH_{+}^{*}(W;\bm{k})\simeq H^{*+1}(W;\bm{k})$. We grade $y_{(g)}$ by the
rational number $2\operatorname{age}(g)$. Under such isomorphism, Proposition
4.4 implies that the coproduct $\Delta(y_{(g)})$ has a minimal rational
grading at least $2n+2\operatorname{age}(g)$. In view of the Lefschetz duality
$H^{*}_{c}(W;{\mathbb{Q}})=\operatorname{Hom}(H^{2n-*}(W;{\mathbb{Q}}),{\mathbb{Q}})$,
we may write
$H^{*}_{c}(W;{\mathbb{Q}})\simeq\oplus_{(g)\in\operatorname{Conj}(G)}\langle\,y^{\vee}_{(g)}\,\rangle$,
where $y^{\vee}_{(g)}$ is the linear dual of $y_{(g)}$ in that basis. We
define a rational grading of $y^{\vee}_{(g)}$ to be
$2n-2\operatorname{age}(g)$ (i.e. $2\operatorname{age}(g^{-1})$). Under such
an isomorphism, the dual of Proposition 4.4 implies that the cup product
$\cup:H^{*}_{c}(W;{\mathbb{Q}})\otimes H^{*}_{c}(W;{\mathbb{Q}})\to
H^{*}_{c}(W;{\mathbb{Q}})$ has the property that the minimal rational grading
of $y^{\vee}_{(g)}\cup y^{\vee}_{(h)}$ is at least
$4n-2\operatorname{age}(g)-2\operatorname{age}(h)$.

Since $S^{2n-1}/G$ is a rational homology sphere, we have that
$H^{*}_{c}(W;{\mathbb{Q}})\to H^{*}(W;{\mathbb{Q}})$ is an isomorphism for
$0<*<2n$ and the cup product length of $W$ is precisely the cup product length
of the monoid $\operatorname{im}(H^{*}_{c}(W;{\mathbb{Q}})\to
H^{*}(W;{\mathbb{Q}}))$ or equivalently
$H^{*}_{c}(W;{\mathbb{Q}})/H^{2n}_{c}(W;{\mathbb{Q}})$. By Remark 3.5,
$y_{(g)}$ in the isomorphism
$H^{*}(W;{\mathbb{Q}})\simeq\oplus_{(g)\in\operatorname{Conj}(G)}\langle\,y{(g)}\,\rangle$
does not have component in $H^{0}(W;{\mathbb{Q}})$ when
$g\neq\operatorname{id}$, while $y_{(\operatorname{id})}$ does. As a
consequence, the cup length product of
$H^{*}_{c}(W;{\mathbb{Q}})/H^{2n}_{c}(W;{\mathbb{Q}})$ is same as the cup
product length on
$\oplus_{(g)\neq(\operatorname{id})\in\operatorname{Conj}(G)}\langle\,y^{\vee}_{(g)}\,\rangle$.
The cup product length of the latter can be estimated by the rational grading.
Since the minimal and maximal rational gradings of $y^{\vee}_{(g)}$ for
$g\neq\operatorname{id}$ are $2(1+\operatorname{md}({\mathbb{C}}^{n}/G))$ and
$2n-2(1+\operatorname{md}({\mathbb{C}}^{n}/G))$ respectively. Hence the claim
follows, as the product does not decrease the rational grading. ∎

By Proposition 3.4, we have that $H^{*}(W)$ is supported in even degrees,
hence the cup product length is tautologically smaller than $n$. Although
Corollary 4.5 only improves the upper bound by $1$ if we only know that
$\operatorname{md}({\mathbb{C}}^{n}/G)>0$, it is crucial in the proof of
Theorem 7.8.

In some cases, the map on the associated graded can be identified with the map
on the original space. Then Proposition 4.4 can recover the coproduct on
cohomology. The following is the simplest case when it happens.

######  Corollary 4.6.

Let ${\mathbb{C}}^{n}/G$ be a terminal singularity of the following cases,
then any exact filling of $(S^{2n-1}/G,\xi_{\rm std})$ has the same coproduct
as ${\mathbb{C}}^{n}/G$ rationally.

  1. (1)

Diagonal actions on ${\mathbb{C}}^{n=2m}$ by finite subgroups of $SU(2)$.

  2. (2)

$G\subset O(n)\subset SU(n)$ with isolated singularity.

###### Proof.

The missed information of the induced map on the associated graded is the
component in the higher filtration of the image. In both cases, we have
$\operatorname{age}(g)=n/2$ for any $g\neq\operatorname{id}$, then there is no
higher filtration for $\Delta(y_{(\operatorname{id})})$ or $\Delta(y_{(g)})$.
∎

Since the meaning of higher filtration depends on the stride length
($\frac{1}{|G|}$) of the filtration in (3.3). Therefore, one can generalize
Corollary 4.6 to other terminal singularities as long as
$\operatorname{age}(g)$ is concentrated around $n/2$ for
$g\neq\operatorname{id}$. Moreover, one can adjust the stride length so that
results in §3 and Proposition 4.4 still hold if we know more about
$\operatorname{md}({\mathbb{C}}^{n}/G)$, which will lead to different
requirement of the concentration. We will not pursue the precise statements
here.

##  5\. Fundamental groups

In this section, we discuss fundamental groups for _strong fillings_ of
contact links of quotient singularities. This requires the definition of
(positive) symplectic cohomology for strong fillings. The existence of
positive symplectic cohomology and tautological long exact sequence is
guaranteed by the asymptotic behaviour lemma [22, Lemma 2.3] instead of the
usual action separation. The transversality issues from sphere bubbles require
the deployment of virtual techniques, for example, Pardon’s setup for
Hamiltonian-Floer cohomology for a general closed symplectic manifold [54]
suffices for the setup for symplectic cohomology used in this paper. If one
wishes to stay within the realm of geometric perturbations, then strong
fillings should be replaced by semi-positive fillings [47, Definition 6.4.1]
in the following results.

######  Proposition 5.1.

Let $W$ be a strong filling of $(S^{2n-1}/G,\xi_{\rm std})$. We write $G_{0}$
as the kernel of $\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$. If
${\mathbb{C}}^{n}/G_{0}$ is terminal, then $\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$
is surjective.

###### Proof.

Assume $\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$ is not surjective. We write
$Q=\pi_{1}(W)/\operatorname{im}(\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W))\neq 0$. We
can consider the symplectic cohomology for the universal cover $\widetilde{W}$
as in [73, §3.3]. Under our assumption, $\partial\widetilde{W}$ is $\\#Q$ many
copies of $(S^{2n-1}/G_{0},\xi_{\rm std})$. Similarly to Proposition 3.3, by
comparing to the natural filling $\bigsqcup_{Q}{\mathbb{C}}^{n}/G_{0}$ of
$\partial\widetilde{W}$, we have that
$SH^{*}_{+}(\widetilde{W};\Lambda)\simeq\prod_{Q}H^{*+1}_{\mathrm{C}R}({\mathbb{C}}^{n}/G_{0};\Lambda)$
where $\Lambda$ is the Novikov field. Moreover, by the same proof of
Proposition 3.4, the composition $SH_{+}^{*}(\widetilde{W};\Lambda)\to
H^{*+1}(\widetilde{W};\Lambda)\to H^{*+1}(\partial\widetilde{W};\Lambda)\to
H^{0}(\partial\widetilde{W})\otimes\Lambda$ is

$SH^{*}_{+}(\widetilde{W};\Lambda)\stackrel{{\scriptstyle\simeq}}{{\to}}\prod_{Q}H^{*+1}_{\mathrm{C}R}({\mathbb{C}}^{n}/G_{0};\Lambda)\twoheadrightarrow\prod_{Q}H^{0}_{\mathrm{C}R}({\mathbb{C}}^{n}/G_{0})\otimes\Lambda=H^{0}(\partial\widetilde{W})\otimes\Lambda=\prod_{Q}\Lambda$

hence surjective. On the other hand, the map above factors through
$H^{0}(\widetilde{W})\otimes\Lambda\to
H^{0}(\partial\widetilde{W})\otimes\Lambda$, which is clearly not surjective,
contradiction. ∎

In some sense, the argument above hints that $(S^{2n-1}/G_{0},\xi_{\rm std})$
should not be the boundary of a connected strong domain with multiple boundary
components, i.e. not co-fillable, see Remark 5.6 for explanations.

If we only consider exact fillings, the un-deformed ring structure on quantum
cohomology allows us to say the following without the terminal assumption.

######  Proposition 5.2.

Let $W$ be an exact filling of $(S^{2n-1}/G,\xi_{\rm std})$ for $n\geq 2$.
Then $\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$ is not injective.

###### Proof.

Assume $\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$ is injective. In the neck-stretching
process, those negative punctures of the top-level curve must be asymptotic to
contractible Reeb orbits on $S^{2n-1}/G$ by the injectivity assumption, as
those orbits are eventually capped off by holomorphic planes in $W$. Note that
all contractible orbits have positive integer SFT degrees, as they correspond
to Reeb orbits on $(S^{2n-1},\xi_{\rm std})$ with the same Conley-Zehnder
index. Similar to Proposition 3.3, the first $|G|+1$ pages of the spectral
sequence are independent of such $W$. Note that
$\pi_{1}(S^{2n-1}/G)\to\pi^{\rm orb}_{1}({\mathbb{C}}^{n}/G)$ is an
isomorphism, we can compare the hypothetical $W$ with the orbifold
${\mathbb{C}}^{n}/G$ and the arguments of Propositions 3.3 and 3.4 go through
and the same conclusions hold for $W$ without the terminal assumption. That is
we have

  1. (1)

the $(|G|+1)$th page of $SH_{+}^{*}(W;{\mathbb{Q}})$ is generated by $x_{(g)}$
(in Theorem 3.2), and the spectral sequence degenerates there;

  2. (2)

$SH_{+}^{*}(W;{\mathbb{Q}})\to H^{*+1}(W;{\mathbb{Q}})$ is an isomorphism.

Therefore elements in $SH_{+}^{*}(W;{\mathbb{Q}})$ are represented by
$x_{(g)}+a$ for $(g)\in\operatorname{Conj}(G)$ and $a$ has a lower filtration
order compared to $x_{(g)}$. Here the homotopy class of $a$ must be the same
as $x_{(g)}$ as $\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$ is injective, which is non-
trivial when $(g)\neq(\operatorname{id})$. This contradicts the fact that
$SH_{+}^{*}(W;{\mathbb{Q}})\to H^{*+1}(W;{\mathbb{Q}})$ is an isomorphism, as
$x_{(g)}+a$ must map to zero in $H^{*+1}(W;{\mathbb{Q}})$ by the restriction
of homotopy classes. ∎

######  Corollary 5.3.

Let ${\mathbb{C}}^{n}/G$ be a terminal singularity, then any exact filling of
$(S^{2n-1}/G,\xi_{\rm std})$ is simply connected.

###### Proof.

Let $W$ be an exact filling. Since ${\mathbb{C}}^{n}/G$ is terminal, by
Proposition 2.12, ${\mathbb{C}}^{n}/G_{0}$ must be terminal as well, where
$G_{0}=\ker(\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W))$. Therefore Proposition 5.1
implies that $\pi_{1}(S^{2n-1}/G)\to\pi_{1}(W)$ is surjective. If
$\pi_{1}(W)\neq 0$, we must have a non-trivial conjugacy class $(g)$, such
that $x_{(g)}$ is not contractible in $W$. However, by
LABEL:{prop:upper_bound}, the element in $SH_{+}^{*}(W;{\mathbb{Q}})$
represented by $x_{(g)}$ plus lower order terms must map to a non-trivial
class in $H^{*}(W;{\mathbb{Q}})$, contradicting the fact that $x_{(g)}$ is not
contractible in $W$. ∎

######  Proposition 5.4.

Let $m\in{\mathbb{N}}$ such that there is no non-trivial square that divides
$m$. Assume ${\mathbb{C}}^{n}/({\mathbb{Z}}/m)$ is terminal, then any strong
filling of $(S^{2n-1}/({\mathbb{Z}}/m),\xi_{\rm std})$ is simply connected.

###### Proof.

Let $W$ be a strong filling. By Proposition 5.1, we have that
${\mathbb{Z}}/m=\pi_{1}(S^{2n-1}/({\mathbb{Z}}/m))\to\pi_{1}(W)$ is
surjective. If $\pi_{1}(W)\neq 0$, then $\pi_{1}(W)={\mathbb{Z}}/p$, where $p$
divides $m$. Therefore, by the universal coefficient theorem, we have
$H^{1}(W;{\mathbb{Z}}/m)\to H^{1}(S^{2n-1}/({\mathbb{Z}}/m);{\mathbb{Z}}/m)$
and $H^{2}(W;{\mathbb{Z}}/m)\to
H^{2}(S^{2n-1}/({\mathbb{Z}}/m);{\mathbb{Z}}/m)$ are both non-trivial and
contain ${\mathbb{Z}}/p\subset{\mathbb{Z}}/m$ in the image. From Remark 2.2,
we have that
$H^{*}(B{\mathbb{Z}}/m;{\mathbb{Z}}/m)=H^{*}(S^{2n-1}/({\mathbb{Z}}/m);{\mathbb{Z}}/m)$
for $*\leq 2n-1$ and the projection is an algebra map. Since
$H^{*}(B{\mathbb{Z}}/m;{\mathbb{Z}}/m)=({\mathbb{Z}}/m)[a,b]$ as an algebra
for $\deg(a)=1$ and $\deg(b)=2$ and $m$ is not divided by $p^{2}$, we get
$\alpha\in H^{1}(W;{\mathbb{Z}}/m),\beta\in H^{2}(W;{\mathbb{Z}}/m)$, such
that $(\alpha\cup\beta^{n-1})|_{\partial W}\neq 0$, i.e.
$H^{2n-1}(W;{\mathbb{Z}}/m)\to
H^{2n-1}(S^{2n-1}/({\mathbb{Z}}/m);{\mathbb{Z}}/m)$ is nontrivial. Hence
$H^{2n}(W,S^{2n-1}/({\mathbb{Z}}/m);{\mathbb{Z}}/m)$ is a non-trivial quotient
of ${\mathbb{Z}}/m$ from the long exact sequence, this is impossible when $W$
is orientable. ∎

C is assembled from Propositions 5.1, 5.2 and 5.4 and Corollary 5.3

######  Remark 5.5.

The $m=2$ case of Proposition 5.4 was proven by Ghiggini and Niederkrüger [37,
Theorem 1.2]. Unlike the $m=2$ case, when $m\geq 3$, there are multiple
${\mathbb{Z}}/m$ actions such that ${\mathbb{C}}^{n}/({\mathbb{Z}}/m)$ is
terminal. There are also many non-terminal examples
${\mathbb{C}}^{2}/({\mathbb{Z}}/m)$, whose contact links admit Weinstein
fillings that are not simply connected, see e.g. [35].

######  Remark 5.6 (Proposition D).

If ${\mathbb{C}}^{n}/G$ is a terminal singularity, the arguments in §3 imply
that there is always a holomorphic curve passing through a point constraint
independent of the point and strong fillings. We claim that
$(S^{2n-1}/G,\xi_{\rm std})$ is not co-fillable. Assume otherwise,
$(S^{2n-1}/G,\xi_{\rm std})\sqcup Y$ can be filled by a connected strong
domain $W$. We can set up our Hamiltonian to be standard on the cylindrical
end of $(S^{2n-1}/G,\xi_{\rm std})$ as in §3 and $0$ on the cylindrical end on
$Y$. By moving the point constraint to the cylindrical end of $Y$, we then
obtain a contradiction with the maximum principle. However, we expect that
$(S^{2n-1}/G,\xi_{\rm std})$ is not co-fillable without any constraint on the
isolated singularity ${\mathbb{C}}^{n}/G$. In dimension $3$, by [62],
$(S^{3}/({\mathbb{Z}}/m),\xi_{\rm std})$ supports a planar open book, hence is
not co-fillable by [33]. In higher dimensions, it is interesting to see
whether $(S^{2n-1}/G,\xi_{\rm std})$ supports an iterated planar open book
[3]. If the answer to this question is affirmative, then $(S^{2n-1}/G,\xi_{\rm
std})$ is not co-fillable by [3]. However, one should not expect the
(iterated) planar property to hold in general, as there are already
counterexamples in dimension $3$ [8]. On the other hand, from the symplectic
cohomology point of view in this paper, when the terminality condition is
removed, we expect fillings of $(S^{2n-1}/G,\xi_{\rm std})$ to have $k$-semi-
dilations [74], see [76] for the special case in complex dimension $2$. The
existence of $k$-semi-dilations, which is also an existence of holomorphic
curves with a point constraint, leads to obstructions to co-fillablity by a
similar argument.

##  6\. Non-existence of exact fillings for real projective spaces

The main result in this section is

######  Theorem 6.1.

For $n=2^{k}\geq 4$, $(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ is not
exactly fillable.

Combining with [72, Theorem 1.1], we have

######  Theorem 6.2 (B).

For any $n\geq 3$, $(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ is not
exactly fillable.

###  6.1. Properties of a hypothetical exact filling

Assume $W$ is a hypothetical exact filling of
$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ for $n=2^{k}>2$. We first list
the properties of $(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ and $W$ from
previous sections.

  1. (1)

By [72, Remark 2.3], the total Chern class of
$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ is trivial.

  2. (2)

By Corollary 5.3, we have $\pi_{1}(W)=0$. As a consequence, we have
$H_{1}(W;{\mathbb{Z}})=\operatorname{Tor}H^{2}(W;{\mathbb{Z}})=0$.

  3. (3)

By Corollary 4.6, we have
$H^{*}(W;{\mathbb{Q}})={\mathbb{Q}}\oplus{\mathbb{Q}}[-2^{k}]$ and
$H^{*}_{c}(W;{\mathbb{Q}})={\mathbb{Q}}[-2^{k}]\oplus{\mathbb{Q}}[-2^{k+1}]$
with a positive intersection form. Therefore we have $H^{2}(W;{\mathbb{Z}})=0$
and hence $c_{1}(W)=0$.

  4. (4)

Let $C$ be the disk bundle of the degree $2$ line bundle over
$\mathbb{C}\mathbb{P}^{n-1}$, which is the natural symplectic cap of
$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$. Then we can glue $C$ to $W$
along $\mathbb{R}\mathbb{P}^{2n-1}$ to obtain a closed symplectic manifold
$X$. Using the rational cohomology, it is easy to see that Euler
characteristic $\chi(X)=n+2$ and signature $\sigma(X)=2$.

######  Proposition 6.3.

We have that $\operatorname{Tor}H^{n}(W;{\mathbb{Z}})\to
H^{n}(\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})$ is zero but
$H^{n}(W;{\mathbb{Z}})\to H^{n}(\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})$ is
surjective.

###### Proof.

Note that we have the following long exact sequence

$0\to H^{n}(W,\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})\to
H^{n}(W;{\mathbb{Z}})\to H^{n}(\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})\to
H^{n+1}(W,\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})\to
H^{n+1}(W;{\mathbb{Z}})\to 0.$ (6.1)

By Lefschetz duality, (6.1) can be rewritten as

$0\to H_{n}(W;{\mathbb{Z}})\to H^{n}(W;{\mathbb{Z}})\to{\mathbb{Z}}/2\to
H_{n-1}(W;{\mathbb{Z}})\to H^{n+1}(W;{\mathbb{Z}})\to 0.$ (6.2)

By the universal coefficient theorem, we have

$H_{n}(W;{\mathbb{Z}})={\mathbb{Z}}\oplus H^{n+1}(W;{\mathbb{Z}}),\quad
H_{n-1}(W;{\mathbb{Z}})=\operatorname{Tor}H^{n}(W;{\mathbb{Z}}).$

Hence (6.2) can be written as

$0\to{\mathbb{Z}}\oplus
H^{n+1}(W;{\mathbb{Z}})\to{\mathbb{Z}}\oplus\operatorname{Tor}H^{n}(W;{\mathbb{Z}})\to{\mathbb{Z}}/2\to\operatorname{Tor}H^{n}(W;{\mathbb{Z}})\to
H^{n+1}(W;{\mathbb{Z}})\to 0.$ (6.3)

If $\operatorname{Tor}H^{n}(W;{\mathbb{Z}})\to
H^{n}(\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})={\mathbb{Z}}/2$ is nonzero,
then we have
$\operatorname{Tor}H^{n}(W;{\mathbb{Z}})=H^{n+1}(W;{\mathbb{Z}})$. On the
other hand, (6.3) implies that the torsion group $H^{n+1}(W;{\mathbb{Z}})$
injects into $\operatorname{Tor}H^{n}(W;{\mathbb{Z}})$. Hence we have
$|\operatorname{Tor}H^{n}(W;{\mathbb{Z}})|\geq 2|H^{n+1}(W;{\mathbb{Z}})|$,
which is a contradiction. Hence $\operatorname{Tor}H^{n}(W;{\mathbb{Z}})\to
H^{n}(\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})={\mathbb{Z}}/2$ is zero.

On the other hand, if $H^{n}(W;{\mathbb{Z}})\to
H^{n}(\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Z}})$ is zero. We have
$|\operatorname{Tor}H^{n}(W;{\mathbb{Z}})|=2|H^{n+1}(W;{\mathbb{Z}})|$ from
the right three terms of (6.3), while
$|\operatorname{Tor}H^{n}(W;{\mathbb{Z}})|=|H^{n+1}(W;{\mathbb{Z}})|$ from the
left two terms of (6.3), contradiction. ∎

Note that the rank of $H^{n}(X;{\mathbb{Z}})$ is two. Let $\mu$ denote a
generator of the free part of $H^{n}(W;{\mathbb{Z}})$888Strictly speaking,
there is no canonical free part as
$0\to\operatorname{Tor}H^{n}(W;{\mathbb{Z}})\to
H^{n}(W;{\mathbb{Z}})\to{\mathbb{Z}}\to 0$ does not have a natural splitting.
But since we only discuss the intersection number, such imprecision does not
matter.. In view of the Mayer-Vietoris sequence for $X=W\cup C$, the free part
of $H^{n}(X;{\mathbb{Z}})$ is generated by $\mu+u^{n/2}$ and $2u^{n/2}$, where
$u\in H^{2}(C;{\mathbb{Z}})=H^{2}(\mathbb{C}\mathbb{P}^{n-1};{\mathbb{Z}})$ is
the generator represented by the first Chern class of ${\mathcal{O}}(1)$. On
the other hand, since
$H^{*}(\mathbb{R}\mathbb{P}^{2n-1};{\mathbb{Q}})=H^{*}(S^{2n-1};{\mathbb{Q}})$,
we have $H^{n}(X;{\mathbb{Q}})=H^{n}(C;{\mathbb{Q}})\oplus
H^{n}(W;{\mathbb{Q}})=H^{n}_{c}(C;{\mathbb{Q}})\oplus
H^{n}_{c}(W;{\mathbb{Q}})$, the later is from the ${\mathbb{Q}}$-isomorphism
$c_{W}:H_{c}^{n}(W;{\mathbb{Q}})\to
H^{n}(W;{\mathbb{Q}}),c_{C}:H_{c}^{n}(C;{\mathbb{Q}})\to
H^{n}(C;{\mathbb{Q}})$ (which is non-trivial on the integer lattice).
Therefore it makes sense to write
$\int_{X}(u^{n/2})^{2}:=\int_{X}0\oplus(u^{n/2})^{2}=\int_{C}(c_{C}^{-1}(u^{n/2}))^{2}=\frac{1}{2}$
as $c_{C}$ is multiplying by $2$ on the integer lattice. Similarly, we write
$\int_{X}\mu^{2}$ to mean $\int_{X}\mu_{2}^{2}\oplus
0=\int_{W}c_{W}^{-1}(\mu)^{2}$.

######  Proposition 6.4.

We have

$\int_{X}\mu^{2}=\frac{1}{2}.$

###### Proof.

Since $\int_{X}(2u^{n/2})^{2}=2$ and $\int_{X}(2u^{n/2})\cup(\mu+u^{n/2})=1$
and $\int_{X}(\mu+u^{n/2})^{2}=\int_{X}\mu^{2}+\frac{1}{2}>\frac{1}{2}$. To
have the intersection being unimodular, we must have
$\int_{X}\mu^{2}=\frac{1}{2}$. ∎

###  6.2. Proof of Theorem 6.1

The proof of Theorem 6.1 seeks contradictions by finding non-integer Chern
numbers using tools like the Hirzebruch signature theorem. As a warm-up, we
first look at the $n=8$ case by direct computations.

####  6.2.1. The $n=8$ case

###### Proof of Theorem 6.1 for $n=8$.

In this case, the $L$-genus is

$\displaystyle L_{4}(X)$ $\displaystyle=$
$\displaystyle\frac{1}{14175}(381p_{4}(X)-71p_{1}(X)p_{3}(X)-19p_{2}^{2}(X)+22p_{1}^{2}(X)p_{2}(X)-3p_{1}^{4}(X)),$
$\displaystyle p_{1}(X)$ $\displaystyle=$ $\displaystyle
c_{1}^{2}(X)-2c_{2}(X),$ $\displaystyle p_{2}(X)$ $\displaystyle=$
$\displaystyle c_{2}^{2}(X)-2c_{1}(X)c_{3}(X)+2c_{4}(X),$ $\displaystyle
p_{3}(X)$ $\displaystyle=$ $\displaystyle
c_{3}^{2}(X)-2c_{2}(X)c_{4}(X)+2c_{1}(X)c_{5}(X)-2c_{6}(X),$ $\displaystyle
p_{4}(X)$ $\displaystyle=$ $\displaystyle
c_{4}^{2}(X)-2c_{3}(X)c_{5}(X)+2c_{2}(X)c_{6}(X)-2c_{1}(X)c_{7}(X)+2c_{8}(X).$

We also have

$c(C)=1+10u+44u^{2}+112u^{3}+182u^{4}+196u^{5}+140u^{6}+64u^{7},u\in
H^{2}(C;{\mathbb{Z}}),$

and $\int_{X}u^{8}=\frac{1}{2}$. We also have $\int_{X}c_{8}(X)=\chi(X)=10$.
Using $c^{{\mathbb{Q}}}_{i}(X)=c^{{\mathbb{Q}}}_{i}(W)\oplus
c^{{\mathbb{Q}}}_{i}(C)$, any Chern number does not involve $c_{4}$ can be
computed in $C$ and $\int_{X}c_{I}(X)c_{4}(X)=\int_{X}c_{I}(C)c_{4}(C)$ for
$c_{I}=\prod_{j=1}c^{i_{j}}_{j}$ if $i_{4}=0$, hence we can get specific
numbers. The only remaining term is $\int_{X}c_{4}^{2}(X)$, which involves the
unknown $c_{4}(W)$. Then the Hirzebruch signature theorem reads

$2=\sigma(X)=\int_{X}L_{4}(X)=\frac{1}{14175}(305\int_{X}c_{4}^{2}(X)-5036092).$

Hence we have that
$\int_{X}c_{4}^{2}(X)=\frac{5064442}{305}\notin{\mathbb{Z}}$, contradiction. ∎

####  6.2.2. The general case of $n=2^{k}>8$

In principle, one can continue the computation above for larger $n$. However,
it is unrealistic to carry out the specific computations, as the complexity
grows exponentially w.r.t. $k$ both in coefficients and Chern numbers. The
computation is simplified significantly if we compare the discrepancy between
the Hirzebruch signature formulae for $X,X^{\prime}$, where $X^{\prime}$ is
the orbifold obtained from gluing $C$ to ${\mathbb{C}}^{n}/({\mathbb{Z}}/2)$
and only look at the factor of $2$. We first set up some modulo-$2$ properties
for later purposes.

######  Proposition 6.5.

For $n=2^{k}$ and $k\geq 2$, we have that the free part of $c_{n/2}(W)=0\mod
4$ for a hypothetical exact filling $W$.

###### Proof.

Following [42], we have $\sigma(X)-\int_{X}v_{n}^{2}=0\mod 8,$ where $v_{n}$
is an integral lift of the middle Wu class. Since $X$ has an almost complex
structure, there are integral lifts of the Wu classes in terms of the Chern
classes by [42, E 1.2]. Namely, the total integral Wu classes have
characteristic series ($x$ is the Chern root)

$1+x+x^{3}+\ldots+x^{2^{m}-1}+\ldots.$

Let $v_{n}$ denote the integral lift of the middle Wu class in
$H^{n}(X;{\mathbb{Q}})$. As $H^{n}(X;{\mathbb{Q}})=H^{n}(W;{\mathbb{Q}})\oplus
H^{n}(C;{\mathbb{Q}})$, we will compute the two components of $v_{n}$.

For the component on $C$, as
$TC|_{\mathbb{C}\mathbb{P}^{n-1}}\oplus\underline{{\mathbb{C}}}={\mathcal{O}}(1)^{n}\oplus{\mathcal{O}}(2)$,
using the characteristic series, we have

$v_{n}|_{C}=\text{ the }u^{2^{k-1}}\text{ coefficient of
}(1+u+u^{3}+\ldots)^{2^{k}}(1+2u+8u^{3}+\ldots).$

Note that $\binom{2^{k}}{i}=0\mod 4$ if $1\leq i<2^{k-1}$ and
$\binom{2^{k}}{2^{k-1}}=2\mod 4$. Therefore, modulo
$\langle\,4,u^{2^{k-1}+1}\,\rangle$, we have

$\displaystyle(1+u+u^{3}+\ldots)^{2^{k}}(1+2u+8u^{3}+\ldots)$ $\displaystyle=$
$\displaystyle(1+2(u+u^{3}+\ldots)^{2^{k-1}})(1+2u)$ $\displaystyle=$
$\displaystyle 1+2u+2u^{2^{k-1}}$

In particular, we have $v_{n}|_{C}=2u^{2^{k-1}}\mod 4$. On the other hand, by
Lemma 6.6 below, we have $v_{n}|_{W}=c_{n/2}(W)$. By Proposition 6.3 and
$c_{n/2}(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})=0$, we have
$c_{n/2}(W)=0\mod 2$. If we write $c_{n/2}(W)=2m\mu$ (the free part only),
then by Proposition 6.4, we have

$\int_{X}v_{n}^{2}=2+2m^{2}\mod 8.$

Since $\sigma(X)=2$, we must have $m=0\mod 2$, i.e. the free part of
$c_{n/2}(W)=0\mod 4$ for the hypothetical exact filling $W$.

∎

######  Lemma 6.6.

The coefficient of $c_{n/2}$ in the integral Wu class $v_{n}$ is $1$.

###### Proof.

We write $n=2m$. Let $f(x)=1+x+x^{3}+\ldots$ be the characteristic series. We
want to write

$f(x_{1})\cdot\ldots\cdot f(x_{2m})=g_{2m}(e_{1},\ldots,e_{2m})$

where $e_{1},\ldots,e_{2m}$ are the elementary symmetric polynomials. Then the
coefficient we are looking for is the coefficient of the monomial $e_{m}$ in
$g_{2m}$. Note that $e_{2m}$ divides $f(x_{1})\cdot\ldots\cdot
f(x_{2m})-g_{2m-1}(e_{1},\ldots,e_{2m-1})$ as $x_{2m}$ divides it. We have

$f(x_{1})\cdot\ldots\cdot f(x_{2m})-g_{2m-1}(e_{1},\ldots,e_{2m-1})=e_{2m}s$

for some symmetric polynomial $s$. As a consequence, the $e_{m}$ coefficient
of $g_{2m}$ is the same as the $e_{m}$ coefficient of $g_{2m-1}$. We can keep
applying the argument for $m-1$ more times and conclude that $e_{m}$
coefficient of $g_{2m}$ is the same as the $e_{m}$ coefficient of $g_{m}$,
which is clearly $1$. ∎

###### Proof of Theorem 6.1.

Let $X^{\prime}$ be the orbifold obtained from gluing
${\mathbb{C}}^{n}/({\mathbb{Z}}/2)$ to $C$. With the orbifold Hirzebruch
signature theorem in [43], we have

$2=\sigma(X)=\int_{X}L(TX),\quad
1=\sigma(X^{\prime})=\int_{X^{\prime}}L(TX^{\prime}),$

Here $\sigma(X^{\prime})$ is the signature on the _de Rham cohomology_ of the
orbifold $X^{\prime}$, which is isomorphic to the real singular cohomology of
the orbit space, _not the Chen-Ruan cohomology_. The $L$-genus of
$TX^{\prime}$ is defined through the orbifold Chern-Weil theory, which can be
expressed in terms of (orbifold) Pontryagin classes and Chern classes with the
same coefficients as in the manifold case. It is important to note that the
orbifold Hirzebruch signature theorem usually involves integrating
characteristic classes over the singular locus. However, it vanishes in our
case, as the corresponding characteristic is given by [5, §6, page 581] (also
see Proposition 7.4) over the isolated singular point. Note that the only
difference between $H^{*}(X;\mathbb{R}),H^{*}(X^{\prime};\mathbb{R})$ is in
the middle degree from the difference between $W$ and
${\mathbb{C}}^{n}/({\mathbb{Z}}/2)$. Hence we know that the difference of
$\int_{X}L(TX)$ and $\int_{X^{\prime}}L(TX^{\prime})$ is given by a multiple
of the integration of $c_{n}$ and a multiple of the integration of
$c^{2}_{n/2}(W)$. For even $m$, we can write the $L$-genus of a complex bundle
in terms of Chern classes as

$L_{m}=a_{m}c_{2m}+b_{m}c_{m}^{2}+\text{other combinations of Chern classes.}$

Then we have

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\int_{X}L(TX)-\int_{X^{\prime}}L(TX^{\prime})$ (6.4)
$\displaystyle=$
$\displaystyle\int_{X}a_{n/2}c_{n}(X)+\int_{X}b_{n/2}c_{n/2}^{2}(W)-\int_{X^{\prime}}a_{n/2}c_{n}(X^{\prime}).$

Now by the Gauss-Bonnet theorem for orbifolds in [59] and the identification
between the Pfaffian polynomial and the top Chern class on a complex vector
space, we have

$\int_{X^{\prime}}c_{n}(X^{\prime})=\chi^{\rm orb}(X^{\prime})=n+\frac{1}{2},$

where the singularity is counted as $1/2$. Since
$\int_{X}c_{n}(X)=\chi(X)=n+2$, and we assume $c_{n/2}(W)=4l\mu$ by
Proposition 6.5, (6.4) reads

$\frac{3}{2}a_{n/2}+8l^{2}b_{n/2}=1.$ (6.5)

Therefore we need to figure out the coefficient $a_{n/2}$ and $b_{n/2}$. First
for even $m$, we write $L_{m}$ as

$\alpha_{m}p_{m}+\beta_{m}p^{2}_{m/2}+\text{other combinations of the
Pontryagin classes.}$

Note that

$p_{m}=2c_{2m}+c_{m}^{2}+\text{other quadratic terms of Chern classes},$
$p_{m/2}=\pm 2c_{m}+\text{other quadratic terms of Chern classes},$

we have $a_{m}=2\alpha_{m}$ and $b_{m}=\alpha_{m}+4\beta_{m}$. Hence (6.5)
becomes

$3\alpha_{n/2}+8l^{2}(\alpha_{n/2}+4\beta_{n/2})=1.$ (6.6)

By [7, (5)], we have

$\displaystyle\alpha_{m}$ $\displaystyle=$
$\displaystyle\frac{2^{2m}(2^{2m-1}-1)}{(2m)!}B_{m},$ $\displaystyle\beta_{m}$
$\displaystyle=$
$\displaystyle\frac{2^{2m-1}}{\pi^{2m}}(-\zeta^{*}(2m)+\zeta^{*}(m)^{2})$
$\displaystyle=$
$\displaystyle-\frac{2^{2m-1}(2^{2m-1}-1)}{(2m)!}B_{m}+\frac{2^{2m-1}(2^{m-1}-1)^{2}}{m!m!}B^{2}_{m/2},$

where $B_{*}$ are the Bernoulli numbers with a positive sign, see Appendix A.
For $m=n/2=2^{k-1}$, (6.6) becomes

$3\cdot
2^{2m}(2^{2m-1}-1)B_{m}+2^{2m+3}\left(-(2^{2m-1}-1)B_{m}+2\cdot(2^{m-1}-1)^{2}\binom{2m}{m}B_{m/2}^{2}\right)l^{2}=(2m)!$

By Theorem A.1, $B_{m}$ in its lowest fraction form can be written as
$N_{m}/(2D_{m})$ for an odd number $D_{m}$. Hence we have

$3\cdot
2^{2m-1}(2^{2m-1}-1)N_{m}D^{2}_{m/2}+2^{2m+2}\left(-(2^{2m-1}-1)N_{m}D_{m/2}^{2}+(2^{m-1}-1)^{2}\binom{2m}{m}N_{m/2}^{2}D_{m}\right)l^{2}=(2m)!D_{m}D_{m/2}^{2}.$

Note that $2^{2m-1}$ divides $(2m)!$ as $2m$ is a power of $2$.999This holds
only if $m$ is a power of $2$, hence it yields a new proof of the main result
in [72]. We have

$8\left((2^{m-1}-1)^{2}\binom{2m}{m}N_{m/2}^{2}D_{m}-(2^{2m-1}-1)N_{m}D_{m/2}^{2}\right)l^{2}=D^{2}_{m/2}\left(\frac{(2m)!}{2^{2m-1}}D_{m}-3(2^{2m-1}-1)N_{m}\right)$
(6.7)

By Lemmas A.2 and A.5 for $m=2^{k-1}>8$, the right side of (6.7) is $-8\mod
64$. While the left side of (6.7) is $8(1-\binom{2m}{m})l^{2}\mod 64$ by Lemma
A.2. By Lemma A.5,

$\binom{2m}{m}=6\mod 8,\quad l^{2}=0,1,4\mod 8,$

hence

$8(1-\binom{2m}{m})l^{2}=0,24,32\mod 64,$

contradiction!

When $m=8$, we have $N_{8}=3617,N_{4}=1,D_{8}=255,D_{4}=15$. Then (6.7)
becomes

$8\cdot 26266354875\cdot l^{2}=15^{2}\cdot 162465228408.$

In particular, $l^{2}$ is not an integer, contradiction.

∎

######  Remark 6.7.

There is a more direct approach to obtain the key relation (6.5) without using
the orbifold Hirzebruch signature theorem and Gauss-Bonnet theorem. Note the
total Chern class of the symplectic cap is in the product form

$c(C)=(1+u)^{n}(1+2u),\mod u^{n+1}.$

Roughly speaking, the $L$-genus $L(TX)$ should be “close” to that of $C$, i.e.
$(\frac{u}{\tanh u})^{n}\frac{2u}{\tanh 2u}$. More precisely, $c(C)=c(X)$
except for $c_{n/2}$ and $c_{n}$. Using that $\int_{X}u^{n}=\frac{1}{2}$, we
have

$\displaystyle\int_{X}(\frac{u}{\tanh u})^{n}\frac{2u}{\tanh 2u}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\cdot\text{ the coefficient of
$u^{n}$ in }(\frac{u}{\tanh u})^{n}\frac{2u}{\tanh 2u}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\mathbf{i}}\oint(\frac{u}{\tanh
u})^{n}\frac{2u}{\tanh 2u}\frac{1}{u^{n+1}}{\rm d}u$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\mathbf{i}}\oint\frac{(1+\tanh^{2}u)}{\tanh^{n+1}u}{\rm
d}u$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\mathbf{i}}\oint\frac{(1+z^{2})}{z^{n+1}(1-z^{2})}{\rm
d}z=1,\qquad(z=\tanh u)$

Since the $u^{n}$ coefficient of $c(C)$ is $2n+1$, by that
$\int_{X}u^{n}=\frac{1}{2}$, the above computes $\int_{X}L(TX)$ using
“$\int_{X}c_{n}(X)=n+\frac{1}{2}$ and $c_{n/2}(X)=0\oplus c_{n/2}(C)$”. If we
use the correct $c_{n/2}$ and that $\int_{X}c_{n}(X)=n+2$, the Hirzebruch
signature theorem becomes

$2=\frac{3}{2}a_{n/2}+b_{n/2}\int_{X}c^{2}_{n/2}(W)+\int_{X}(\frac{u}{\tanh
u})^{n}\frac{2u}{\tanh 2u},$

that is precisely (6.5).

However, the orbifold approach generalizes to a general quotient singularity
without any reference to a particular symplectic cap $C$, while the direct
computation above requires knowledge of $C$.

####  6.2.3. The $n=4$ case

To prove the $n=4$ case, the Hirzebruch signature theorem alone (or with the
modulo $8$ property of the modified signature in [42]) will not give us a
contradiction. We also need to use that the $\hat{A}$-genus is an integer for
spin manifolds.

###### Proof of Theorem 6.1 for $n=4$.

The second $L$-genus is given by

$L_{2}(X)=\frac{1}{45}(7p_{2}(X)-p^{2}_{1}(X)).$

Since $X$ is an almost complex manifold, hence the Pontryagin classes can be
expressed using Chern classes as

$p_{1}(X)=c^{2}_{1}(X)-2c_{2}(X),\quad
p_{2}(X)=c_{2}^{2}(X)-2c_{1}(X)c_{3}(X)+2c_{4}(X).$

As a consequence, we have

$L_{2}(X)=\frac{1}{45}(3c_{2}^{2}(X)-14c_{1}(X)c_{3}(X)+14c_{4}(X)-c_{1}^{4}(X)+4c_{1}^{2}(X)c_{2}(X)).$

On one hand, we have

$c(C)=(1+u)^{4}(1+2u)\mod u^{4}=1+6u+14u^{2}+16u^{3},u\in
H^{2}(C;{\mathbb{Z}})$

We have $\int_{X}u^{4}=\frac{1}{2}$ when we view $u\in
H^{2}(C;{\mathbb{Z}})\subset H^{2}(C;{\mathbb{Q}})\simeq
H^{2}_{c}(C;{\mathbb{Q}})$. Finally, we have $\int_{X}c_{4}(X)=6$ and
$\int_{X}c^{2}_{1}(X)c_{2}(X)=\int_{X}c_{1}^{2}(C)c_{2}(C)$ using the natural
inclusion from $c^{{\mathbb{Q}}}_{i}(X)=c^{{\mathbb{Q}}}_{i}(W)\oplus
c^{{\mathbb{Q}}}_{i}(C)$. Then by the Hirzebruch signature theorem, we have

$\int_{X}L_{2}(X)=\sigma(X)=2.$

Hence we can compute that $\int_{X}c_{2}^{2}(X)=106$. As a consequence, we
have $\int_{W}c_{2}^{2}(W)=8$. Since $p_{1}(W)=2c_{2}(W)$, we have
$\int_{W}p_{1}^{2}(W)=32$.

Now we consider a new closed manifold $Y=W\cup_{\phi}W$, where
$\phi:\mathbb{R}\mathbb{P}^{7}\to\mathbb{R}\mathbb{P}^{7}$ is the orientation
reversing diffeomorphism induced from
$(z_{1},z_{2},z_{3}.z_{4})\to(\overline{z}_{1},z_{2},z_{3},z_{4})$. Then we
have $\sigma(Y)=2$ and $\int_{Y}p_{1}^{2}(Y)=2\int_{W}p_{1}^{2}(W)=64$. Hence
from the Hirzebruch signature theorem

$\frac{1}{45}\int_{Y}7p_{2}(Y)-p_{1}^{2}(Y)=2,$

we have $\int_{Y}p_{2}(Y)=22$.

We claim that $Y$ is a spin manifold. To prove $w_{2}(Y)=0$, it is sufficient
to prove $\langle\,w_{2}(Y),A\,\rangle=0$ for any $A\in
H_{2}(Y;{\mathbb{Z}}/2)$. Since $W$ is simply connected, we have that $Y$ is
also simply connected. Then by the universal coefficient theorem, Hurewicz
theorem, and Whitney embedding theorem, we have that $H_{2}(Y;{\mathbb{Z}}/2)$
is generated by embedded spheres. Therefore we have that
$H^{2}(Y;{\mathbb{Z}}/2)$ is generated by embedded spheres contained in one of
the two copies of $W$ and the glued sphere from two disks in each $W$ whose
boundary is nontrivial in $\pi_{1}(\mathbb{R}\mathbb{P}^{7})$. Assume $A\in
H_{2}(Y;{\mathbb{Z}}/2)$ is represented by a map $f:S^{2}\to Y$, then
$\langle\,w_{2}(Y),A\,\rangle=0$ is equivalent to that $f^{*}TY$ is trivial.
If $f$ is contained in one of $W$, since $c_{1}^{{\mathbb{Z}}}(W)=0$, we have
$f^{*}TY$ is trivial. If $f$ is the glued sphere, we may assume the separating
circle $C$ is the circle in the $z_{1}$ complex line. We can fix a
trivialization of $TY|_{C}$, which is derived from a trivialization of the
normal bundle of $C$ in $\mathbb{R}\mathbb{P}^{7}$, a chosen orientation of
$C$, and outward normal vectors of the first copy of $W$. Since $\phi$ maps
$C$ to $C$ in an orientation reversing manner, $\phi$ push the trivialization
to the same (up to homotopy as it is multiplying the constant map
$S^{1}\to\operatorname{diag}(-1,-1,1,\ldots,1)$) trivialization on the second
copy of $W$. Therefore the second Stiefel-Whitney class of $f^{*}TY$ is
measured by the sum of the discrepancy of the trivialization of $f^{*}TY$ on
the two disks with the above trivialization, which is clearly a multiple of
$2$, hence zero. Therefore $Y$ is a spin manifold. Now by the Atiyah-Singer
index theorem for Dirac operators, we have

${\mathbb{Z}}\ni\int_{Y}\hat{A}(Y)=\frac{1}{5760}\int_{Y}-4p_{2}(Y)+7p_{1}^{2}(Y)=\frac{1}{16}$

which is a contradiction. ∎

###  6.3. Unique exact orbifold fillings

######  Proposition 6.8.

Let $M$ be an exact cobordism from $(S^{2n-1},\xi_{\rm std})$ to itself for
$n\geq 2$, then $M$ is an $h$-cobordism.

###### Proof.

Since $M$ glued with ${\mathbb{D}}^{2n}$ is an exact filling of
$(S^{2n-1},\xi_{\rm std})$, which is diffeomorphic to ${\mathbb{D}}^{2n}$.
Hence we have $\pi_{1}(M)=0$ and $H^{*}(M,\partial_{-}M;{\mathbb{Z}})=0$ by
the tautological long exact sequence of pairs and excision. Then by Lefschetz
duality and the universal coefficient theorem, we have
$H_{*}(M,\partial_{-}M;{\mathbb{Z}})=H_{*}(M,\partial_{+}M;{\mathbb{Z}})=0$.
Then the Hurewicz theorem implies that $M$ is an $h$-cobordism. ∎

######  Theorem 6.9 (I).

Let $W$ be an exact orbifold filling of $(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm
std})$ for $n\geq 3$ odd. Then $W$ is diffeomorphic to
${\mathbb{D}}^{2n}/({\mathbb{Z}}/2)$.

###### Proof.

Let $W$ be an exact orbifold filling, by [38, Theorem G], there is exactly one
orbifold point modelled on ${\mathbb{C}}^{n}/({\mathbb{Z}}/2)$. Then, outside
a small neighborhood of the singularity, we get an exact cobordism $M$ with
both concave boundary $\partial_{-}M$ and convex boundary $\partial_{+}M$ are
$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$. By the same argument of
Proposition 5.1, we know that $\pi_{1}(\mathbb{R}\mathbb{P}^{2n-1})\to\pi^{\rm
orb}_{1}(W)$ is surjective. Since
$\pi_{1}(\mathbb{R}\mathbb{P}^{2n-1})\to\pi_{1}^{\rm
orb}({\mathbb{C}}^{n}/({\mathbb{Z}}/2))$ is an isomorphism, by the van-Kampen
theorem, we have $\pi^{\rm orb}_{1}(W)=\pi_{1}(M)$. Therefore we have
$\pi^{\rm orb}_{1}(W)=\pi_{1}(M)=0$ or $\pi^{\rm
orb}_{1}(W)=\pi_{1}(M)={\mathbb{Z}}/2$. If $\pi^{\rm
orb}_{1}(W)=\pi_{1}(M)=0$, then $H^{2}(M;{\mathbb{Z}})$ has no torsion. Since
$c_{1}(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})\neq 0$ when $n$ is odd, we
must have $\operatorname{rank}H^{2}(M;{\mathbb{Z}})\geq 1$. We can stack
several copies of $M$ together to get an orbifold filling of
$(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$, such that the cobordism $M$
from deleting the orbifold point satisfying
$\operatorname{rank}H^{2}(M;{\mathbb{Z}})\geq 3$.

Now we use the homotopy argument in the proof of [38, Theorem B], namely we
consider the moduli space of thrice-punctured sphere solving the perturbed
Cauchy-Riemann equation with two positive punctures and one negative puncture,
i.e. the moduli space defining the pair-of-pants product. By Proposition 3.4,
$x_{(\operatorname{id})}+a$ represents a class in $SH_{+}^{*}(W;{\mathbb{Q}})$
such that $x_{(\operatorname{id})}+a$ is sent to $|G|$ in the composition
$SH_{+}^{*}(W;{\mathbb{Q}})\to H^{*+1}_{\mathrm{C}R}(W;{\mathbb{Q}})\to
H^{0}_{\mathrm{C}R}(W;{\mathbb{Q}})$. We plug in $x_{(\operatorname{id})}+a$
as one of the inputs and a constant orbit $y$ representing a class in
$H^{2}(M;{\mathbb{Q}})\subset H^{2}_{\mathrm{C}R}(W;{\mathbb{Q}})$ as another
input. Then by analyzing the degeneration, the output is a cochain $z$ from
the symplectic cohomology cochain complex, such that

$\delta(z)=m(\delta_{+,0}(x_{(\operatorname{id})}+a),y)$ (6.8)

where $\delta_{+,0}$ is the cochain map defining
$SH_{+}^{*}(W;{\mathbb{Q}})\to H_{\mathrm{C}R}^{*+1}(W;{\mathbb{Q}})$ and $m$
is the map defining the pair-of-pants product. By our assumption,
$\delta_{+,0}(x_{(\operatorname{id})}+a)$ is $|G|$ plus constant orbits
representing a class $A$ in the kernel of
$H^{*}_{\mathrm{C}R}(W;{\mathbb{Q}})\to H^{0}_{\mathrm{C}R}(W;{\mathbb{Q}})$.
As both $\delta_{+,0}(x_{(\operatorname{id})}+a)$, $y$ are constant orbits,
and $m(\delta_{+,0}(x_{(\operatorname{id})}+a),y)$ also consists of constant
orbits. Moreover, since $y$ is from $M$, no constant holomorphic curve
concentrated at the singularity appears in the above analysis. We write
$z=z_{+}+z_{0}$, where $z_{+}$ is the positive cochain part and $z_{0}$ is the
Morse theory part. Then (6.8) implies that

$\delta_{+}(z_{+})=0,\quad\delta_{+,0}(z_{+})=m(|G|+A,y)\in
H^{*}_{\mathrm{C}R}(W;{\mathbb{Q}})$

where $\delta_{+}$ is the differential on the positive cochain complex. Since
$m(|G|+A,y)$ does not involve constant holomorphic curves concentrated at the
singularity, $m(|G|+A,y)$ is the cup-product of $y$ and $|G|+A^{\prime}$ in
the cohomology $H^{*}(W;{\mathbb{Q}})\subset
H_{\mathrm{C}R}^{*}(W;{\mathbb{Q}})$, where $A^{\prime}$ is the
$H^{*}(W;{\mathbb{Q}})$-component of $A$. In particular, $m(|G|+A,\cdot)$ is
an injective map from $H^{*}(M;{\mathbb{Q}})$. Therefore we have that $\dim
SH^{*}_{+}(W;{\mathbb{Q}})$ is at least $\dim H^{*}(M;{\mathbb{Q}})\geq 3$ by
$z_{+}$ above. This contradicts with Proposition 3.3.101010We get an instant
contradiction if Proposition 3.4 holds for exact orbifolds, this boils down to
establishing the pair-of-pants product for exact orbifolds.

When $\pi^{\rm orb}_{1}(W)=\pi_{1}(M)={\mathbb{Z}}/2$. In this case
$\pi_{1}(\mathbb{R}\mathbb{P}^{2n-1})\to\pi^{\rm
orb}_{1}(W)=\pi_{1}(M)={\mathbb{Z}}/2$ is an isomorphism. We claim that
${\mathbb{Z}}/2=\pi_{1}(\partial_{-}M)\to\pi_{1}(W)$ is also an isomorphism.
Otherwise, the map is zero. We can consider the universal cover of $M$, which
an exact cobordism with convex boundary $(S^{2n-1},\xi_{\rm std})$ and concave
end $(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm
std})\sqcup(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$. This would yield an
exact orbifold filling of $(S^{2n-1},\xi_{\rm std})$ with two orbifold points,
contradicting [38, Theorem C]. Now since
${\mathbb{Z}}/2=\pi_{1}(\partial_{-}M)\to\pi_{1}(W)$ is an isomorphism, the
universal cover $\widetilde{M}$ is an $h$-cobordism by Proposition 6.8. In
particular, $M$ itself an $h$-cobordism. Since the Whitehead group of
${\mathbb{Z}}/2$ is trivial, $M$ is an $s$-cobordism, hence a trivial
cobordism as $n\geq 3$. As a consequence, we have that $W$ is diffeomorphic to
${\mathbb{D}}^{n}/({\mathbb{Z}}/2)$. ∎

##  7\. More non-existence results

Propositions 3.4 and 4.4 provide evidences for the conjecture that exact
orbifold fillings of $(S^{2n-1}/G,\xi_{\rm std})$ might be unique, namely
${\mathbb{C}}^{n}/G$, when it is a terminal singularity. If this conjecture is
true, then in particular, $(S^{2n-1}/G,\xi_{\rm std})$ has no exact filling
when ${\mathbb{C}}^{n}/G$ is terminal. In addition to the case of real
projective spaces discussed in the previous section, we give various but non-
exhaustive lists of cases where we can establish the non-existence of exact
fillings using the topological information obtained in previous sections. The
following are some of the arguments we use to arrive at a contradiction
assuming there is an exact filling $W$:

  1. (1)

From enough non-vanishing maps $H^{*}(W;{\mathbb{Z}})\to
H^{*}(S^{2n-1}/G;{\mathbb{Z}})$, we obtain a contradiction with $\dim
H^{*}(W;{\mathbb{Q}})=|\operatorname{Conj}(G)|$ using Lefschetz duality and
the universal coefficient theorem. This can be achieved by finding enough non-
vanishing Chern classes of $(S^{2n-1}/G,\xi_{\rm std})$, which is the method
used in [72] or by exploiting the non-trivial product structure on
$H^{*}(S^{2n-1}/G;{\mathbb{Z}})$ as well as the cup product length estimate in
§4, e.g. Theorem 7.8.

  2. (2)

Let $X$ be the compact symplectic manifold obtained by gluing a symplectic cap
of dimension $4k$ and consider its signature $\sigma(X)$. Then a potential
contradiction of having non-integral characteristic classes may arise using
the the Hirzebruch signature theorem. Moreover, by [42], we have
$\sigma(X)-\int_{X}v^{2}_{2k}=0\mod 8$, where $v^{2}_{2k}$ is an integral lift
of the middle Wu class, which always exists for almost complex manifolds. This
also provides a place to look for a contradiction.

  3. (3)

If $c_{1}^{{\mathbb{Q}}}(W)=0$, then the boundary rational grading is a global
rational grading and the isomorphism $SH^{*}_{+}(W;{\mathbb{Q}})\simeq
H^{*+1}(W;{\mathbb{Q}})$ is a graded isomorphism. We can get a contradiction
if $SH^{*}_{+}(W;{\mathbb{Q}})$ is not supported in integer degrees.

###  7.1. Preliminary facts

######  Proposition 7.1.

Assume ${\mathbb{C}}^{n}/G$ is an isolated quotient singularity. Let $W$ be a
compact oriented manifold such that $\partial W=S^{2n-1}/G$. Then
$H^{i}(W;{\mathbb{Q}})=H^{2n-i}(W;{\mathbb{Q}})$ for $0<i<2n$.

###### Proof.

Note that we have the long exact sequence

$\ldots\to H^{*}_{c}(W;{\mathbb{Q}})\to H^{*}(W;{\mathbb{Q}})\to
H^{*}(\partial W;{\mathbb{Q}})\to\ldots$

Since $H^{*}(S^{2n-1}/G;{\mathbb{Q}})=H^{*}(S^{2n-1};{\mathbb{Q}})$, we have
$H^{i}_{c}(W;{\mathbb{Q}})=H^{i}(W;{\mathbb{Q}})$ when $0<i<2n$. Then by
Lefschetz duality, we have
$H^{i}(W;{\mathbb{Q}})=H^{2n-i}_{c}(W;{\mathbb{Q}})=H^{2n-i}(W;{\mathbb{Q}})$
for $0<i<2n$. ∎

######  Proposition 7.2.

Assume ${\mathbb{C}}^{n}/G$ is an isolated quotient singularity. Suppose that
both $H^{*}(W;{\mathbb{Q}})$ and $H^{*}(BG;{\mathbb{Z}})$ are supported in
even degrees. Let $\phi^{*}:H^{*}(W;{\mathbb{Z}})\to
H^{*}(S^{2n-1}/G;{\mathbb{Z}}),\phi^{*}_{\operatorname{Tor}}:\operatorname{Tor}H^{*}(W;{\mathbb{Z}})\to
H^{*}(S^{2n-1}/H;{\mathbb{Z}})$ denote the maps given by fullbacks along the
inclusion of the boundary. For $0<i<n$, we have

$|\operatorname{im}\phi^{2n-2i}_{\operatorname{Tor}}|=|\operatorname{coker}\phi^{2i}|.$

###### Proof.

By Proposition 2.1, $H^{*}(S^{2n-1}/G;{\mathbb{Z}})=H^{*}(BG;{\mathbb{Z}})$
whenever $*<2n-1$. Therefore, by our assumption that $H^{*}(BG;{\mathbb{Z}})$
is supported in even degrees, we have exact sequences

$0\to H^{2i}(W,\partial W;{\mathbb{Z}})\to H^{2i}(W;{\mathbb{Z}})\to
H^{2i}(S^{2n-1}/G;{\mathbb{Z}})\to H^{2i+1}(W,\partial W;{\mathbb{Z}})\to
H^{2i+1}(W;{\mathbb{Z}})\to 0,$ $0\to H^{2n-2i}(W,\partial W;{\mathbb{Z}})\to
H^{2n-2i}(W;{\mathbb{Z}})\to H^{2n-2i}(S^{2n-1}/G;{\mathbb{Z}})\to
H^{2n-2i+1}(W,\partial W;{\mathbb{Z}})\to H^{2n-2i+1}(W;{\mathbb{Z}})\to 0.$

Let $k=\dim H^{2i}(W;{\mathbb{Q}})=\dim H^{2n-2i}(W;{\mathbb{Q}})$. By
Lefschetz duality and the universal coefficient theorem, we have

$H^{2i}(W,\partial
W;{\mathbb{Z}})=H_{2n-2i}(W;{\mathbb{Z}})={\mathbb{Z}}^{k}\oplus\operatorname{Tor}H^{2n-2i+1}(W;{\mathbb{Z}})={\mathbb{Z}}^{k}\oplus
H^{2n-2i+1}(W;{\mathbb{Z}}),$ $H^{2n-2i}(W,\partial
W;{\mathbb{Z}})=H_{2i}(W;{\mathbb{Z}})={\mathbb{Z}}^{k}\oplus\operatorname{Tor}H^{2i+1}(W;{\mathbb{Z}})={\mathbb{Z}}^{k}\oplus
H^{2i+1}(W;{\mathbb{Z}}),$ $H^{2i+1}(W,\partial
W;{\mathbb{Z}})=H_{2n-2i-1}(W;{\mathbb{Z}})=\operatorname{Tor}H^{2n-2i}(W;{\mathbb{Z}}),$
$H^{2n-2i+1}(W,\partial
W;{\mathbb{Z}})=H_{2i-1}(W;{\mathbb{Z}})=\operatorname{Tor}H^{2i}(W;{\mathbb{Z}}).$

Hence the two long exact sequences become

$0\to{\mathbb{Z}}^{k}\oplus
H^{2n-2i+1}(W;{\mathbb{Z}})\to{\mathbb{Z}}^{k}\oplus\operatorname{Tor}H^{2i}(W;{\mathbb{Z}})\to
H^{2i}(S^{2n-1}/G;{\mathbb{Z}})\to\operatorname{Tor}H^{2n-2i}(W;{\mathbb{Z}}),\to
H^{2i+1}(W;{\mathbb{Z}})\to 0,$ $0\to{\mathbb{Z}}^{k}\oplus
H^{2i+1}(W;{\mathbb{Z}})\to{\mathbb{Z}}^{k}\oplus\operatorname{Tor}H^{2n-2i}(W;{\mathbb{Z}})\to
H^{2n-2i}(S^{2n-1}/G;{\mathbb{Z}})\to\operatorname{Tor}H^{2i}(W;{\mathbb{Z}})\to
H^{2n-2i+1}(W;{\mathbb{Z}})\to 0.$

From the first exact sequence, we have

$|\operatorname{Tor}H^{2n-2i}(W;{\mathbb{Z}})|=|H^{2i+1}(W;{\mathbb{Z}})|\cdot|\operatorname{coker}\phi^{2i}|.$

From the second exact sequence, we have a short exact sequence,

$0\to
H^{2i+1}(W;{\mathbb{Z}})\to\operatorname{Tor}H^{2n-2i}(W;{\mathbb{Z}})\to\operatorname{im}\phi^{2n-2i}_{\operatorname{Tor}}\to
0,$

hence

$|\operatorname{Tor}H^{2n-2i}(W;{\mathbb{Z}})|=|H^{2i+1}(W;{\mathbb{Z}})|\cdot|\operatorname{im}\phi^{2n-2i}_{\operatorname{Tor}}|,$

which implies the claim. ∎

That $H^{*}(W;{\mathbb{Q}})$ is supported in even degrees holds for our
hypothetical fillings if the singularity is terminal by Proposition 3.4 and
that $H^{*}(BG;{\mathbb{Z}})$ is supported in even degrees holds for cyclic or
$A,D,E$ type groups. A special case of Proposition 7.2 is when $n$ is even and
$i=n/2$. If $|H^{n}(S^{2n-1}/G;{\mathbb{Z}})|$ is not a square, then we have
$|\operatorname{im}\phi^{n}_{\operatorname{Tor}}|\neq|\operatorname{im}\phi^{n}|$,
i.e. $H^{n}(W;{\mathbb{Q}})\neq 0$. If one can argue that
$\phi^{2i},\phi^{2n-2i}$ are both surjective and
$H^{2i}(S^{2n-1}/G;{\mathbb{Z}})=H^{2n-2i}(S^{2n-1}/G;{\mathbb{Z}})\neq
0$,111111The first equality always holds by Poincaré duality and the universal
coefficient theorem. then we must have
$H^{2i}(W;{\mathbb{Q}})=H^{2n-2i}(W;{\mathbb{Q}})\neq 0$. This is the argument
for [72, Theorem 1.1]. More generally, we have

######  Proposition 7.3.

Under the same assumptions of Proposition 7.2. Suppose we have
$H^{2i}(S^{2n-1}/G;{\mathbb{Z}})=H^{2n-2i}(S^{2n-1}/G;{\mathbb{Z}})={\mathbb{Z}}/m$
where $1<i<n$. Let $p$ be a prime number that divides $m$. If we know that
both $\operatorname{im}\phi^{2i},\operatorname{im}\phi^{2n-2i}$ contain
elements which are non-zero modulo $p$, then
$H^{2i}(W;{\mathbb{Z}}),H^{2n-2i}(W;{\mathbb{Z}})$ can not be torsion.

###### Proof.

Under the assumption, we know that
$\operatorname{coker}\phi^{2i},\operatorname{coker}\phi^{2n-2i}$ can only be
${\mathbb{Z}}/k$, where $k$ is a factor of $m$ and is coprime to $p$. If
$H^{2n-2i}(W;{\mathbb{Z}})$ is torsion, by Proposition 7.2, we know that
$|\operatorname{im}\phi^{2n-2i}|=|\operatorname{im}\phi^{2n-2i}_{\operatorname{Tor}}|=|\operatorname{coker}\phi^{2i}|$
is coprime to $p$, contradicting with
$|\operatorname{im}\phi^{2n-2i}|\cdot|\operatorname{coker}\phi^{2n-2i}|=m$,
which is divisible by $p$. ∎

Finally, we recall the orbifold Hirzebruch signature theorem for orbifolds
with isolated singularities following [5, 43]. Assume $G\subset U(n)$, such
that ${\mathbb{C}}^{n}/G$ is an isolated singularity. For each $g\in
G\backslash\\{\operatorname{id}\\}$, $g$ can be diagonalized as
$\operatorname{diag}(\theta^{1}_{g},\ldots,\theta^{n}_{g})$ for
$0<\theta^{j}_{g}<2\pi$. Then we define

$L_{{\mathbb{C}}^{n}/G}:=\frac{1}{|G|}\sum_{\begin{subarray}{c}g\in
G\backslash\\{\operatorname{id}\\}\\\
g\neq-\operatorname{id}\end{subarray}}\prod_{j=1}^{n}(\mathbf{i}\tan(\theta^{j}_{g}/2))^{-1}=\sum_{\begin{subarray}{c}(g)\in\operatorname{Conj}(G)\backslash\\{(\operatorname{id})\\}\\\
(g)\neq(-\operatorname{id})\end{subarray}}\frac{1}{|C_{g}|}\prod_{j=1}^{n}(\mathbf{i}\tan(\theta^{j}_{g}/2))^{-1}.$

Here $\tan(\pi/2)$ is defined to be $\infty$ (this also explains why we drop
$g=-\operatorname{id}$). In particular, when
$G={\mathbb{Z}}/2=\\{\operatorname{id},-\operatorname{id}\\}$, we define
$L_{{\mathbb{C}}^{n}/G}=0$.

######  Proposition 7.4 ([43]).

Assume $X$ is $4n$ dimensional oriented orbifold with only isolated
singularities modeled on ${\mathbb{C}}^{2n}/G_{i}$ with the complex
orientation for $1\leq i\leq k$. Then we have

$\sigma(X)=\int_{X}L(TX)+\sum_{i=1}^{k}L_{{\mathbb{C}}^{2n}/G_{i}}.$

###  7.2. General results