Centralizers of Hamiltonian circle actions on rational ruled surfaces

Pranav Chakravarthy

Department of Mathematics
The Hebrew University of Jerusalem
Israel
<EMAIL_ADDRESS>
Martin Pinsonnault

Department of Mathematics
University of Western Ontario
Canada
<EMAIL_ADDRESS>
###### Abstract

In this paper, we compute the homotopy type of the group of equivariant
symplectomorphisms of $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ under the presence of
Hamiltonian group actions of the circle $S^{1}$. We prove that the group of
equivariant symplectomorphisms are homotopy equivalent to either a torus, or
to the homotopy pushout of two tori depending on whether the circle action
extends to a single toric action or to exactly two non-equivalent toric
actions. This follows from the analysis of the action of equivariant
symplectomorphisms on the space of compatible and invariant almost complex
structures $\mathcal{J}^{S^{1}}_{\omega}$. In particular, we show that this
action preserves a decomposition of $\mathcal{J}^{S^{1}}_{\omega}$ into strata
which are in bijection with toric extensions of the circle action. Our results
rely on $J$-holomorphic techniques, on Delzant’s classification of toric
actions and on Karshon’s classification of Hamiltonian circle actions on
$4$-manifolds.

###### Acknowledgements

The paper contains results from the first author’s PhD thesis at the
University of Western Ontario under the supervision of the second author. The
first author would like to thank Yael Karshon and Mohan Swaminathan for
helpful discussions during the completion of the project. The first author
would also like to thank the Hebrew University of Jerusalem where this project
was completed. The second author is supported by the Natural Science and
Engineering Research Council of Canada.

###### Contents

  1. 1 Introduction
  2. 2 Hamiltonian torus actions on ruled $4$-manifolds
    1. 2.1 Preliminaries on Hamiltonian actions
    2. 2.2 Torus actions on $S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$
      1. 2.2.1 Symplectic forms on $S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$
      2. 2.2.2 Toric actions on $S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$
      3. 2.2.3 Hamiltonian circle actions on $S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$
    3. 2.3 Equivalent circle actions
      1. 2.3.1 Equivalent circle actions in $\mathbb{T}^{2}_{m}$
      2. 2.3.2 Toric extensions of circle actions
  3. 3 Action of $\operatorname{Symp}^{S^{1}}(S^{2}\times S^{2},\omega_{\lambda})$ on $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$
    1. 3.1 $J$-Holomorphic Preliminaries
    2. 3.2 $J$-holomorphic spheres in $S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$
    3. 3.3 Intersection of $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ with the strata
    4. 3.4 Symplectic actions of compact abelian groups on $\mathbb{R}^{4}$
    5. 3.5 Homotopical description of $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{k}$
      1. 3.5.1 Notation
      2. 3.5.2 Case 1: $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times S^{2},\omega_{\lambda})$ action on $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{2s}$ with $s\neq 0$
      3. 3.5.3 Case 2: $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times S^{2},\omega_{\lambda})$ action on $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{0}$
  4. 4 The homotopy type of the symplectic centralisers of $S^{1}(a,b;m)$
    1. 4.1 When $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ is homotopy equivalent to a single symplectic orbit
    2. 4.2 The two orbits case
      1. 4.2.1 Homological injectivity
      2. 4.2.2 Cohomology module of the centralizer
      3. 4.2.3 The homotopy pushout $T_{m}\leftarrow S^{1}(\pm 1,b;m)\to T_{m^{\prime}}$
      4. 4.2.4 Homotopy type of $S^{1}(\pm 1,b;m)$ equivariant symplectomorphisms
    3. 4.3 Centralizers of Hamiltonian $S^{1}$ actions on $S^{2}\times S^{2}$
  5. 5 Partition of the space of invariant almost-complex structures
    1. 5.1 Space of invariant complex structures
    2. 5.2 Construction of Equivariant moduli spaces
    3. 5.3 Isotropy representations
      1. 5.3.1 Even Hirzebruch surfaces and their isometry groups
      2. 5.3.2 Even isotropy representations and codimension calculation
  6. 6 Odd Hirzebruch surfaces
    1. 6.1 Homotopy type of $\operatorname{Symp}(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$
    2. 6.2 Isometry groups of odd Hirzebruch surfaces
      1. 6.2.1 Odd isotropy representations and codimension calculation
  7. A Equivariant Gauge Groups
  8. B Equivariant Gompf argument
  9. C Equivariant Differential Topology
  10. D Alexander-Eells isomorphism

##  Chapter 1 Introduction

Let $(M,\omega)$ be a closed symplectic manifold. The group
$\operatorname{Symp}(M,\omega)$ of symplectomorphisms, equipped with the
standard $C^{\infty}$-topology, is an infinite dimensional Fréchet Lie group.
A fundamental theorem of Banyaga [Ban88] states that the isomorphism type of
$\operatorname{Symp}(M,\omega)$ as a discrete group determines the conformal
symplectomorphism type of $(M,\omega)$. However, the ways topological
properties of $\operatorname{Symp}(M,\omega)$ relate to the geometry of
$(M,\omega)$ remain largely unknown. To measure our level of ignorance, it
suffices to observe that the following two questions are almost completely
open in all dimensions $2n\geq 4$:

  1. Q1.

At the most basic level, how does the homotopy type of
$\operatorname{Symp}(M,\omega)$ depend on $\omega\,$?

  2. Q2.

Similarly, suppose $(M,\omega)$ admits a symplectic or Hamiltonian actions by
a compact Lie group $G$ (possibly finite), viewed here as a Lie subgroup
$G\subset\operatorname{Symp}(M,\omega)$. What are the homotopy types of the
centralizer and of the normalizer $N(G)$ in
$C(G)\subset\operatorname{Symp}(M,\omega)\,$? Can we understand the inclusions

$G\hookrightarrow\operatorname{Symp}(M,\omega)\quad\text{~{}and~{}}\quad
C(G)\hookrightarrow N(G)\hookrightarrow\operatorname{Symp}(M,\omega)$

from a homotopy theoretic point of view?

Regarding the first question, most efforts as been devoted to the study of
symplectomorphism groups of rational $4$-manifolds. Following the seminal work
of M. Gromov [Gro85] who showed that the group of compactly supported
symplectomorphisms of $\mathbb{R}^{4}$ is contractible, the homotopical
properties of the group of symplectomorphisms of $\mathbb{C}P^{2}$,
$S^{2}\times S^{2}$ and of the $k$-fold symplectic blow-ups
$\mathbb{C}P^{2}\\#k\overline{\mathbb{C}P}^{2}$, $k\leq 5$, were studied in
several papers such as [Abr98], [AGK09], [AM00], [Pin08b], [AG04], [AP13],
[AE19], and [LLW22]. In particular, for $\mathbb{C}P^{2}$, $S^{2}\times
S^{2}$, and $\mathbb{C}P^{2}\\#k\overline{\mathbb{C}P^{2}}$, $k\leq 3$, the
rational homotopy type of $\operatorname{Symp}(M,\omega)$ can be described
precisely in terms of the cohomology class $[\omega]$. For $k\geq 4$, partial
results are known, mostly for $\pi_{0}(\operatorname{Symp}(M,\omega))$ and
$\pi_{1}(\operatorname{Symp}(M,\omega))$. It is worth pointing out that all
these results rely on special properties of $J$-holomorphic curves in
symplectic $4$-manifolds and, as such, do not generalize readily to higher
dimensions.

The second question can be partially answered in the case of Hamiltonian toric
actions using moment map techniques, see [Pin08a]. If a torus
$\mathbb{T}^{n}\subset\operatorname{Ham}(M^{2n},\omega)$ acts effectively on
$(M^{2n},\omega)$ with moment map $\mu:M^{2n}\to\mathbb{R}^{n}$, then

  1. 1.

the centralizer $C(\mathbb{T}^{n})$ is equal to the group of all
symplectomorphisms $\phi$ that preserve the moment map, that is, such that
$\mu\circ\phi=\mu$.

  2. 2.

$C(\mathbb{T}^{n})$ is a maximal torus in
$\operatorname{Symp}(M^{2n},\omega)$, that is, a maximal, connected, and
abelian subgroup of $\operatorname{Symp}(M^{2n},\omega)$. In particular, since
toric manifolds are simply-connected,
$C(\mathbb{T}^{n})\subset\operatorname{Ham}(M^{2n},\omega)$.

  3. 3.

$C(\mathbb{T}^{n})$ deformation retracts onto $\mathbb{T}^{n}$. In particular,
the homotopy type of the centraliser of a toric action is independent of the
action.

  4. 4.

Moreover, the Weyl group
$W(\mathbb{T}^{n}):=N(\mathbb{T}^{n})/C(\mathbb{T}^{n})$ is always
finite111Futhermore, as for maximal tori in compact Lie groups, it can be
shown that the number of conjugacy classes of toric centralizers is finite,
and that each $C(\mathbb{T}^{n})$ is flat and totally geodesic in
$\operatorname{Symp}(M^{2n},\omega)$ for the $L^{2}$ metric.

On the other hand, even for toric actions, very little is known about the
homotopy theoretic properties of the inclusions
$G\hookrightarrow\operatorname{Symp}(M,\omega)$ or $C(G)\hookrightarrow
N(G)\hookrightarrow\operatorname{Symp}(M,\omega)$. Two noteworthy exceptions
are i) the works of McDuff-Slimowitz [MS01] and McDuff-Tolman [MT06] who show
that under some rather mild conditions, Hamiltonian $G$ actions induce
injective maps
$\pi_{1}(G)\hookrightarrow\pi_{1}(\operatorname{Ham}(M^{2n},\omega))$, and ii)
the results on symplectomorphism groups of rational surface mentionned above
that, as a byproduct, allow one to understand the subrings of
$\pi_{*}(\operatorname{Symp}(M,\omega))$ or
$H_{*}(\operatorname{Symp}(M,\omega))$ that the various Hamiltonian actions on
$(M,\omega)$ generate.

In this paper we combine pseudo-holomorphic curve techniques with moment map
techniques to determine the homotopy type of equivariant symplectomorphisms of
$S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ under
the presence of a Hamiltonian circle action. Our main result are Theorem 4.7
and Theorem 6.7 that give the full homotopy type of the centralizer
$C(S^{1})\subset\operatorname{Symp}(M,\omega)$ for all choices of symplectic
forms and of Hamiltonian circle actions on these two $4$-manifolds. Contrary
to the toric case, the homotopy type of the stabilizer $C(S^{1})$ is not
constant and depends, essentially, on whether the circle action extends to a
single toric action or to two distinct toric actions. In the former case,
apart from a few exceptional circle actions that must be treated separately,
$C(S^{1})$ retracts onto the unique torus $\mathbb{T}^{2}$ the circle extends
to. In the latter case, the two toric actions $\mathbb{T}^{2}_{1}$,
$\mathbb{T}^{2}_{2}$ that extend the circle action to do not commute, even up
to homotopy. We show that the Pontryagin products of the generators of
$H_{1}(\mathbb{T}^{2}_{1})$ and $H_{1}(\mathbb{T}^{2}_{2})$ generate a
subalgebra $P^{alg}\subset H_{*}(C(S^{1}))$ that contains classes of arbitrary
large degrees. In particular, $C(S^{1})$ does not have the homotopy type of a
finite dimensional $H$-space. Moreover, looking at the action of the
centralizer $C(S^{1})$ on the space of invariant almost complex structures, we
prove that there is an equality of Pontryagin rings $P^{alg}=H_{*}(C(S^{1}))$.
This homology equivalence implies that $C(S^{1})$ has the homotopy type of a
pushout $\mathbb{T}^{2}_{1}\leftarrow S^{1}\to\mathbb{T}^{2}_{2}$ in the
category of topological groups. Finally, when viewed as a bare topological
space, we show that $C(S^{1})$ is homotopy equivalent to the product $\Omega
S^{3}\times S^{1}\times S^{1}\times S^{1}$.

It seems likely that our results on centralizers of Hamiltonian circle actions
could be proven using moment map techniques alone. The main advantage of
introducing pseudo-holomorphic curve techniques is that our setting
generalises to actions of any compact, possibly finite, abelian group
$A\subset\operatorname{Ham}(M,\omega)$. For instance, in the companion paper
[CP], we use the same framework to determine the homotopy type of the
centralizers of most finite cyclic subgroups $\mathbb{Z}_{n}$ acting on
$(S^{2}\times S^{2},\omega_{\lambda})$ and
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$ by
Hamiltonian diffeomorphisms.

In an effort to make the paper understandable to both equivariant geometers
and symplectic topologists, we include more details than would be needed in a
document aimed at either audience alone. The paper is structured as follows:

In Chapter 2, we recall T. Delzant’s equivariant classification of toric
actions in terms of moment polytopes, and Y. Karshon’s equivariant
classifications of Hamiltonian circle actions on $4$-manifolds in terms of
labelled graphs. We then determine all possible toric extensions, up to
equivalences, of an arbitrary Hamiltonian circle actions on $S^{2}\times
S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$.

The crux of the paper lies in Chapters 3 and 4 in which we adapt the framework
of [AM00] to study groups of equivariant symplectomorphisms in the presence of
an $S^{1}$ action. In particular we show that the space of invariant almost
complex structures $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ decomposes into
disjoint strata, each of them being homotopy equivalent to an orbit of the
centralizer, with stabilizer homotopy equivalent to $S^{1}$ equivariant Kähler
isometries (see Theorems 3.43 and 3.51). In Chapter 3, we show that the number
of invariant strata in the decomposition corresponds to the number of toric
extensions of the given circle action (Proposition 3.22). In particular we
prove that $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ decomposes into either one
or two strata, and that the later case occurs only for an exceptional family
of circle actions on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ (Corollaries 3.23,3.25 and
3.24). In Chapter 4, using techniques similar to the ones developed in [AG04]
we compute the homotopy type of $\operatorname{Symp}^{S^{1}}(S^{2}\times
S^{2},\omega_{\lambda})$ for all Hamiltonian circle actions on $(S^{2}\times
S^{2},\omega_{\lambda})$.

In Chapter 5, we prove $S^{1}$ equivariant analogues of some technical lemmas
of [AGK09] involving deformation theory. We use these results to prove that in
the case the circle admits two distinct toric extensions, the stratum with
positive codimension in $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ is always of
codimension two.

Finally in Chapter 6, we carry out a similar analysis on the manifold
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ and obtain the homotopy type of
$\operatorname{Symp}^{S^{1}}(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$
for all Hamiltonian circle actions on
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$.

##  Chapter 2 Hamiltonian torus actions on ruled $4$-manifolds

###  2.1 Preliminaries on Hamiltonian actions

Let $G$ be a compact Lie group, possibly finite, acting effectively and
symplectically on a symplectic manifold $(M,\omega)$. Every such action
$\rho:G\times M\to M$ induces an injective homomorphism
$G\hookrightarrow\operatorname{Symp}(M,\omega)$ and, in particular, defines a
subgroup $G\subset\operatorname{Symp}(M,\omega)$. Let $\mathfrak{g}$ denote
the Lie algebra of $G$ and $\mathfrak{g}^{*}$ be it’s dual. Given
$Y\in\mathfrak{g}$, we denote by $\overline{Y}$ the corresponding fundamental
vector field on $M$.

######  Definition 2.1.

The action of $G$ is called Hamiltonian if

  * •

$G$ is finite and the induced homomorphism $G\to\operatorname{Symp}(M,\omega)$
takes values in the subgroup of Hamiltonian diffeomorphisms
$\operatorname{Ham}(M,\omega)$, or

  * •

$\dim G\geq 1$ and there exists a moment map, that is, a smooth map
$\mu:M\to\mathfrak{g}^{*}$ such that

    1. 1.

$d\mu_{p}(X_{p})(Y)=\omega(X,\overline{Y})$ for all $X\in T_{p}M$ and
$Y\in\mathfrak{g}$, and

    2. 2.

the map $\mu$ is equivariant with respect to the $G$ action on $M$ and the
coadjoint action on $\mathfrak{g}^{*}$.

In particular,
$G\subset\operatorname{Ham}(M,\omega)\subset\operatorname{Symp}(M,\omega)$.

######  Remark 2.2.

We note that in this definition, the moment map $\mu:M\to\mathfrak{g}^{*}$ is
an auxiliary structure whose sole existence is required. There is no claim
about its unicity. For instance, the moment map of a Hamiltonian torus action
is only defined up to adding a constant $c\in\mathfrak{t}^{*}$. However, there
are situations where the choice of a specific moment map is needed. For this
reason, we will always distinguish between the structure $(M,\omega,\rho)$
given by a Hamiltonian action alone, from the structure $(M,\omega,\rho,\mu)$
in which a moment map $\mu$ is chosen.

######  Definition 2.3.

We will say that two actions $\rho_{i}:G\to\operatorname{Symp}(M,\omega)$ are
symplectically equivalent if the corresponding subgroups $\rho_{i}(G)$ belong
to the same conjugacy class with respect to the action of
$\operatorname{Symp}(M,\omega)$.

Given a symplectic action $\rho:G\times M\to M$, let $C(G)$ be the centralizer
of the corresponding subgroup $G\subset\operatorname{Symp}(M,\omega)$, that
is,

$C(G)=\\{\phi\in\operatorname{Symp}(M,\omega)~{}|~{}\phi\circ
g\circ\phi^{-1}=g,~{}\forall g\in G\\}$

and let $N(G)$ be its normalizer, namely,

$N(G)=\\{\phi\in\operatorname{Symp}(M,\omega)~{}|~{}\phi\circ
G\circ\phi^{-1}=G\\}$

Clearly, equivalent actions have isomorphic centralizers and normalizers.

In this paper, we will investigate the homotopy type of the centralizers
$C(T)$ of Hamiltonian torus actions on some $4$-manifolds. We start by giving
a simple characterization of symplectomorphisms commuting with a Hamiltonian
torus action.

######  Lemma 2.4.

Let $(M,\omega)$ be a compact symplectic manifold equipped with a Hamiltonian
torus action $\rho:T\times M\to M$ with moment map $\mu$. Then $\phi\in C(T)$
if, and only if, $\mu\circ\phi=\mu$ and
$\phi\in\operatorname{Symp}(M,\omega)$.

###### Proof.

$(\Leftarrow)$ Suppose $\phi\in\operatorname{Symp}(M,\omega)$ satisfies
$\mu\circ\phi=\mu$. Let $X\in\mathfrak{t}$ and let $\overline{X}$ denote the
associated fundamental vector field. We then have

$\omega(d\phi^{-1}(\overline{X}),Y)=\phi^{*}\omega(\overline{X},d\phi(Y))=\omega(\overline{X},d\phi(Y)=d\mu(d\phi(Y))=d\mu(Y)=\omega(\overline{X},Y)$

for any vector field $Y$, which implies $d\phi(\overline{X})=\overline{X}$ for
all $X\in\mathfrak{t}$. Consequently, $\phi$ commutes with the action.

$(\Rightarrow)$ If the action $\rho$ has moment map $\mu$, then
$\mu\circ\phi^{-1}$ is a moment map for the conjugate action
$\phi^{-1}\circ\rho\circ\phi$. If $\phi^{-1}\circ\rho\circ\phi=\rho$, this
implies $\mu\circ\phi^{-1}=\mu+C$ for some constant $C\in\mathfrak{t}$. Since
the moment images $\mu(M)$ and $\mu\circ\phi(M)$ are compact and coincide,
this constant must be $0$. ∎

The image of the moment map of a torus action have special convexity
properties.

######  Theorem 2.5.

(Atiyah-Guillemin-Sternberg) Let $(M,\omega)$ be a symplectic manifold on
which a torus $\mathbb{T}^{d}$ is acting with moment map $\mu$. Then the image
of $\mu$ is a convex polytope of $\mathfrak{t}^{*}$ whose vertices are images
of the fixed points of the torus action.

As usual, we call an effective Hamiltonian torus action _toric_ if the torus
acting is half the dimension of the manifold $M$. By a theorem of Delzant, the
moment map image $\Delta:=\mu(M)$ of a toric action determines the symplectic
manifold $(M,\omega)$, the action $\rho$, and the moment map $\mu$ up to
equivariant symplectomorphisms.

######  Theorem 2.6.

(Delzant [Del88]) Let $(M_{i},\omega_{i},\rho_{i},\mu_{i})$, $i=1,2$, be two
toric manifolds of dimension $2n$ with moment maps
$\mu_{i}:M_{i}\to\mathfrak{t}^{*}$. If the two moment polytopes
$\mu_{i}(M_{i})$ coincide, then there exists a $\mathbb{T}^{n}$-equivariant
symplectomorphism $\phi:(M_{1},\omega_{1})\to(M_{2},\omega_{2})$ such that
$\phi^{*}\mu_{2}=\mu_{1}$. Conversely, the moment map images of two
equivariantly symplectomorphic toric structures
$(M_{i},\omega_{i},\rho_{i},\mu_{i})$, $i=1,2$, coincide.

Choose an identification $\mathfrak{t}\simeq\mathbb{R}^{n}$ such that
$\ker(\exp:\mathfrak{t}\to\mathbb{T}^{n})\simeq\mathbb{Z}^{n}$. The moment
polytopes of toric actions on manifolds of dimension $2n$ – called Delzant
polytopes of dimension $n$ – are completely characterized by the following
three properties, see [Del88]:

  * •

(simplicity) there are exactly $n$ edges meeting at each vertex;

  * •

(rationality) the edges meeting at any vertex $p$ are of the form $p+tu_{i}$,
$t\in[0,\ell_{i}]$, where $\ell_{i}\in\mathbb{R}$ and
$u_{i}\in\mathbb{Z}^{n}$;

  * •

(smoothness) for each vertex $p$, the corresponding vectors
$u_{1},\ldots,u_{n}$ can be chosen to be a basis of $\mathbb{Z}^{n}$ over
$\mathbb{Z}$.

This provides a purely combinatorial description of toric structures. Finally,
if we are only interested in the subgroup
$\mathbb{T}^{n}\subset\operatorname{Ham}(M^{2n},\omega)$ associated to a toric
action, that is, if we disregard both the parametrization
$\mathbb{T}^{n}\hookrightarrow\operatorname{Ham}(M^{2n},\omega)$ and the
moment map, Delzant’s classification theorem yields a bijection

$\displaystyle\\{\text{Inequivalent toric actions
on~{}}2n\text{-manifolds}\\}$ $\displaystyle\updownarrow$
$\displaystyle\\{\text{Delzant polytopes in~{}}\mathbb{R}^{n}\text{~{}up
to~{}}\operatorname{AGL}(n;\mathbb{Z})\text{~{}action}\\}$

As the next proposition shows, the centralizer and normalizer of symplectic
toric actions are easy to describe in terms of moment maps. In particular, the
homotopy type of $C(T)$ does not depend on the toric structure.

######  Proposition 2.7.

Consider an effective toric action
$\rho:\mathbb{T}^{n}\to\operatorname{Symp}(M^{2n},\omega)$ with moment map
$\mu:M\to\mathfrak{t}$. Denote by $T$ the corresponding torus in
$\operatorname{Symp}(M,\omega)$, by $C(T)$ its centralizer, by $N(T)$ its
normalizer, and by $W(T)=N(T)/C(T)$ its Weyl group.

  1. 1.

The normalizer $C(T)$ deformation retracts onto $\mathbb{T}^{n}$. In
particular, $C(T)\subset\operatorname{Ham}(M,\omega)$.

  2. 2.

The normalizer $N(T)$ is equal to the group of all symplectomorphisms $\psi$
such that $\mu\circ\psi=\Lambda\circ\mu$ for some
$\Lambda\in\operatorname{AGL}(n;\mathbb{Z})$. In particular, the Weyl group
$W(T)$ is finite.

###### Proof.

This is a slightly stronger version of Proposition 3.21 in [Pin08a], and only
the first statement requires some additional justification. By Lemma 2.4,
$\phi\in C(T)$ iff $\phi\in\operatorname{Symp}(M,\omega)$ and
$\mu=\mu\circ\phi$. Since the action is toric, each fiber of the moment map
consists in a single orbit. It follows that there is a unique map
$g_{\phi}:\mu(M)=\Delta\to\mathbb{T}^{n}$ such that $\phi$ can be written as

$\phi(m)=g_{\phi}(\mu(m))\cdot m$

Pick a base point $b\in\Delta$ and consider the evaluation fibration

$\displaystyle C_{b}(T)\to C(T)$ $\displaystyle\to\mathbb{T}^{n}$
$\displaystyle\phantom{C_{b}(T)\to}\phi$ $\displaystyle\mapsto g_{\phi}(b)$

whose fiber is the subgroup

$C_{b}(T)=\\{\phi\in
C(T)~{}|~{}\phi=\operatorname{id}\text{~{}on~{}}\mu^{-1}(b)\\}$

Consider $\phi\in C_{b}(T)$. Since $\Delta$ is contractible, the function
$g_{\phi}$ has a unique lift $\tilde{g}_{\phi}:\Delta\to\mathfrak{t}$ such
that $\tilde{g}_{\phi}(b)=0$, and $g_{\phi}=\exp\circ\tilde{g}_{\phi}$.
Setting $\tilde{g}_{\phi,s}=s\tilde{g}_{\phi}$ for $s\in[0,1]$, we obtain a
retraction of $C_{b}(T)$ onto $\\{\operatorname{id}\\}$ through
diffeomorphisms $\phi_{s}$ leaving $\mu$ invariant and such that
$\phi_{s}=\operatorname{id}$ on $\mu^{-1}(b)$. Applying an equivariant Moser
isotopy to the family $\phi_{s}$, this retraction is seen to be homotopic to a
retraction in $C_{b}(T)$. This shows that $C_{b}(T)$ is contractible, which in
turns implies the result. ∎

For Hamiltonian torus actions $T^{d}\to\operatorname{Ham}(M^{2n},\omega)$ for
which the torus $T^{d}$ has dimension $d<n$, there is no simple description of
the homotopy type of the symplectic centralizer $C(T)$. As we will see, this
already happens in the simplest case, namely, for Hamiltonian circle actions
on $4$-manifolds.

In [Kar99], Y. Karshon established an equivariant classification of
Hamiltonian circle actions on 4-dimensional symplectic manifolds, similar to
Delzant classification, in which the moment map image is replaced by labelled
graphs. More precisely, given any Hamiltonian $S^{1}$ action on a 4-manifold
$(M,\omega)$ with moment map $\mu$, one can construct a labelled graph as
follows:

  * •

Each component of the fixed point set corresponds to a unique vertex of the
graph.

  * •

Each vertex is labeled by the value of the moment map on the corresponding
fixed point component.

  * •

An extremal vertex that corresponds to a fixed symplectic surface S is said to
be ”fat”. Two additional labels are attached to fat vertices: the genus of
that surface, and its normalized symplectic area.

  * •

Two vertices are connected by an edge if and only if the corresponding
isolated fixed points are connected by a $\mathbb{Z}_{k}$-sphere i.e by a
$S^{1}$ invariant sphere on which $S^{1}$ acts with a global stabilizer
$\mathbb{Z}_{k}$, $k\geq 2$.

  * •

Each edge is labelled by the isotropy weight $k\geq 2$ of the corresponding
$\mathbb{Z}_{k}$ sphere.

Labeled graphs associated to effective Hamiltonian circle actions are
characterized by the following properties. If we order the vertices according
to their moment map labels, then

  * •

there are exactly two extremal vertices;

  * •

fat vertices are extremal, and if the graph contains two fat vertices, then
their genus label must coincide;

  * •

the area label of any fat vertex must be strictly positive;

  * •

a vertex is connected to no more than two edges, and no edge is connected to a
fat vertex;

  * •

the moment map labels must be strictly monotone along each chain of edges;

  * •

if $e_{1},\ldots,e_{\ell}$ is a chain of edges, and if $k_{1},\ldots,k_{\ell}$
are the orders of their stabilizers, then $\gcd(k_{i},k_{i+1})=1$ for
$i=1,\ldots\ell-1$, and $(k_{i-1}+k_{i+1})/k_{i}$ is an integer for
$i=2,\ldots,\ell-1$.

We call such graphs _admissible_. Just as Delzant polytopes classify toric
structures up to symplectomorphisms, admissible labelled graphs classify
Hamiltonian $S^{1}$ actions with moment maps.

######  Theorem 2.8.

(Karshon [Kar99]) Each admissible labelled graph determines a Hamiltonian
circle action $\rho$ with moment map $\mu$ on a symplectic $4$ manifold
$(M,\omega)$. The tuple $(M,\omega,\rho,\mu)$ is unique up to
$S^{1}$-equivariant symplectomorphisms preserving the moment map. Conversely,
the labelled graphs of two equivariantly symplectomorphic Hamiltonian circle
actions with moment maps are isomorphic.

Again, if one is only interested in conjugacy classes of circles in
Hamiltonian groups, Karshon’s classification implies that there is a bijection

$\displaystyle\\{\text{Inequivalent Hamiltonian circle actions
on~{}}4\text{-manifolds}\\}$ $\displaystyle\updownarrow$
$\displaystyle\\{\text{Admissible labelled graphs up
to~{}}\operatorname{AGL}(1;\mathbb{Z})\text{~{}action on moment map
labels}\\}$

######  Remark 2.9.

  * •

Note that when $M$ is compact, it is always possible to canonically assign a
moment map to an Hamiltonian circle action by imposing one of the
normalizations $\min\mu=0$ or $\int_{M}\mu=0$.

  * •

Note also that in Karshon’s classification, it is not necessary to keep track
of invariant spheres on which the restricted action is effective. In
particular, these spheres do not appear in the labelled graphs. This point
will be important in our analysis.

The symplectic slice theorem implies that a Hamiltonian circle action on a
$4$-manifold has a simple local description near any fixed point.

######  Proposition 2.10.

[see [Kar99] Corollary A.7 or [LT97] Lemma A.1] Let $p$ be a fixed point of a
Hamiltonian circle action on a $4$-manifold $(M,\omega)$. Then there exist
complex coordinates $(z,w)$ on a neighborhood of $p$ in $M$, and unique
integers $\alpha$ and $\beta$, called the isotropy weights at $p$, such that

  1. 1.

the circle action is $t\cdot(z,w)=(t^{\alpha}z,t^{\beta}w)$,

  2. 2.

the symplectic form is $\omega=\frac{i}{2}(dz\wedge d\bar{z}+dw\wedge
d\bar{w})$,

  3. 3.

the moment map is
$\mu(z,w)=\mu(p)+\frac{\alpha}{2}|z|^{2}+\frac{\beta}{2}|w|^{2}$.

Although complex coordinates are used in the definition of weights, the
unordered set $\\{\alpha,\beta\\}$ is a symplectic invariant of the circle
action at the fixed point $p$. This already imposes some restrictions on
equivariant symplectomorphisms.

######  Corollary 2.11.

Let $\phi\in C(S^{1})$ be an $S^{1}$-equivariant symplectomorphism. Then
$\phi$ acts on the fixed point set preserving their weights as unordered sets.
In particular, equivalent actions have the same set of weights.∎

As explained in [Kar99] pages 6-7, the isotropy weights of a Hamiltonian
circle action can be read from its labelled graph as follows:

  * •

for $k\geq 2$, a fixed point has an isotropy weight $-k$ if and only if it is
the north pole of a $\mathbb{Z}_{k}$ -sphere, and it has an isotropy weight
$k$ if and only if it is the south pole of a $\mathbb{Z}_{k}$ -sphere;

  * •

an interior fixed point has one positive weight and one negative weight, a
maximal fixed point has both weights non-positive, a minimal fixed point has
both weights non-negative;

  * •

a fixed point has a weight $0$ if and only if it lies on a fixed surface.

We end this section by stating two simple lemmata that will be useful in
computing homology classes of invariant spheres directly from labelled graphs.

######  Lemma 2.12 (Lemma 5.4 in [Kar99]).

Let $S^{1}$ act on $S^{2}$ by rotating it $k$ times while fixing the north and
south poles. Suppose that the action lifts to a complex line bundle $E$ over
$S^{2}$. Then $S^{1}$ acts linearly on the fibers over the north and south
poles; let $\alpha$ and $\beta$ be the weights for these actions. Then

$\alpha-\beta=-ek$

where $e$ is the self intersection of the zero section.∎

######  Lemma 2.13.

Let $S^{1}$ act on $S^{2}$ by rotating it $k$ times while fixing the north and
south poles. Let $\mu(n)$ denote the value of the momentum map at the north
pole and $\mu(s)$ the value of the momentum map at the south pole. Then the
symplectic area of the $S^{2}$ is given by $\frac{\mu(n)-\mu(s)}{k}$.∎

###  2.2 Torus actions on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$

We now combine the _equivariant_ classification theorems of Delzant and
Karshon, together with the Lalonde-McDuff classification of symplectic forms
on ruled surfaces, to describe all inequivalent Hamiltonian circle actions on
$S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ and
their relations to toric actions.

####  2.2.1 Symplectic forms on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$

We equip $S^{2}\times S^{2}$ with the product symplectic form
$\omega_{\lambda}=\lambda\sigma\oplus\sigma$, where $\sigma$ is the standard
area form on $S^{2}$ normalized such that $\sigma(S^{2})=1$. If we think of
$S^{2}\times S^{2}$ as a trivial fiber bundle, $\omega_{\lambda}$ gives area
$1$ to the fibers, while the area of horizontal sections is $\lambda$.
Similarly, if we view $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ as the
non-trivial $S^{2}$ bundle over $S^{2}$, we define an analogous form
$\omega_{\lambda}$ which gives area $1$ to the fibers and area $\lambda-1$ to
symplectic sections of self-intersection $-1$, that is, to sections homologous
to the exceptional divisor. From an homological point of view, if $F$ denotes
the homology class of a fiber in either $S^{2}\times S^{2}$ or
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, if $B$ denotes the class of a
section of self-intersection $0$ in $S^{2}\times S^{2}$, if $E$ denotes the
class of the exceptional divisor in
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, and if $L$ denotes the class
of a line in $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, then
$[\omega_{\lambda}]F=1$, $[\omega_{\lambda}]B=\lambda$,
$[\omega_{\lambda}]L=\lambda$ and $[\omega_{\lambda}]E=\lambda-1$. Note that
with our conventions on $\omega_{\lambda}$ for the non-trivial bundle
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, $\lambda$ must be strictly
greater than 1 for the symplectic curve in class $E$ to have positive area. We
can now state the Lalonde-McDuff classification theorem.

######  Theorem 2.14 (Lalonde-McDuff [LM96]).

Any symplectic form on $S^{2}\times S^{2}$ or
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ is conformally diffeomorphic to
a form $\omega_{\lambda}$ with $\lambda\geq 1$. Moreover, any two cohomologous
forms are diffeomorphic.

####  2.2.2 Toric actions on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$

It is well known that, up to symplectic equivalence, the only toric actions on
$S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ are
given by the standard torus actions on the Hirzebruch surfaces. We now review
the main points of the argument.

Recall that the $m^{\text{th}}$ Hirzebruch surface $W_{m}$ is the complex
submanifold of $\mathbb{C}P^{1}\times\mathbb{C}P^{2}$ defined by the equation

$W_{m}:=\left\\{\left(\left[x_{1},x_{2}\right],\left[y_{1},y_{2},y_{3}\right]\right)\in\mathbb{C}P^{1}\times\mathbb{C}P^{2}~{}|~{}x^{m}_{1}y_{2}-x_{2}^{m}y_{1}=0\right\\}$

The projection map
$\mathbb{C}P^{1}\times\mathbb{C}P^{2}\longrightarrow\mathbb{C}P^{1}$ gives
$W_{m}$ the structure of a $\mathbb{C}P^{1}$ bundle over $\mathbb{C}P^{1}$
which is diffeomorphic (but not complex isomorphic) to $S^{2}\times S^{2}$ if
$m$ is even and diffeomorphic to the non-trivial $S^{2}$ bundle over $S^{2}$
i.e $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ if $m$ is odd. Thus, for
each $m$ we have an integrable complex structure $J_{m}$ induced on
$S^{2}\times S^{2}$ or $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$. When
$m=2k$ even, choose $\lambda>k$, then we can endow
$\mathbb{C}P^{1}\times\mathbb{C}P^{2}$ with the symplectic form
$(\lambda-k)\sigma_{1}\oplus\sigma_{2}$, where $\sigma_{1}$ is the standard
symplectic form on $\mathbb{C}P^{1}$ with area 1 and $\sigma_{2}$ is the
standard symplectic form on $\mathbb{C}P^{2}$ such that $\sigma_{2}(L)=1$,
where $L$ is the class of the line in $\mathbb{C}P^{2}$. Restricting this
symplectic form to $W_{m}$ makes it a symplectic manifold. Similarly when
$m=2k+1$, choose $\lambda>k+1$, then we can analogously define the form
$(\lambda-(k+1))\sigma_{1}\oplus\sigma_{2}$ when $m$ is odd. With these
choices of symplectic forms, the Lalonde-McDuff classification theorem 2.14
implies that $W_{m}$ is symplectomorphic to $(S^{2}\times
S^{2},\omega_{\lambda})$ if $m$ is even and to
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$ when $m$ is
odd.

Given an integer $m\geq 0$, the torus $\mathbb{T}^{2}$ acts on
$\mathbb{C}P^{1}\times\mathbb{C}P^{2}$ by setting

$\left(u,v\right)\cdot\left(\left[x_{1},x_{2}\right],\left[y_{1},y_{2},y_{3}\right]\right)=\left(\left[ux_{1},x_{2}\right],\left[u^{m}y_{1},y_{2},vy_{3}\right]\right)$

This action leaves $W_{m}$ invariant and preserves both the complex and the
symplectic structures. Its restriction to $W_{m}$ defines a toric action that
we denote $\mathbb{T}^{2}_{m}$.

When $m$ is even, the image of the moment map is the polytope of Figure 2.1

$Q=(0,1)$$P=(0,0)$$R=(\lambda-\frac{m}{2},1)$$S=(\lambda+\frac{m}{2},0)$$D_{m}=B-\frac{m}{2}F$$F$$F$$B+\frac{m}{2}F$
Figure 2.1: Even Hirzebruch polygon

where the labels along the edges refer to the homology classes of the
$\mathbb{T}^{2}$ invariant spheres in $S^{2}\times S^{2}$, and where the
vertices $P$,$Q$,$R$,$S$ are the fixed points of the torus action. Similarly,
when $m$ is odd, the moment map image is given in Figure 2.2

$Q=(0,1)$$P=(0,0)$$R=(\lambda-\frac{m+1}{2},1)$$S=(\lambda+\frac{m-1}{2},0)$$D_{m}=B-\frac{m+1}{2}F$$F$$F$$B+\frac{m-1}{2}F$
Figure 2.2: Odd Hirzebruch polygon

where $B$ now refers to the homology class of a line $L$ in
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, $E$ is the class of the
exceptional divisor, and $F$ is the fiber class $L-E$.

We define the zero-section $s_{0}$ to be

$\displaystyle s_{0}:\mathbb{C}P^{1}$ $\displaystyle\to W_{m}$
$\displaystyle[x_{1};x_{2}]$ $\displaystyle\mapsto\\{[x_{1},x_{2}],[0;0;1]\\}$

and the section at infinity $s_{\infty}$ to be

$\displaystyle s_{\infty}:\mathbb{C}P^{1}$ $\displaystyle\to W_{m}$
$\displaystyle[x_{1};x_{2}]$
$\displaystyle\mapsto\\{[x_{1},x_{2}],[x^{m}_{1};x^{m}_{2};0]\\}$

The image of $s_{0}$ is an invariant and holomorphic sphere homologous to
either $D_{m}:=B-\frac{m}{2}F$ in $S^{2}\times S^{2}$ or to
$D_{m}:=B-\frac{m+1}{2}F$ in $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$,
depending on the parity of $m$. Similarly, $s_{\infty}$ defines an invariant
and holomorphic sphere that represents either $B+\frac{m}{2}F$ in $S^{2}\times
S^{2}$ or $B+\frac{m+1}{2}F$ in
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$. Finally, the homology class
$F$ can be represented by an invariant fibre such as
$\\{[1,0],[y_{1},0,y_{3}]\\}$.

It is well known (see [Aud04]) that a Delzant polygon
$\Delta\subset\mathbb{R}^{2}$ with $e\geq 3$ edges defines a toric
$4$-manifold $M_{\Delta}$ whose second Betti number is
$b_{2}(M_{\Delta})=e-2$. It follows that the moment polytope of any toric
action on either $S^{2}\times S^{2}$ or
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ is a quadrilateral. It is easy
to see that any Delzant quadrilateral is equivalent to one of the above
Hirzebruch trapezoids up to $\operatorname{AGL}(2,\mathbb{Z})$ action and up
to rescaling. It follows from Delzant’s classification that any toric action
on $S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ is
equivalent to an action of the above form. In particular, the equivalence
classes of toric actions on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ are completely characterized by
the existence of invariant spheres in specific homology classes.

######  Lemma 2.15.

Up to equivalence, the toric action $\mathbb{T}^{2}_{m}$, $m\geq 1$, is
characterised by the existence of an invariant, embedded, symplectic sphere
$C_{m}$ with self intersection $-m$. The toric action $\mathbb{T}^{2}_{0}$ is
characterized by the existence of invariant, embedded, symplectic spheres
representing the classes $B$ and $F$.∎

######  Lemma 2.16.

Write $\lambda=\ell+\delta$ with $\ell$ an integer and $0<\delta\leq 1$. Then,
up to symplectomorphisms and reparametrizations,

  * •

if $\lambda\geq 1$, there are exactly $\ell+1$ inequivalent toric actions on
$(S^{2}\times S^{2},\omega_{\lambda})$ given by the even Hirzebruch actions
$\mathbb{T}^{2}_{2k}$ with $0\leq k\leq\ell$, and

  * •

if $\lambda>1$, there are exactly $\ell$ inequivalent toric actions on
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$ given by the
odd Hirzebruch actions $\mathbb{T}^{2}_{2k+1}$ with $0\leq k\leq\ell-1$.

###### Proof.

Write $m=2k$ or $m=2k+1$ with $k\geq 0$. As seen above, any toric action on
$S^{2}\times S^{2}$ and $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ is
$\mathbb{T}^{2}$-equivariantly symplectomorphic to one of the actions
$\mathbb{T}^{2}_{m}$. The invariant symplectic sphere $D_{k}$ in class $B-kF$
on $S^{2}\times S^{2}$ or $L-(k+1)F$ in
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ must have positive area, that
is,

$0<\omega_{\lambda}(B-kF)=\lambda-k=\ell+\delta-k\quad\text{~{}or~{}}\quad
0<\omega_{\lambda}(L-(k+1)F)=\lambda-(k+1)F=\ell+\delta-(k+1)$

The result follows. ∎

######  Proposition 2.17.

The Weyl group
$W(\mathbb{T}^{2}_{m})=N(\mathbb{T}^{2}_{m})/C(\mathbb{T}^{2}_{m})$ is
isomorphic to

$W(\mathbb{T}^{2}_{m})\simeq\begin{cases}\text{the dihedral
group~{}}D_{8},&\text{~{}when~{}}m=0\text{~{}and~{}}\lambda=1;\\\ \text{the
dihedral group~{}}D_{2},&\text{~{}when~{}}m=0\text{~{}and~{}}\lambda>1;\\\
\text{the dihedral group~{}}D_{1}\simeq\mathbb{Z}_{2},&\text{~{}when~{}}m\geq
1.\end{cases}$

###### Proof.

This follows directly from Proposition 2.7 and from the description of
Hirzebruch trapezoids. Indeed, for $m=0$ and $\lambda=1$, the moment map
polygon is a unit square whose symmetries can be realized by elements of
$\operatorname{AGL}(2,\mathbb{Z})$. The same holds for $m=0$ and $\lambda>1$,
with the only difference that the moment polygone is now a rectangle with
sides of lengths $1$ and $\lambda$. Finaly, for $m\geq 1$, if we write $m=2k$
or $m=2k+1$, the only non-trivial element of
$\operatorname{AGL}(2,\mathbb{Z})$ that leaves the standard Hirzebruch
trapezoid invariant is

$\Lambda=\begin{pmatrix}-1&-m\\\
0&1\end{pmatrix}+\begin{pmatrix}\lambda+k,0\end{pmatrix}$

which is an element of order 2. ∎

####  2.2.3 Hamiltonian circle actions on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$

A list of all possible Hamiltonian circle action on $S^{2}\times S^{2}$ or
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ comes easily from an extension
theorem due to Y. Karshon.

######  Theorem 2.18 (Karshon [Kar02], Theorem 1).

Any symplectic $S^{1}$ action on $(S^{2}\times S^{2},\omega_{\lambda})$ and
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$ extends to
an Hamiltonian toric action.

Consequently, every symplectic $S^{1}$ action on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ is given by an embedding

$\displaystyle S^{1}$ $\displaystyle\hookrightarrow\mathbb{T}^{2}_{m}$
$\displaystyle t$ $\displaystyle\mapsto(t^{a},t^{b})$

which is itself characterized by a unique triple of numbers
$(a,b;m)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}_{\geq 0}$. Since we are
only interested in effective actions, this translates numerically into the
condition $\gcd(a,b)=1$. We shall always assume this unless otherwise stated.

In order to construct the graphs of the circle action $S^{1}(a,b;m)$, we claim
that it is enough to compute the isotropy weights at every fixed points.
Indeed, given an Hirzebruch trapezoid, the pre-image of a vertex under the
moment map is a toric fixed point, and the pre-image of an edge is an
invariant embedded two-sphere connecting two fixed points. When we view the
space as a Hamiltonian $S^{1}$-space, such a two-sphere is either fixed by the
action, or is a $\mathbb{Z}_{k}$ sphere, or has trivial global isotropy. These
three possibilities are completely determined by the weights of the
$S^{1}(a,b;m)$ action at its two fixed points. We can thus construct the
graphs starting from the fixed points and adding edges according to the
weights. If there is a fixed surface, its area label can be read from the
Hirzebruch trapezoid. Finally, the moment map labels can be added starting
with the minimal vertex (characterised by having two positive weights) that we
label $\mu=0$, and then using Lemma 2.13 to compute the moment labels for the
remaining interior fixed points, and for the maximal fixed point
(characterised by having two negative weights).

Now, for the Hirzebruch actions $\mathbb{T}^{2}_{m}=(S^{1}\times S^{1})_{m}$,
each $S^{1}$ factor defines two weights at each of the four fixed points
$P,Q,R,S$. These weights are listed in table 2.1. Further, if the the
$\mathbb{T}^{2}_{m}$ action has weights $\\{\alpha_{1},\beta_{1}\\}$ and
$\\{\alpha_{2},\beta_{2}\\}$ at a fixed point $x$, then the restricted
$S^{1}(a,b;m)$ action has weights

$\left\\{a\alpha_{1}+b\alpha_{2},a\beta_{1}+b\beta_{2}\right\\}$

at $x$. This gives the weights of the $S^{1}(a,b;m)$ actions at the fixed
points $P,Q,R,S$.

Vertex |  Weights for $\mathbb{T}^{2}_{m}$ action |  Weights for the $S^{1}(a,b;m)$ action
---|---|---
P |  $\big{\\{}\\{1,0\\},\\{0,1\\}\big{\\}}$ |  $\\{a,b\\}$
Q |  $\big{\\{}\\{1,0\\},\\{0,-1\\}\big{\\}}$ |  $\\{a,-b\\}$
R |  $\big{\\{}\\{-1,m\\},\\{0,-1\\}\big{\\}}$ |  $\\{-a,am-b\\}$
S |  $\big{\\{}\\{-1,-m\\},\\{0,1\\}\big{\\}}$ |  $\\{-a,-am+b\\}$
Table 2.1: Weights of $\mathbb{T}^{2}_{m}$ and $S^{1}(a,b;m)$ actions

In Figures 2.3 and 2.4, we present the graphs of circles actions
$S^{1}(a,b;m)$ on $(S^{2}\times S^{2},\omega_{\lambda})$ and
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$. As before,
we write $m=2k$ or $m=2k+1$, and in order to deal with even and odd Hirzebruch
surfaces simultaneously, we introduce the symbol $\epsilon_{m}:=m\mod 2$. Each
label $\mu$ represents the value of the moment map normalized by setting
$\min\mu=0$, while $A$ is the area label of a fixed surface. All fixed
surfaces have genus 0. Note that when the isotropy label on an edge is 1, then
the circle action on the corresponding invariant sphere has no isotropy, and
we erase that edge from the graph. Note also that the identification of the
fixed points with the vertices $P,Q,R,S$ of the Hirzebruch trapezoid is there
for convenience only and is not part of the data.

For actions with fixed surfaces, one of the weights $a$, $b$, or $am-b$ must
be zero. Since $\gcd(a,b)=1$, the possible triples are $(\pm 1,0;m)$, $(0,\pm
1;m)$, and $(\pm 1,\pm m;m)$.

mRS$\mu=\lambda+k$$\mu=0$$A=1$$\mu=\lambda-k-\epsilon_{m}$ (a) Graph of
$S^{1}(1,0;m)$

mSR$\mu=\lambda+k$$\mu=0$$A=1$$\mu=m$ (b) Graph of $S^{1}(-1,0;m)$

$\mu=1$$\mu=0$$A=\lambda-k-\epsilon_{m}$$A=\lambda+k$ (c) Graph of
$S^{1}(0,1;m)$

$\mu=1$$\mu=0$$A=\lambda+k$$A=\lambda-k-\epsilon_{m}$ (d) Graph of
$S^{1}(0,-1;m)$

mPQ$\mu=\lambda+k$$\mu=0$$A=1$$\mu=m$ (e) Graph of $S^{1}(1,m;m)$

mPQ$\mu=\lambda+k$$\mu=0$$A=1$$\mu=\lambda-k-\epsilon_{m}$ (f) Graph of
$S^{1}(-1,-m;m)$

Figure 2.3: Graphs of circle actions with non-isolated fixed points

For actions whose fixed points are isolated, none of the weights are zero. The
graphs then depend on the signs of $a$, $b$, and $am-b$, as these signs
determine which fixed point is minimal and which one is maximal.

(a) *

$b$$a$$a$$am-b$$\mu=a(\lambda+k)$$\mu=b$$\mu=b+a(\lambda-k-\epsilon_{m})$$\mu=0$QRSP
(g) When $a>0,~{}b>0$, and $am-b>0$

QSR$\mu=b+a(\lambda-k-\epsilon_{m})$$\mu=a(\lambda+k)$$\mu=b$P$b$$a$$a$$b-am$$\mu=0$
(h) When $a>0,~{}b>0$ and $am-b<0$

$\mu=a(\lambda+k)-b$$\mu=-b$$\mu=a(\lambda-k-\epsilon_{m})$$\mu=0$$a$$-b$$am-b$$a$PRSQ
(i) When $a>0,~{}b<0$

$\mu=b-a(\lambda+k)$$\mu=b-am$$\mu=-a(\lambda+k)$$\mu=0$$-a$$b-am$$b$$-a$RPQS
(j) When $a<0,~{}b>0$

$\mu=-b-a(\lambda-k-\epsilon_{m})$$\mu=am-b$$\mu=-a(\lambda-k-\epsilon_{m})$$\mu=0$$-a$$am-b$$-b$$-a$SQPR
(k) When $a<0,~{}b<0$, and $am-b>0$

$b-am$$-a$$-a$$-b$$\mu=-a(\lambda+k)$$\mu=b-am$$\mu=b-a(\lambda+k-\epsilon_{m})$$\mu=0$PQRS
(l) When $a<0,~{}b<0$, and $am-b<0$

Figure 2.4: Graphs of circle actions whose fixed points are all isolated

###  2.3 Equivalent circle actions

A single equivalence class of Hamiltonian circle actions on $S^{2}\times
S^{2}$ or $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ may be represented by
more than one triple $(a,b;m)$, and this happens whenever the associated
labelled graphs are the same up to the action of
$\operatorname{AGL}(1,\mathbb{Z})$ on the moment map labels.

####  2.3.1 Equivalent circle actions in $\mathbb{T}^{2}_{m}$

The action $S^{1}(-a,-b;m)$ is equivalent to $S^{1}(a,b;m)$ since they only
differ by a reparametrization of the circle. It follows that in Figure 2.3,
the graphs (A), (C), and (E) are
$\operatorname{AGL}(1,\mathbb{Z})$-equivalent, respectively, to the graphs
(B), (D), and (F). Similarly, the graphs (G), (H), and (I) in Figure 2.4 are
$\operatorname{AGL}(1,\mathbb{Z})$-equivalent, respectively, to the graphs
(L), (K), and (J). Other examples of equivalent subcircles in
$\mathbb{T}^{2}_{m}$ are obtained by letting the normalizer
$N(\mathbb{T}^{2}_{m})$ act on the circle subgroups of $\mathbb{T}^{2}_{m}$.
For instance, $S^{1}(a,b;m)$ is conjugated to $S^{1}(-a,b-am;m)$ by the
adjoint of the element $\Lambda\in N(\mathbb{T}^{2}_{m})$ of Proposition 2.17.

######  Proposition 2.19.

The only subcircles of $\mathbb{T}^{2}_{m}$ that are equivalent to
$S^{1}(a,b;m)$ are the elements of the orbit of $S^{1}(a,b;m)$ under the
action of the Weyl group $W(\mathbb{T}^{2}_{m})$ and their reparametrizations.

###### Proof.

We have to show that if $S^{1}(a,b;m)$ and $S^{1}(c,d;m)$ are in the same
conjugacy class, then there is an element of $N(\mathbb{T}^{2}_{m})$ taking
one circle to the other. As conjugation of actions corresponds to uniform
translation of the moment map labels, this will follow from a systematic
inspection of the possible labelled graphs.

The first cases to consider are actions $S^{1}(a,b;m)$ whose fixed points are
all isolated, which means that the weights $a$, $b$, and $am-b$ are all non-
zero. Two equivalent graphs arising from normalized moment maps must have the
same moment map values at the maximum, and their respective weights at the
maximum and at the minimum must also coincide.

Type of graph | Weights at max | Weights at min | Moment label at max
---|---|---|---
G: $a>0$, $b>0$, $am-b>0$ |  $-a$, $b-am$ |  $a$, $b$ | $a(\lambda+k)$
H: $a>0$, $b>0$, $am-b<0$ |  $-a$, $am-b$ |  $a$, $b$ | $b+a(\lambda-k-\epsilon_{m})$
I: $a>0$, $b<0$ |  $-a$, $b-am$ |  $a$, $-b$ | $a(\lambda+k)-b$
J: $a<0$, $b>0$ |  $a$, $-b$ |  $-a$, $b-am$ | $b-a(\lambda+k)$
K: $a<0$, $b<0$, $am-b>0$ |  $a$, $b$ |  $-a$, $am-b$ | $-b-a(\lambda-k-\epsilon_{m})$
L: $a<0$, $b<0$, $am-b<0$ |  $a$, $b$ |  $-a$, $b-am$ | $-a(\lambda+k)$
Table 2.2: Weights and moment map values for graphs (G) – (L)

We start by assuming $a>0$, $b>0$, and $m>0$, which implies that the graph of
$S^{1}(a,b;m)$ is of type (G) or (H).

Assume the graph of $S^{1}(a,b;m)$ is of type (G), that is, $am-b>0$, and
suppose $S^{1}(c,d;m)$, $(c,d)\neq(a,b)$, is in the same conjugacy class.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is also of type (G). Looking at the
weights at the minimum, we see that $(c,d)=(b,a)$ is the only non-trivial
possibility. Looking at the moment map values at the maximum, we conclude that
$a(\lambda+k)=b(\lambda+k)$, that is, $a=b$, contradicting the fact that
$(a,b)\neq(c,d)$.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is of type (H). Looking again at the
weights at the minimum, we see that $c=b$, and $d=a$, and that $a\neq b$ as
before. The sets of weights at the maximum must also be equal, that is,
$\\{-a,b-am\\}=\\{-c,cm-d\\}=\\{-b,bm-a\\}$, which implies that $a=a-bm$,
contradicting the fact that $bm\neq 0$.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is of type (I), which implies $c>0$,
$d<0$. Looking at the weights at the minimum, we must have either
$(c,d)=(a,-b)$ or $(c,d)=(b,-a)$. In the former case, looking at the moment
map values at the maximum, we conclude that
$a(\lambda+k)=c(\lambda+k)-d=a(\lambda+k)+b$, that is, $b=0$, which is a
contradiction. In the later case, the weights at the maximum become
$\\{-c,d-cm\\}=\\{-b,-a-bm\\}$, and we must have
$\\{-b,-a-bm\\}=\\{-a,b-am\\}$ as sets. Since $a+bm\neq a$, we must have
$b=a$, which implies $a=b=1$, $c=1$, and $d=-1$. The moment map values at the
maximum are then $a(\lambda+k)=(\lambda+k)$ and
$c(\lambda+k)-d=(\lambda+k)+1$, which are not equal.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is of type (J), which implies $c<0$,
$d>0$. Looking at the weights at the minimum, we must have
$\\{a,b\\}=\\{-c,d-cm\\}$ as sets. So, either $(c,d)=(-a,b-am)$ or
$(c,d)=(-b,a-bm)$. In the first case, $b-am<0$, which is impossible at the
minimum. In the second case, looking at the moment map values at the maximum,
we conclude that
$a(\lambda+k)=d-c(\lambda+k)=a-bm+b(\lambda+k)=a+b(\lambda-k)$. This implies
$a(\lambda+k-1)/(\lambda+k)=b$. As $a$ and $b$ are non-zero integers, this is
impossible.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is of type (K), which implies $c<0$,
$d<0$, and $cm-d>0$. Comparing the weights at the minimum, we see that either
$(c,d)=(-a,-am-b)$ or $(c,d)=(-b,-bm-a)$ with $a\neq b$. In the former case,
comparing the weights at the maximum yields
$\\{-a,b-am\\}=\\{c,d\\}=\\{-a,-am-b\\}$. This implies $b=0$, which is
excluded. In the former case, we must have $a\neq b$ and
$\\{-a,b-am\\}=\\{c,d\\}=\\{-b,-bm-a\\}$, which again implies $b=0$.

  * •

Finally, suppose the graph of $S^{1}(c,d;m)$ is of type (K), which implies
$c<0$, $d<0$, and $cm-d<0$. Comparing the weights at the minimum, we see that
either $(c,d)=(-a,b-am)$ or $(c,d)=(-b,a-bm)$. In the former case, we get the
conjugate circle $S^{1}(-a,b-am;m)$. In the later case, the moment map values
at the maximum gives $a(\lambda+k)=-c(\lambda+k)=b(\lambda+k)$, that is,
$a=b$. Then $(c,d)=(-a,b-am)$ and $S^{1}(c,d;m)=S^{1}(-a,b-am;m)$ as before.

We conclude that the only circle action $S^{1}(c,d;m)$ conjugated to an action
$S^{1}(a,b;m)$ of type (G) is $S^{1}(-a,b-am;m)$.

Assume now that the graph of $S^{1}(a,b;m)$ is of type (H), that is, $a>0$,
$b>0$, and $am-b<0$. By transitivity, we already know that an action
$S^{1}(c,d;m)$ in the same conjugacy class cannot be of type (G) or (L).

  * •

Suppose the graph of $S^{1}(c,d;m)$ is also of type (H). Comparing the weights
at the minimum, we see that we must have $(c,d)=(b,a)$. Looking at the weights
at the maximum, we must have $\\{-a,am-b\\}=\\{-c,cm-d\\}$ as sets. Since
$(c,d)\neq(a,b)$, the only possibility is $a=d-cm=a-bm$, which implies $b=0$.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is of type (I). Comparing the weights at
the minimum, we see that we must have either $(c,d)=(a,-b)$ or $(c,d)=(b,-a)$
with $a\neq b$. In the first case, comparing the weights at the maximum, we
should have $\\{-a,am-b\\}=\\{-c,d-cm\\}=\\{-a,-b-am\\}$. This implies $a=0$,
which is excluded. Similarly, in the second case, we get
$\\{-a,am-b\\}=\\{-c,d-cm\\}=\\{-b,-b-am\\}$ with $a\neq b$. This again
implies $a=0$.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is of type (J). Looking at the weights at
the minimum, we must have $\\{a,b\\}=\\{-c,d-cm\\}$ as sets. So, either
$(c,d)=(-a,b-am)$ or $(c,d)=(-b,a-bm)$. In the first case, we obtain the
conjugate action $S^{1}(-a,b-am;m)$. In the second case, looking at the
weights at the maximum, we must have $\\{-a,am-b\\}=\\{c,-d\\}=\\{-b,bm-a\\}$.
If $a=b=1$, this gives us the conjugate action $S^{1}(-1,1-m;m)$. If $a=a-bm$,
then $b=0$, which is excluded.

  * •

Finally, suppose the graph of $S^{1}(c,d;m)$ is of type (K). Looking at the
weights at the minimum, we must have $\\{a,b\\}=\\{-c,cm-d\\}$ as sets. So,
either $(c,d)=(-a,-b-am)$ or $(c,d)=(-b,-a-bm)$. Looking at the weights at the
maximum, we must also have $\\{-a,am-b\\}=\\{c,d\\}$. If $(c,d)=(-a,-b-am)$,
then $am-b=-am-b$, which implies $a=0$. Instead, if $(c,d)=(-b,-a-bm)$, then
either $a=b=1$, which is equivalent to the previous case, or $a=a+bm$ which
forces $b=0$.

We conclude that $S^{1}(-a,b-am;m)$ is the only subcircle of
$\mathbb{T}^{2}_{m}$ conjugated to an action $S^{1}(a,b;m)$ of type (H).

Assume now that the graph of $S^{1}(a,b;m)$ is of type (I), that is, $a>0$,
and $b<0$. By transitivity, we already know that an action $S^{1}(c,d;m)$ in
the same conjugacy class cannot be of type (G), (H), (J), or (L). By symmetry
of Table 2.2 under a change of sign of the pair $(a,b)$, and comparing with
actions of types (H) and (J), we see that the only action $S^{1}(c,d;m)$ of
type (K) conjugated to $S^{1}(a,b;m)$ is $S^{1}(-a,b-am;m)$.

  * •

Suppose the graph of $S^{1}(c,d;m)$ is also of type (I). Comparing the weights
at the minimum, we see that we must have $(c,d)=(-b,-a)$. Looking at the
weights at the maximum, we must also have
$\\{-a,b-am\\}=\\{-c,d-cm\\}=\\{b,-a+bm\\}$. If $b=-a$, then
$(a,b)=(1,-1)=(c,d)$. If $b=b-am$, then we must have $a=0$, which is
impossible.

This shows that for an action $S^{1}(a,b;m)$ of type (I), the only other
action $S^{1}(c,d;m)$ in the same conjugacy class is $S^{1}(-a,b-am;m)$.

This concludes the proof of the proposition for actions $S^{1}(a,b;m)$ whose
fixed points are all isolated, in the case $m>0$. When $m=0$ and $\lambda>1$,
the graphs (G) and (L) no longer exist, and the four remaining graphs (H)–(K)
only differ by the action of $D_{2}$ on the pair $(a,b)$, that is, by a change
of sign of $a$ or $b$. When $\lambda=1$, we can also interchange $a$ and $b$
to obtain an equivalent graph, which defines an action of $D_{4}$. Finally,
the case of actions with non-isolated surfaces involves the graphs (A)–(F) and
is simpler. The details are left to the reader. ∎

####  2.3.2 Toric extensions of circle actions

######  Definition 2.20.

We shall say a circle action $S^{1}(a,b;m)$ _extends_ to a toric action
$\mathbb{T}^{2}_{n}$ if it is $S^{1}$-equivariantly symplectomorphic to a
circle action of the form $S^{1}(c,d;n)$.

In this section, we determine all possible toric extensions of $S^{1}(a,b;m)$.
We begin with the exceptional case $\lambda=1$.

######  Proposition 2.21.

Consider $(S^{2}\times S^{2},\omega_{\lambda})$ with $\lambda=1$. Then the
only Hamiltonian circle actions are of the form $S^{1}(a,b;0)$. In particular,
they can only extend to the torus $\mathbb{T}^{2}_{0}$.

###### Proof.

This follows directly from Theorem 2.16. ∎

By Lemma 2.15, a toric extension of $S^{1}(a,b;m)$ to $\mathbb{T}^{2}_{n}$,
$n\geq 1$, implies the existence of an invariant sphere of self-intersection
$-n$ and of positive symplectic area. These two numerical invariants can be
read from labelled graphs using Lemma 2.12 and Lemma 2.13. As we will see,
this imposes enough conditions on the triple $(a,b;m)$ to determine all
possible embeddings $S^{1}(a,b;m)\hookrightarrow\mathbb{T}^{2}_{n}$, $n\neq
m$.

######  Proposition 2.22.

Consider a Hamiltonian circle action $S^{1}(a,b;m)$ with $\lambda>1$. Under
the following numerical conditions on $a,b,m,\lambda$, the circle action only
extends to the toric action $\mathbb{T}^{2}_{m}$:

  * •

when $a\neq\pm 1$;

  * •

when $b=0$ or $b=am$;

  * •

when $a=\pm 1$ and $2\lambda\leq|2b-am|+\epsilon_{m}$.

In all other cases, the circle action may extend to at most two inequivalent
toric actions, namely,

  * •

when $a=\pm 1$, $2\lambda>|2b-am|+\epsilon_{m}$, and $b\not\in\\{0,am\\}$, the
circle action $S^{1}(a,b;m)$ only extends to the toric action
$\mathbb{T}^{2}_{m}$ and, possibly, to $\mathbb{T}^{2}_{|2b-am|}$.

###### Proof.

Suppose $S^{1}(a,b;m)$ is equivariantly symplectomorphic to $S^{1}(c,d;n)$ for
some $n\neq m$. Note that we necessarily have $m=n\mod 2$. By assumption, the
two actions share the same normalized labelled graph.

We first consider an action $S^{1}(a,b;m)$ whose fixed points are all
isolated. For the moment, let’s also assume that $m\neq 0$ and $n\neq 0$. We
observe that if the $\mathbb{T}^{2}_{n}$ invariant curve $C_{n}$ of self-
intersection $-n$ has non-trivial isotropy, then it must appear as some edge
in the graph of $S^{1}(a,b;m)$. As the edge connecting the vertices $Q$ and
$R$ is the only one that corresponds to an invariant sphere of negative self-
intersection, namely $-m$, we conclude that $m=n$. This contradiction shows
that $C_{n}$ must have trivial isotropy. By symmetry, the same is true of the
invariant curve $C_{m}$ of self-intersection $-m$. It follows that $a=\pm 1$
and $c=\pm 1$.

Assume $a=1$. Because there are no fixed surfaces, we know that $b\neq 0$ and
$m-b\neq 0$. Figure 2.5 shows the three possible graphs for $S^{1}(1,b;m)$,
which are then of types (G), (H), or (I). The dashed edges represent the
possible locations of the $\mathbb{T}^{2}_{n}$ invariant curves $C_{n}$ and
$C_{-n}$. We can compute the self-intersection of $C_{n}$ and $C_{-n}$ by
applying Lemma 2.12 to the normal bundle of these invariant spheres, and using
the configurations of weights shown in Figure 2.6. The self-intersection of
the curve between $P$ and $R$ is $2b-m$, while the self-intersection of the
curve between $Q$ and $S$ is $m-2b$. Set $n=|2b-m|$. For the toric action
$\mathbb{T}^{2}_{n}$ to exist, we must also have
$0<2\omega_{\lambda}(C_{n})=2\lambda-|2b-m|-\epsilon_{m}$. We conclude that
the action $S^{1}(1,b;m)$, $m\neq 0$, may extend to $\mathbb{T}^{2}_{|2b-m|}$
whenever $2\lambda>|2b-m|+\epsilon_{m}$, and that this is the only other
possible toric extension.

(a) *

$b$$m-b$$\mu=(\lambda+k)$$\mu=b$$\mu=b+(\lambda-k-\epsilon_{m})$$\mu=0$QRSP
(g) When $a=1,~{}b>0$, and $m-b>0$

PQSR$\mu=b+(\lambda-k-\epsilon_{m})$$\mu=(\lambda+k)$$\mu=b$$b$$b-m$$\mu=0$
(h) When $a=1,~{}b>0$ and $m-b<0$

$\mu=(\lambda+k)-b$$\mu=-b$$\mu=(\lambda-k-\epsilon_{m})$$\mu=0$$-b$$m-b$PRSQ
(i) When $a=1,~{}b<0$

Figure 2.5: Graphs of $S^{1}(1,b;m)$ with possible locations of $C_{n}$ and
$C_{-n}$ $1\,$$b$$\,-1$$m-b$$P$$R$$1$$1\,$$-b$$\,-1$$b-m$$Q$$S$$1$ Figure 2.6:
Configurations of weights along $C_{\pm n}$ when $a=1$

When $a=-1$, the same arguments apply to $S^{1}(-1,b;m)$ whose possible graphs
are now of types (J), (K), and (L). The self-intersections of the curves
$C_{\pm 1}$ are now $\pm|2b+m|$. We conclude that the action $S^{1}(-1,b;m)$
may extend to $\mathbb{T}^{2}_{|2b+m|}$ whenever $m\neq 0$ and
$2\lambda>|2b+m|+\epsilon_{m}$.

In the special case $m=0$, any invariant sphere appearing in the graph of
$S^{1}(a,b;0)$ has zero self-intersection. If $S^{1}(a,b;0)$ extends to
$\mathbb{T}^{2}_{n}$ with $n\geq 2$, it again follows that the invariant curve
$C_{n}$ of self-intersection $-n$ must have trivial isotropy. Consequently,
$c=\pm 1$, and at least one of the weights $a$ or $b$ must be $\pm 1$. If
$a=1$, the possible graphs of $S^{1}(1,b;0)$ are the graphs (H) and (I) of
Figure 2.5. Then, the same argument as before shows that the self-intersection
of invariant curves are $0$ and $\pm 2b$. Consequently, the action
$S^{1}(1,b;0)$ may extend to $\mathbb{T}^{2}_{|2b|}$ provided $2\lambda>|2b|$.
Similarly, when $a=-1$, the action $S^{1}(-1,b;0)$ may extend to
$\mathbb{T}^{2}_{|2b|}$ whenever $2\lambda>|2b|$.

When $m=0$ and $b=1$, then the possible self-intersections of invariant curves
are $0$ and $\pm 2a$. However, the two possible graphs are now of type (H) and
(J). In both cases, the area of the tentative invariant curve $C_{\pm n}$
connecting the two interior fixed points is negative. It follows that it is
impossible to have $m=0$ and $b=1$. Similarly, if $b=-1$, the possible graphs
are of type (I) and (K) and, as before, the area of the tentative invariant
curve $C_{\pm n}$ connecting the two interior fixed points is negative.
Consequently, we cannot have $m=0$ and $b=\pm 1$. This concludes the proof of
the statement for actions whose fixed points are all isolated.

(a) *

QSR$\mu=1+a(\lambda-k-\epsilon_{m})$$\mu=a(\lambda+k)$$\mu=1$P$a$$a$$\mu=0$
(h) When $m=0$, $a>0,~{}b=1$

$\mu=1-a(\lambda+k)$$\mu=1$$\mu=-a(\lambda+k)$$\mu=0$$-a$$-a$RPQS (j) When
$m=0$, $a<0,~{}b=1$

Figure 2.7: Graphs of $S^{1}(a,1;0)$ with impossible locations of $C_{n}$ and
$C_{-n}$

Now suppose $S^{1}(a,b;m)$ has non-isolated fixed points, that is, $a=0$, or
$b=0$, or $b=am$. As the moment map values of graphs (A), (B), (E), and (F)
depend only on $m$, it is impossible to get the same graphs from another
action $S^{1}(c,d;n)$ with $n\neq m$. The same remark applies to the area
labels of graphs (C) and (D), showing that $S^{1}(a,b;m)$ only extends to
$\mathbb{T}^{2}_{m}$. ∎

It remains to investigate whether the action $S^{1}(\pm 1,b;m)$ extends to
$\mathbb{T}^{2}_{|2b-m|}$ when $2\lambda>|2b-am|+\epsilon_{m}$ and
$b\not\in\\{0,am\\}$. A straightforward but lengthy comparison of the possible
graphs of $S^{1}(\pm 1,b;m)$ and $S^{1}(\pm,d;|2b-am|)$ shows that this is
always the case and yields the equivalences stated in the following two
corollaries. The graphs giving the first equivalence are shown in Figure 2.8.
The other cases are left to the reader.

######  Corollary 2.23.

Consider the action $S^{1}(1,b;m)$ and suppose $2\lambda>|2b-m|+\epsilon_{m}$.
Then under the following numerical conditions on $b$ and $m$, the
$S^{1}(1,b;m)$ action extends to the toric action $\mathbb{T}^{2}_{|2b-m|}$
and is equivariantly symplectomorphic to the following subcircle in
$\mathbb{T}^{2}_{|2b-m|}$ :

  1. 1.

if $b>0$ and $b>m$, then $S^{1}(1,b;m)$ is equivalent to $S^{1}(1,b;|2b-m|)$;

  2. 2.

if $b>0$, $m>b$, and $2b-m<0$, then $S^{1}(1,b;m)$ is equivalent to
$S^{1}(1,-b;|2b-m|)$;

  3. 3.

if $b>0$, $m>b$, and $2b-m>0$, then $S^{1}(1,b;m)$ is equivalent to
$S^{1}(1,b;|2b-m|)$;

  4. 4.

finally, if $b<0$, then $S^{1}(1,b;m)$ is equivalent to $S^{1}(1,-b;|2b-m|)$.∎

######  Corollary 2.24.

Consider the $S^{1}$ actions $S^{1}(-1,b;m)$ on $(S^{2}\times
S^{2},\omega_{\lambda})$ and suppose $2\lambda>|2b+m|$. Then under the
following numerical conditions on $b$ and $m$, the $S^{1}(-1,b;m)$ action
extends to the toric action $\mathbb{T}^{2}_{|2b+m|}$ and is equivariantly
symplectomorphic to the following subcircle in $\mathbb{T}^{2}_{|2b+m|}$ :

  1. 1.

if $b<0$ and $m>-2b$, then $S^{1}(-1,b;m)$ is equivalent to
$S^{1}(-1,-b;|2b+m|)$;

  2. 2.

if $b<0$, $m>-b$, and $-2b>m$, then $S^{1}(-1,b;m)$ is equivalent to
$S^{1}(-1,b;|2b+m|)$;

  3. 3.

if $b<0$ and $-b>m$, then $S^{1}(-1,b;m)$ is equivalent to
$S^{1}(-1,b;|2b+m|)$;

  4. 4.

if $b>0$, then $S^{1}(-1,b;m)$ is equivalent to $S^{1}(-1,-b;|2b+m|)$.∎

$S^{1}(1,b;m)$ of type (H)

(a) *

$S^{1}(1,b;2b-m)$ of type (G)

(b) *

[.45]
QSR$\mu=b+(\lambda-k-\epsilon_{k})$$\mu=(\lambda+k)$$\mu=b$P$b$$a=1$$a=1$$b-m$$\mu=0$
[.45]
$b$$a=1$$a=1$$(2b-m)-b=b-m$$\mu=\lambda+\frac{(2b-m)-\epsilon_{k}}{2}=b+\lambda-k-\epsilon_{k}$$\mu=b$$\mu=b+\lambda-\frac{(2b-m)-\epsilon_{k}}{2}-\epsilon_{k}$$=\lambda+k$$\mu=0$QRSP

Figure 2.8: The equivalence $S^{1}(1,b;m)\sim S^{1}(1,b;2b-m)$ when $b>m$

##  Chapter 3 Action of $\operatorname{Symp}^{S^{1}}(S^{2}\times
S^{2},\omega_{\lambda})$ on $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$

In this chapter we show that the space of $S^{1}$ invariant compatible almost
complex structures $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ can be decomposed
into strata each of which being homotopy equivalent to a homogeneous space
under the action of the equivariant symplectomorphism group.

###  3.1 $J$-Holomorphic Preliminaries

We first recall a few facts about compatible almost complex structures and
associated $J$-holomorphic curves.

######  Definition 3.1 (Compatible almost complex structures).

An almost complex structure $J$ on a symplectic manifold $(M,\omega)$ is said
to be compatible with $\omega$ if $\omega(u,Ju)>0$ and
$\omega(Ju,Jv)=\omega(u,v)$ for all non-zero $u,v\in T_{*}M$.

######  Lemma 3.2.

The space $\mathcal{J}_{\omega}=\mathcal{J}(M,\omega)$ of all compatible
almost complex structures on a symplectic manifold $(M,\omega)$ is non-empty
and contractible.

######  Definition 3.3.

$J$-holomorphic spheres: Let $(M,\omega)$ be a symplectic manifold endowed
with a compatible almost complex structure $J$. A rational $J$-holomorphic
map, also called a parametrized $J$-holomorphic sphere, is a $C^{\infty}$ map

$u:({S}^{2},j)\longrightarrow(M,\omega,J)$

satisfying the Cauchy-Riemann equation

$\bar{\partial}_{J}(u)=\frac{1}{2}(du\circ j-J\circ du)=0$

where $j$ is the usual complex structure on the sphere. The image of a
$J$-holomorphic rational map is called a rational $J$-holomorphic curve or
simply a $J$-curve.

######  Definition 3.4 (Multi-covered and simple maps).

We say that a $J$-holomorphic map $u:\mathbb{C}P^{1}\longrightarrow(M,J)$ is
multi-covered if $u={u}^{\prime}\circ f$, where
$f:\mathbb{C}P^{1}\to\mathbb{C}P^{1}$ is a holomorphic map of degree greater
than one and where $u^{\prime}:\mathbb{C}P^{1}\to(M,J)$ is a $J$-holomorphic
map. We call a $J$-holomorphic map simple if it is not multi-covered.

######  Remark 3.5.

We usually assume that a $J$-holomorphic map is somewhere injective, meaning
that $\exists z\in{S}^{2}$ such that $du_{z}\neq 0$ and $u^{-1}u(z)=z$. In
particular, somewhere injective maps do not factor through multiple covers
$h:S^{2}\to S^{2}$.

######  Definition 3.6 (Moduli spaces of $J$-holomorphic maps or curves).

Let $(M,\omega)$ be a symplectic manifold and let $J\in\mathcal{J}_{\omega}$.
Given $A\in H_{2}(M,\mathbb{Z})$ we denote by $\widetilde{\mathcal{M}}(A,J)$
the space of all $J$-holomorphic, somewhere injective maps representing the
homology class $A$. The Mobius group $G=\operatorname{PSL}(2,\mathbb{C})$ acts
freely on this space by reparametrization and the quotient space
$\mathcal{M}(A,J):=\widetilde{\mathcal{M}}(A,J)/G$ is called the moduli space
of (unparametrised) $J$-curves in class $A$.

In dimension 4, the geometric properties of $J$-holomorphic curves are, to a
large extend, controlled by homological data. As a result, many properties of
complex algebraic curves in complex algebraic surfaces extend to
$J$-holomorphic curves in $4$-dimensional symplectic manifolds. Below we list
the key properties of $J$-holomorphic curves we will be relying on.

######  Theorem 3.7 (Positivity).

Let $(M,\omega)$ be a 4-dimensional symplectic manifold. If a homology class A
$\in H_{2}(M,\mathbb{Z})$ is represented by a nonconstant J-curves for some
$J\in\mathcal{J_{\omega}}$ then $\omega(A)>0$.

######  Theorem 3.8 (Fredholm property and automatic regularity.).

Let $(M,\omega)$ be a 4-dimensional symplectic manifold. Then the universal
moduli space

$\widetilde{\mathcal{M}}(A,\mathcal{J}_{\omega}):=\bigcup_{J\in{\mathcal{J}_{\omega}}}\widetilde{\mathcal{M}}(A,J)$

with $C^{l}$-topology ($l\geq 2$) is a smooth Banach manifold and the
projection map

$\pi_{A}:\widetilde{\mathcal{M}}(A,\mathcal{J}_{\omega})\longrightarrow\mathcal{J}_{\omega}$

is a Fredholm map of index $2(c_{1}(A)+2)$ where $c_{1}\in
H^{2}(M,\mathbb{Z})$ is the first chern class of $(TM,J)$ (note that the Chern
class is independent of choice of $J\in\mathcal{J}_{\omega}$). An almost
complex structure is said to be regular for the class $A$ if it is a regular
value for the projection $\pi_{A}$. If this is the case then the moduli spaces
$\widetilde{\mathcal{M}}(A,J)$ and $\mathcal{M}(A,J)$ are smooth manifolds of
dimensions $2(c_{1}(A)+2)$ and $2(c_{1}(A)-1)$ respectively. The set of
regular values $\mathcal{J}\in\mathcal{J}_{\omega}$ is a subset of second
category and is denoted by $\mathcal{J}_{\omega}^{\textrm{reg}}(A)$. If $J\in
J_{\omega}$ is integrable and $S$ is an embedded $J$-holomorphic sphere with
self-intersection number $[S]\cdot[S]\geq-1$, then $J$ is regular for the
class $[S]$. In dimension $4$, the same conclusion holds without the
integrability assumption.

######  Definition 3.9 (Cusp Curves).

Let $(M,\omega)$ be a symplectic manifold. Let $J\in J_{\omega}$. A
$J$-holomorphic cusp curve $C$ is a connected finite union of $J$-holomorphic
curves

$C=C_{1}\cup C_{2}\ldots\cup C_{k}$

where $C_{i}=u_{i}(\mathbb{C}P^{1})$ and $u_{i}:\mathbb{C}P^{1}\to(M,J)$ is a
(possibly multi-covered) $J$-holomorphic map.

######  Theorem 3.10 (Gromov’s compactness theorem).

Let $(M,\omega)$ be a compact symplectic manifold. Let
$J_{n}\in\mathcal{J}_{\omega}$ be a sequence converging to $J$ in the
$C^{\infty}$ topology and let $S_{i}$ be $J_{i}$-holomorphic spheres of
bounded symplectic area $\omega(S_{i})$. Then there is a subsequence of the
$S_{i}$ which converges weakly to a $J$-holomorphic curve or cusp-curve $S$.
In particular if all the $S_{i}$’s belong to the class $A$, then $S$ also
belongs the class $A$, and any cusp curve defines a homological decomposition
of $A=\sum_{i}A_{i}$ such that $\omega(A_{i})>0$.

######  Theorem 3.11 (Positivity of intersections).

Let $J\in\mathcal{J}_{\omega_{\lambda}}$ and $A$, $B$ be two distinct
$J$-holomorphic curves in a $4$-dimensional manifold. Then they intersect at
only finitely many points and each point contributes positively to the
intersection multiplicity $[A]\cdot[B]$. Moreover, $[A]\cdot[B]=1$ iff the
curves intersect transversally at exactly one point, while $[A]\cdot[B]=0$ iff
the curves are disjoint.

As a corollary of Positivity of intersections we have the following result
under the presence of a group action.

######  Corollary 3.12.

Let $(M,\omega)$ be a symplectic 4-manifold and let $G$ be a compact Lie group
acting symplectically on $M$. Suppose that $G$ acts trivially on homology. Let
$\mathcal{J}^{G}$ denote the space of $\omega$ tame (or compatible) almost
complex structures and let $C$ be a $J$ holomorphic curve for some
$J\in\mathcal{J}^{G}$. Then,

  1. 1.

if C has negative self intersection, then $g\cdot C=C$ for all $g\in G$.

  2. 2.

If C has zero self intersection, then $g\cdot C=C$ or $g\cdot C\cap
C=\emptyset$ for all $g\in G$.

######  Theorem 3.13 (Adjunction formula).

Let $u:({S}^{2},j)\longrightarrow(M^{4},J)$ be a somewhere injective
$J$-holomorphic map representing the homology class $A$ in a $4$-dimensional
manifold. Define the virtual genus of $A$ as

$g_{v}(A)=1+\frac{1}{2}([A]\cdot[A]-c_{1}(A))$

where $c_{1}(A)=\langle c_{1}(TM,J),A\rangle$. Then $g_{v}(A)\geq 0$ with
equality if, and only if, the map $u$ is an embedding.

###  3.2 $J$-holomorphic spheres in $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$

In this section, will show how the existence of certain $J$-holomorphic
spheres in ruled $4$-manifolds induces a natural partition of the space
$\mathcal{J}_{\omega}$. We shall state all the relevant results, and refer the
reader to Chapter 9 of [MS12] for more details.

######  Corollary 3.14.

Let $\lambda=l+\delta$ where $l\in\mathbb{N}$ and $0<\delta\leq 1$. Then we
have

  1. 1.

Any $J$-holomorphic representative of the class $F$ is a simple curve.

  2. 2.

The only $J$-holomorphic decomposition of the class $B$ are of the form
$B=(B-kF)+kF$, where $0\leq k\leq\ell$. In this decomposition, the
$J$-holomorphic representative of the class $(B-kF)$ is an embedded sphere,
while the class $kF$ may be represented by a collection of (possibly multiply
covered) spheres representing multiples of the class $F$.∎

######  Proposition 3.15.

The moduli space $\widetilde{\mathcal{M}}(F,J)\neq\emptyset$ for all $J\in
J_{\omega}$. In particular, for every compatible almost complex structure
$J\in\mathcal{J}_{\omega_{\lambda}}$, and for any given point $p\in
S^{2}\times S^{2}$, there is a unique embedded $J$-holomorphic sphere
representing the class $F$ passing through $p$.∎

We also have an analogous result for
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$.

######  Theorem 3.16.

Given any point $p\in\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, and any
$J\in\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$, there is a $J$-holomorphic curve
in the class $F:=L-E$ passing through $p$. ∎

The following Theorem due to Abreu and McDuff [AM00] tells us about the
decomposition of the space of compatible almost complex structures on
$(S^{2}\times S^{2},\omega_{\lambda})$ into finitely many strata.

######  Theorem 3.17.

Let $\mathcal{J}_{\omega_{\lambda}}$ denote the space of all compatible almost
complex structures (not necessarily invariant) for the form
$\omega_{\lambda}$, on $S^{2}\times S^{2}$ then the space
$\mathcal{J}_{\omega_{\lambda}}$ admits a finite decomposition into disjoint
Fréchet manifolds of finite codimensions

$\mathcal{J}_{\omega_{\lambda}}=U_{0}\sqcup U_{2}\sqcup U_{4}\ldots\sqcup
U_{2n}$

where $2n=\lceil 2\lambda\rceil-1$ and $\lceil\lambda\rceil$ is the unique
integer $l$ such that $l<\lambda\leq l+1$ and where

$U_{2i}:=\left\\{J\in\mathcal{J}_{\omega_{\lambda}}~{}|~{}D_{2i}=B-iF\in
H_{2}(S^{2}\times S^{2},\mathbb{Z})\text{~{}is represented by a
$J$-holomorphic sphere}\right\\}$

A completely analogous description holds true for
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$.

######  Theorem 3.18.

Let $\mathcal{J}_{\omega_{\lambda}}$ denote the space of all compatible almost
complex structures (not necessarily invariant) for the form
$\omega_{\lambda}$, then the space $\mathcal{J}_{\omega_{\lambda}}$ admits a
finite decomposition into disjoint Fréchet manifolds of finite codimensions

$\mathcal{J}_{\omega_{\lambda}}=U_{1}\sqcup U_{3}\sqcup U_{5}\ldots\sqcup
U_{2n-1}$

where $2n=\lceil 2\lambda\rceil-1$, and $\lceil\lambda\rceil$ is the unique
integer $l$ such that $l<\lambda\leq l+1$ and where

$U_{2i-1}:=\left\\{J\in\mathcal{J}_{\omega_{\lambda}}~{}|~{}D_{2i-1}=B-iF\in
H_{2}(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\mathbb{Z})\text{~{}is
represented by a $J$-holomorphic sphere}\right\\}$

######  Remark 3.19.

We label the strata in $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ by the homological self-
intersection of the classes $D_{s}$.

######  Remark 3.20.

We note that for both $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, there is a canonical
integrable almost complex structure $J_{s}$ in the strata $U_{s}$ coming from
realizing $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ as the $s^{\text{th}}$\-
Hirzebruch surface $W_{s}$ of section 1.2. Further recall that associated to
each $J_{s}$ we have a unique $J_{s}$-holomorphic Hamiltonian toric action
$\mathbb{T}_{s}$. Thus the set of possible equivalence classes of toric
actions on $S^{2}\times S^{2}$ and
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ are in one-to-one
correspondence with the strata in the decomposition of
$\mathcal{J}_{\omega_{\lambda}}$. This fact will be crucial in our later
analysis of centralizer subgroups.

###  3.3 Intersection of $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ with the
strata

In this section we shall answer the question as to which strata the space of
invariant almost complex structures for a given action $(S^{2}\times
S^{2},\omega_{\lambda})$ or
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$ intersects.

In what follows, we will use the following simple observation several times.
Let $(M,\omega)$ be a simply connected symplectic $4$-manifold. There is a
left-exact sequence

$1\to\operatorname{Symp}_{h}(M,\omega)\to\operatorname{Symp}(M,\omega)\to\operatorname{Aut}_{c_{1},\omega}\left(H_{2}(M,\mathbb{Z})\right)$
(3.1)

where $\operatorname{Symp}_{h}(M,\omega)$ is the subgroup of
symplectomorphisms acting trivially on homology, and where
$\operatorname{Aut}_{c_{1},\omega_{\lambda}}\left(H_{2}(M,\mathbb{Z})\right)$
is the group of automorphisms of $H_{2}(M,\mathbb{Z})$ that preserve the
intersection product and the Poincaré duals of the cohomology classes
$c_{1}(TM)$ and $[\omega_{\lambda}]$. This later group is the identity for
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$ with
$\lambda\geq 1$ and for $(S^{2}\times S^{2},\omega_{\lambda})$ with
$\lambda>1$. In the case of $(S^{2}\times S^{2},\omega_{\lambda})$ with
$\lambda=1$, the group
$\operatorname{Aut}_{c_{1},\omega_{\lambda}}\left(H_{2}(M,\mathbb{Z})\right)$
is equal to $\mathbb{Z}_{2}$ and is generated by the symplectomorphism that
swaps the two $S^{2}$ factors. Consequently, for any symplectically ruled
rational surface, the above sequence is also right-exact and splits.

######  Lemma 3.21.

We have the following equalities among symplectomorphism groups:

  * •

$\operatorname{Symp}(S^{2}\times
S^{2},\omega_{\lambda})=\operatorname{Symp}_{h}(S^{2}\times
S^{2},\omega_{\lambda})\ltimes\mathbb{Z}_{2}$ when $\lambda=1$,

  * •

$\operatorname{Symp}(S^{2}\times
S^{2},\omega_{\lambda})=\operatorname{Symp}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$ when $\lambda>1$, and

  * •

$\operatorname{Symp}_{h}(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})=\operatorname{Symp}(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$
for all $\lambda\geq 1$.

∎

######  Proposition 3.22.

Fix a circle action $S^{1}(a,b;m)$. Then the space of invariant almost complex
structures $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ intersects the stratum
$U_{n}$ iff the circle action $S^{1}(a,b;m)$ extends to the toric action
$\mathbb{T}^{2}_{n}$.

###### Proof.

If $\lambda=1$, the result is immediate as the only toric action is
$\mathbb{T}^{2}_{0}$.

Let $\lambda>1$ and suppose the action $S^{1}(a,b;m)$ extends to
$\mathbb{T}^{2}_{n}$. This means that there exists a circle action
$S^{1}(c,d;n)\subset\mathbb{T}^{2}_{n}$ which is equivariantly
symplectomorphic to the action $S^{1}(a,b;m)$ via some symplectomorphism
$\phi$. Let $J_{n}$ denote the standard complex structure in the stratum
$U_{n}$. By Lemma 3.21, when $\lambda>1$, the group
$\operatorname{Symp}_{h}(S^{2}\times S^{2},\omega_{\lambda})$ is equal to
$\operatorname{Symp}(S^{2}\times S^{2},\omega_{\lambda})$ so that $\phi$
preserves homology. Consequently, $\phi^{*}J_{n}$ belongs to the stratum
$U_{n}$ and is invariant with respect to the $S^{1}(a,b;m)$ action, showing
that the space $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ intersects the stratum
$U_{n}$.

Conversely, suppose that $J\in\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{n}$. If $n\geq 1$, then there exists an invariant, embedded, symplectic
sphere $C_{n}$ of self-intersection $-n$. Arguing as in Proposition 2.22 and
Corollaries 2.23 and 2.24, we conclude that $S^{1}(a,b;m)$ must extend to the
torus $\mathbb{T}^{2}_{n}$. If $n=0$, there there exist invariant spheres
representing the homology classes $B$ and $F$ passing to a common fixed point.
Again, this implies that $S^{1}(a,b;m)$ extends to the torus
$\mathbb{T}^{2}_{0}$. Alternatively, we can adapt the proofs of Lemma 3.31 in
the present document and of Proposition 4.7 in [CK19] to show that if an
invariant $J$ is in the stratum $U_{n}$ the circle action $S^{1}(a,b;m)$
extends to the toric action $\mathbb{T}^{2}_{n}$. ∎

The following corollaries immediately follow from Proposition 2.22 and
Corollaries 2.23 and 2.24.

######  Corollary 3.23.

Suppose $\lambda=1$. Then the space $\mathcal{J}_{\omega_{1}}$ of compatible,
almost-complex structures on $(S^{2}\times S^{2},\omega_{1})$ is made of only
one stratum. In particular, any Hamiltonian circle action extends to the toric
action $\mathbb{T}^{2}_{0}$ and the subspace
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ of $S^{1}$ invariant almost-complex
structures is contractible. ∎

######  Corollary 3.24.

Suppose $\lambda>1$. Consider an $S^{1}(a,b;m)$ action $(S^{2}\times
S^{2},\omega_{\lambda})$ or
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$ depending on
whether $m$ is even or odd. Under the following numerical conditions on
$a,b,m,\lambda$, the space $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ intersects
only the stratum $U_{m}$:

  * •

when $a\neq\pm 1$;

  * •

when $b=0$ or $b=am$;

  * •

when $a=\pm 1$ and $2\lambda\leq|2b-m|+\epsilon_{m}$.∎

######  Corollary 3.25.

Consider the $S^{1}(\pm 1,b,m)$ action on $(S^{2}\times
S^{2},\omega_{\lambda})$ or
$(\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}},\omega_{\lambda})$, then
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ intersects exactly two strata. More
precisely,

  1. 1.

if $a=1$, $b\neq\\{0,m\\}$, $\lambda>1$, and $2\lambda>|2b-m|+\epsilon_{m}$,
then the space of $S^{1}(1,b;m)$-equivariant almost complex structures
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ intersects the two strata $U_{m}$ and
$U_{|2b-m|}$.

  2. 2.

If $a=-1$, $b\neq\\{0,-m\\}$, $\lambda>1$, and $2\lambda>|2b+m|+\epsilon_{m}$,
then the space of $S^{1}(-1,b;m)$-equivariant almost complex structures
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ intersects the two strata $U_{m}$ and
$U_{|2b+m|}$.∎

###  3.4 Symplectic actions of compact abelian groups on $\mathbb{R}^{4}$

In order to study the action of the the equivariant symplectomorphism group
$\operatorname{Symp}_{h}^{S^{1}}(S^{2}\times S^{2},\omega_{\lambda})$ on each
invariant stratum $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{k}$, we will
need to understand the equivariant topology of linearised symplectic actions.
The main result is Theorem 3.29 which is an equivariant version of Gromov’s
Theorem on the contractibility of the group of compactly supported
symplectomorphisms of star-shaped domains of $\mathbb{R}^{4}$. This relies on
Lemma 3.26 and Proposition 3.27 that were proven by W. Chen in the unpublished
manuscript [Che]. For completeness, we shall reproduce their proof here.

Let $G$ be a compact group acting effectively and symplectically on
$\mathbb{C}^{2}=\mathbb{R}^{4}$ with the symplectic form
$\omega_{0}:=dx_{1}\wedge dy_{1}+(dx_{2}\wedge dy_{2})$. We say it acts
linearly if it acts as a subgroup of
$\operatorname{U}(2)\subset\operatorname{Sp}(4)$.

######  Lemma 3.26 (Chen).

Let $G$ be a compact group acting linearly on $(\mathbb{R}^{4},\omega_{0})$.
Suppose $V$ is a $G$ invariant, compact, and star-shaped neighbourhood of $0$.
Let $f:\mathbb{R}^{4}\setminus V\to\mathbb{R}^{4}$ be a $G$-equivariant
symplectic embedding which is the identity near infinity. Then, for every
$G$-invariant neighbourhood $W$ of $V$, there exists a $G$-equivariant
symplectomorphism $g:\mathbb{R}^{4}\to\mathbb{R}^{4}$ such that
$g|_{\mathbb{R}^{4}\setminus W}=f$.

###### Proof.

As $0\in int(V)$ and as $f$ is the identity near infinity, there exist $T>0$
such that $f(Tx)=Tx$ for all $x\in\mathbb{R}^{4}\setminus~{}V$. Define
$f_{t}(x)=\frac{f(tx)}{t}$ for $1\leq t\leq T$. As the $G$ action is linear,
we observe that $f_{t}$ is equivariant for all $t\in[1,T]$. By construction,
$f_{1}=f$, $f_{T}=id$ and $f_{t}^{*}\omega_{0}=\omega_{0}$ for all $t$. Thus
there are compact sets $V_{t}=f_{t}(V)$ and open neighbourhoods
$W_{t}=f_{t}(W)$ of $V_{t}$ such that the restriction
$f_{t}:\mathbb{R}^{4}\setminus V\to\mathbb{R}^{4}\setminus V_{t}$ and
$f_{t}:\mathbb{R}^{4}\setminus W\to\mathbb{R}^{4}\setminus W_{t}$ are
diffeomorphic. As $G$ acts linearly, each of the sets $W_{t}$ and $V_{t}$ are
$G$-invariant.

Define $X_{t}$ as the vector field that satisfies
$\frac{d}{dt}f_{t}=X_{t}\circ f_{t}$ and consider the one form
$\alpha_{t}=i_{X_{t}}\omega_{0}$. As $f_{t}$ is $G$ equivariant and
symplectic, both $X_{t}$ and $\alpha_{t}$ are $G$-invariant. Let
$H_{t}:\mathbb{R}^{4}\setminus V_{t}\to\mathbb{R}$ be a one parameter family
of Hamiltonians that are $G$-invariant and that satisfy $\alpha_{t}=dH_{t}$.
Since $f_{t}$ is the identity near infinity, this implies that $H_{t}$ is
constant near infinity and we can take this constant to be 0.

Finally, we can take a family of $G$-invariant bump functions
$\rho_{t}:\mathbb{R}^{4}\to[0,1]$ such that $\rho_{t}\equiv 0$ in a
neighbourhood of $V_{t}$ and $\rho_{t}\equiv 1$ on $\mathbb{R}^{4}\setminus
W_{t}$. Then the Hamiltonian $\rho_{t}H_{t}:\mathbb{R}^{4}\to\mathbb{R}$ is
defined on the whole $\mathbb{R}^{4}$ and is also $G$-invariant. The
Hamiltonian isotopy $g_{t}$ generated by $\rho_{t}H_{t}$ is $G$ equivariant
for all $1\leq t\leq T$ and satisfies the properties $g_{T}=\operatorname{id}$
and $g_{1}|_{\mathbb{R}^{4}\setminus W}=f$. Consequently, $g_{1}$ is the
symplectomorphism $g$ we were looking for. ∎

Any unitary representation of a compact abelian group $G$ on $\mathbb{C}^{2}$
induces a splitting into eigenspaces
$\mathbb{C}^{2}=\mathbb{C}_{1}\bigoplus\mathbb{C}_{2}$. This simple fact turns
out to be essential in our treatment of the equivariant Gromov’s Theorem.
Consequently, from now on, we assume that the group $G$ is abelian.

######  Proposition 3.27 (Chen).

Let $V\subset\mathbb{R}^{4}$ be a compact star-shaped neighbourhood of
$\\{0\\}$ and let $\omega$ be a symplectic form on $\mathbb{R}^{4}$ such that
$\omega=\omega_{0}$ outside some smaller open neighbourhood of the origin
$U\subset V$. Let $G$ be a compact abelian group acting on $(V,\omega)$ via
symplectomorphisms that are linear near the boundary of V. Then the G action
is conjugate to a linear symplectic action of $G$ on $(V,\omega_{0})$ by a
diffeomorphism $\Phi$ which is the identity near the boundary and which
satisfies $\Phi^{*}\omega=\omega_{0}$.

###### Proof.

Identify $\mathbb{R}^{4}$ with $\mathbb{C}^{2}$. The linear action near
$\partial V$ extends to a unitary action on $\mathbb{R}^{4}$. As $G$ is
abelian this linear action splits into two eigenspaces
$\mathbb{C}_{1}\oplus\mathbb{C}_{2}$. We can then compactify $\mathbb{C}^{2}$
along these eigenspaces to obtain $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$
(diffeomorphic to $S^{2}\times S^{2}$) equipped with two symplectic forms
$\tilde{\omega}$ and $\tilde{\omega}_{0}$ induced from $\omega$ and
$\omega_{0}$. The compactified space inherits two actions of $G$, namely,
$\rho:G\hookrightarrow\operatorname{Symp}(S^{2}\times S^{2},\tilde{\omega})$
coming from extending the $G$ action on $\mathbb{R}^{4}$ and
$\rho_{\operatorname{lin}}:G\hookrightarrow\operatorname{Symp}(S^{2}\times
S^{2},\tilde{\omega})$ which extends the linear action near $\partial V$.

By construction, there exists a star-shaped subset $V_{1}\subset V$ such that
both the action $\rho$ and $\rho_{\operatorname{lin}}$ agree on
$\mathbb{C}^{2}\setminus V_{1}$. Note that the point at infinity
$p:=(\infty,\infty)$ is a fixed point for both the action $\rho$ and
$\rho_{lin}$. Choose a $\tilde{\omega}$ compatible $G$-invariant almost
complex structure $J$ on $S^{2}\times S^{2}$ which is standard in some
neighborhood $X_{\epsilon}:=(S^{2}\times D_{\epsilon})\cup(S^{2}\times
D_{\epsilon})$ of the wedge $(S^{2}\times\\{\infty\\})\vee(\\{\infty\\}\times
S^{2})$ (where $D_{\epsilon}$ is a small disk of radius $\epsilon$ at
$\infty$). As $(S^{2}\times\\{\infty\\})$ and $(\\{\infty\\}\times S^{2})$ are
$J$-holomorphic spheres representing the classes $B$ and $F$ respectively, we
conclude that $J$ belongs to the stratum $U_{0}$ so that there exists
foliations $\mathcal{B}_{J}$ and $\mathcal{F}_{J}$ by embedded $J$-holomorphic
spheres in the classes $B$ and $F$. Given any $q=(z,w)\in S^{2}\times S^{2}$,
let $u_{w}$ denote the unique curve in the class B passing through $(0,w)$
and, similarly, let $v_{z}$ be the curve in class F passing through $(z,0)$.
We can define a self-map of $S^{2}\times S^{2}$ by setting

$\displaystyle\Psi_{J}:S^{2}\times S^{2}$ $\displaystyle\longrightarrow
S^{2}\times S^{2}$ $\displaystyle(z,w)$ $\displaystyle\longmapsto u_{w}\cap
v_{z}$

As explain in [MS12] Chapter 9, this map $\Psi_{J}$ is a diffeomorphism.
Moreover, $\Psi_{J}$ is equivariant with respect to the linear action
$\rho_{\operatorname{lin}}$ on the domain and the action $\rho$ on the
codomain. Furthermore, as $J$ is the standard complex structure in a
neighbourhood $X_{\epsilon}$ of the wedge, $\Psi_{J}$ is the identity near the
base point $p$. We now modify $\Psi_{J}$ in order to make it the identity in
the neighbourhood $X_{\epsilon}$. As $J$ is the standard complex structure on
$X_{\epsilon}$ we can write

$\Psi_{J}(z,w)=\begin{cases}(z,\phi_{2}(z,w))&\text{~{}if~{}}z\in
D_{\epsilon}\subset S^{2}\times S^{2}\\\ (\phi_{1}(z,w),w)&\text{~{}if~{}}w\in
D_{\epsilon}\subset S^{2}\times S^{2}\end{cases}$

where $\phi_{1}(z,0)=z$ and $\phi_{2}(0,w)=w$ for all $z,w\in S^{2}$. Choose a
$G$ equivariant (for the $G$ action on $\\{\infty\\}\times S^{2})$) smooth map
$\beta_{1}:S^{2}\longrightarrow S^{2}$ such that $\beta_{1}(z)=z$ for all
$z\in D_{\epsilon}$ and $\beta_{1}=\infty$ in a neighbourhood $D_{\delta}$
contained in $D_{\epsilon}$ and such that $\det(d\beta_{1}(z))\geq
0~{}\forall~{}z\in S^{2}$. Similarly define a $G$ equivariant map (for the $G$
action on $S^{2}\times\\{\infty\\}$) $\beta_{2}:S^{2}\to S^{2}$ satisfying
analogous conditions as $\beta_{1}$. Then we define a modification of
$\Psi_{J}$ by setting

$\Psi^{\prime}_{J}(z,w)=\begin{cases}\Psi_{J}&\text{~{}if~{}}z\in(S^{2}\times
S^{2})\setminus X_{\epsilon}\\\
(z,(\phi_{2}(\beta_{1}(z),w))&\text{~{}if~{}}z\in D_{\epsilon}\\\
(\phi_{1}(z,\beta_{2}(w)),w)&\text{~{}if~{}}w\in D_{\epsilon}\end{cases}$

This modification $\Psi^{\prime}_{J}$ is the identity in a smaller
neighbourhood $X_{\delta}:=(S^{2}\times D_{\delta})\cup(S^{2}\times
D_{\delta})\subset X_{\epsilon}$.

The submanifolds $\\{z\\}\times S^{2}$ and $S^{2}\times\\{w\\}$ for all
$z,w\in S^{2}$ are symplectic for the form
${\Psi^{\prime}_{\\!J}}^{*}\tilde{\omega}$ and hence
$\tilde{\omega}_{0}\wedge{\Psi^{\prime}_{\\!J}}^{*}\tilde{\omega}>0$. Thus the
path
$\omega_{t}:=t\tilde{\omega}+(1-t){\Psi^{\prime}_{\\!J}}^{*}\tilde{\omega}$ is
a path of non-degenerate symplectic forms for all $t\in[0,1]$. We can then
apply an equivariant Moser isotopy to get an equivariant diffeomorphism
$\alpha$ of $S^{2}\times S^{2}$ such that
$\alpha^{*}{\Psi^{\prime}_{\\!J}}^{*}\tilde{\omega}=\tilde{\omega}_{0}$.
Further, as ${\Psi^{\prime}_{\\!J}}^{*}\tilde{\omega}=\tilde{\omega}_{0}$ on
$X_{\delta}$, the restriction of $\alpha$ to $X_{\delta}$ is the identity. We
then set $\tilde{\Psi}_{J}:=\Psi^{\prime}_{\\!J}\circ\alpha$.

The restriction of $\tilde{\Psi}_{J}:\mathbb{C}^{2}\to\mathbb{C}^{2}$ gives us
a map which is $G$-equivariant with respect to the action
$\rho_{\operatorname{lin}}$ on the domain and $\rho$ on the codomain. As noted
before there exists a star-shaped subset $V_{1}\subset V$ such that both the
action $\rho$ and $\rho_{\operatorname{lin}}$ agree on
$\mathbb{C}^{2}\setminus V_{1}$. We can choose $V_{1}$ such that $0\in
int(V_{1})$. We now apply Theorem 3.26 to the map
$f:={\tilde{\Psi}_{J}}^{-1}|_{\mathbb{C}^{2}\setminus V_{1}}$ and we choose W
in Theorem 3.26 to be a $G$-invariant open subset of $V$ which contains
$V_{1}$. Let $g:\mathbb{C}^{2}\to\mathbb{C}^{2}$ be an equivariant
symplectomorphism as in Theorem 3.26 such that $g|_{\mathbb{R}^{4}\setminus
W}=f$, then the map $\Phi:=\left(\tilde{\Psi}_{J}\circ
g\right)\Big{|}_{V}:V\to V$ is identity near the boundary and satisfies
$\Phi^{*}\omega=\omega_{0}$ and is $G$-equivariant where the action of G on
the domain of $\Phi$ is the linear action $\rho_{\operatorname{lin}}$ while on
the range of $\Phi$ it is the $G$ action $\rho$ on V that we started out with.
Thus $\Phi$ is the required equivariant symplectomorphism that linearizes the
given $G$ action and takes the form $\omega$ to $\omega_{0}$. ∎

Consider a polydisk $D^{2}\times D^{2}$ whose product structure is compatible
with the eigenspace decomposition
$\mathbb{R}^{4}=\mathbb{C}^{2}=\mathbb{C}_{1}\oplus\mathbb{C}_{2}$. Consider
the symplectic form $\omega$ such that $\omega=\omega_{0}$ outside of some
smaller polydisk of the form $D_{r}\times D_{r}\subset D^{2}\times D^{2}$ for
some radius $r$.

######  Theorem 3.28 (Equivariant Gromov’s Theorem for Polydisks).

Let $G$ be an abelian group. Let $\omega$ be a symplectic form on $D^{2}\times
D^{2}$ which is equal to $\omega_{0}$ near the boundary. Let $G$ act
symplectically on $(D^{2}\times D^{2},\omega)$ and suppose the action is
linear near the boundary. Then the group
$\operatorname{Symp}_{c}^{G}(D^{2}\times D^{2},\omega)$ of equivariant
symplectomorphisms that are equal to the identity near the boundary of
$D^{2}\times D^{2}$ is non-empty and contractible.

###### Proof.

As the $G$ action outside of $D_{r}\times D_{r}\subset\mathbb{R}^{4}$ is
linear, we can extend this $G$ action to the whole of $\mathbb{R}^{4}$. Then
we can compactify each eigenspace of $\mathbb{C}$ to an $S^{2}$ and hence this
$G$ action extends to a symplectic action $S^{2}\times S^{2}$ with respect to
the form $\tilde{\omega}$ induced by $\omega$.

By Proposition 3.27 we can conjugate our $G$ action by a symplectomorphism
which is identity near the boundary to get a linear $G$ action on the whole of
$V$. As any two conjugate topological subgroups are homeomorphic, we shall
just study the homotopy type of the compactly supported equivariant
symplectomorphism group
$\operatorname{Symp}_{c,\operatorname{lin}}^{G}(D^{2}\times D^{2},\omega)$ for
the linear $G$ action on $V$.

Let $\mathcal{J}^{G}_{\omega}$ be the non-empty and contractible space of all
equivariant almost complex structures on $D^{2}\times D^{2}$ that are
compatible with $\omega$ and are the standard split almost complex structure
$J_{0}$ outside of $D_{r}\times D_{r}$. As they are the standard almost
complex structures outside of a neighbourhood these almost complex structure
extend to $S^{2}\times S^{2}$ and are compatible with $\tilde{\omega}_{0}$.
Further once we pick a base point $p=(\infty,\infty)\in S^{2}\times S^{2}$ and
identify $D^{2}\times D^{2}$ with the complement of a standard neighborhood
$X_{\epsilon}:=(S^{2}\times D_{\epsilon})\cup(S^{2}\times D_{\epsilon})$ of
the wedge $(S^{2}\times\\{\infty\\})\vee(\\{\infty\\}\times S^{2})\subset
S^{2}\times S^{2}$ (note that the wedge point $(\infty,\infty)$ is a fixed
point for the extended action of G on $S^{2}\times S^{2}$), then any element
$J\in\mathcal{J}^{G}_{\sigma}$ extends to a equivariant almost complex
structure of $S^{2}\times S^{2}$ which is the standard product complex
structure on $S^{2}\times S^{2}$ on a neighbourhood $X_{\epsilon}$ of the
wedge $(S^{2}\times\\{y\\})\vee(\\{x\\}\times S^{2})\subset S^{2}\times
S^{2}$. Conversely any such equivariant almost complex structure compatible
with $\tilde{\omega}$ that is standard in some neighbourhood of the wedge
$(S^{2}\times\\{\infty\\})\vee(\\{\infty\\}\times S^{2})\subset S^{2}\times
S^{2}$ gives us an element of $\mathcal{J}^{G}_{\omega}$ .

In order to show that
$\operatorname{Symp}_{c,\operatorname{lin}}^{G}(D^{2}\times D^{2},\omega)$ is
contractible, we shall prove that it is homotopy equivalent to the
contractible space $\mathcal{J}^{G}_{\omega}$.

Define the map $\tilde{\Psi}_{J}$ as in the proof of Proposition 3.27. Thus we
have a map

$\displaystyle\tau:\mathcal{J}^{G}_{\omega}$
$\displaystyle\longrightarrow\operatorname{Symp}_{c,\operatorname{lin}}^{G}(D^{2}\times
D^{2},\omega)$ $\displaystyle J$ $\displaystyle\longmapsto\tilde{\Psi}_{J}$

To prove that $\tau$ is a homotopy equivalence we construct a homotopy inverse
as follows. Fix a $J^{\prime}\in\mathcal{J}^{G}_{\omega}$, then the homotopy
inverse $\beta$ is defined as,

$\displaystyle\beta:\operatorname{Symp}_{c,\operatorname{lin}}^{G}(D^{2}\times
D^{2},\omega)$ $\displaystyle\longrightarrow\mathcal{J}^{G}_{\omega}$
$\displaystyle\phi$ $\displaystyle\longmapsto\phi_{*}J^{\prime}$

By construction we see that $\tau(\beta(\phi))=\operatorname{id}$ and the
other direction is homotopic to the identity as $J^{G}_{\omega}$ is
contractible. ∎

We shall repeatedly use the following theorem in our analysis of the homotopy
type of the equivariant symplectomorphism groups of $S^{2}\times S^{2}$.

######  Theorem 3.29.

(Equivariant Gromov’s Theorem) Let $(V,\omega)$ be an compact star-shaped
symplectic domain of $\mathbb{R}^{4}$ such that $0\in int(V)$ and let $\omega$
be such that $\omega=\omega_{0}$ near the boundary of $V$. Let $G$ be a
compact abelian group that acts symplectically and linearly near the boundary
and that sends the boundary to itself, then the space of equivariant
symplectomorphisms that act as identity near the boundary (denoted by
$\operatorname{Symp}_{c}^{G}(V,\omega)$) is non-empty and contractible.

###### Proof.

By Proposition 3.27 we can conjugate our $G$ action by a symplectomorphism
which is identity near the boundary to get a linear $G$ action on the whole of
$V$ and such that it takes the form $\omega$ to $\omega_{0}$. As the homotopy
type of the two conjugate equivariant symplectomorphism group is the same
(they are in fact homeomorphic), we shall just study the homotopy type of the
compactly supported equivariant symplectomorphism group for the linear $G$
action on $(V,\omega_{0})$. We denote this group by
$\operatorname{Symp}_{c,\operatorname{lin}}^{G}(V,\omega_{0})$.

Choose real numbers $r>0$ and $T>1$ , $D_{r}\times D_{r}$ is a polydisk of
radius $r$, such that $\frac{1}{T}V\subset D_{r}\times D_{r}\subset int(V)$,
and consider the family of maps
$F_{t}:\operatorname{Symp}_{c,\operatorname{lin}}^{G}(V,\omega_{0})\to\operatorname{Symp}_{c,\operatorname{lin}}^{G}(V,\omega_{0})$
for $1\leq t\leq T$ defined by $F_{t}(\phi)(x)=\frac{\phi(tx)}{t}$ for all
$x\in V$.

Then we have that $F_{t}$ is $G$ equivariant for all $1\leq t\leq T$,
$F_{1}(\phi)=\phi$ for all
$\phi\in\operatorname{Symp}_{c,\operatorname{lin}}^{G}(V,\omega_{0})$,
$F_{t}(id)=id$ for all t, and
$F_{T}\left(\operatorname{Symp}_{c,\operatorname{lin}}^{G}(V,\omega_{0})\right)\subset\operatorname{Symp}_{c,\operatorname{lin}}^{G}(D_{r}\times
D_{r},\omega)$.

The proof of Theorem 3.28 tells us that the inclusion
$i:\operatorname{Symp}_{c,\operatorname{lin}}^{G}(D_{r}\times
D_{r},\omega)\hookrightarrow\operatorname{Symp}_{c,\operatorname{lin}}^{G}(V,\omega_{0})$
is contractible. Hence we can fix a contraction $\alpha_{t}$ for $T\leq t\leq
T+1$ such that $\alpha_{T}=i$ and $\alpha_{T+1}(\phi)=id$ for all
$\phi\in\operatorname{Symp}_{c,\operatorname{lin}}^{G}(D_{r}\times
D_{r},\omega)$. Then the concatenation

$\tilde{F_{t}}:=\begin{cases}F_{t}~{}~{}1\leq t\leq T\\\
F_{T}\circ\alpha_{t}~{}~{}T\leq t\leq T+1\end{cases}$

gives us a retraction of
$\operatorname{Symp}_{c,\operatorname{lin}}^{G}(V,\omega_{0})$ to $id$ and
hence $\operatorname{Symp}_{c}^{G}(V,\omega)$ is contractible. ∎

######  Remark 3.30.

To our knowledge, it is not known whether an equivariant version of Gromov’s
Theorem on holds true for non-abelian compact groups.

###  3.5 Homotopical description of
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{k}$

We now consider the action of the group of equivariant symplectomorphisms
$\operatorname{Symp}_{h}^{S^{1}}(S^{2}\times S^{2},\omega_{\lambda})$ on the
contractible space $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$ of invariant,
compatible, almost-complex structures, and we investigate the orbit-type
stratification of this action up to homotopy. In the case of the nontrivial
bundle $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, the analysis of the
action of the centralizer on the stratification will be postponed to Section
6.

####  3.5.1 Notation

We shall use the following notation in the rest of the document. Let $M$
denote the manifolds $S^{2}\times S^{2}$ or
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$. Let $G$ be a compact abelian
group acting symplectically on $(M,\omega_{\lambda})$. Let $p_{0}$ be a fixed
point for the group action. Given a $G$ invariant symplectic curve C, and a
$\omega_{\lambda}$ orthogonal $G$ invariant sphere $\overline{F}$ in the
homology class $F$ that intersects C at a point $p_{0}$, we define the
following spaces:

  * •

$N(C)$:= The symplectic normal bundle to a symplectic submanifold $C$.

  * •

$\operatorname{Symp}^{G}_{h}(M,\omega_{\lambda})$ := The group of $G$
equivariant symplectomorphisms on $(M,\omega_{\lambda})$ that acts trivially
on homology.

  * •

$\operatorname{Stab}^{G}(C)$ := The group of all
$\phi\in\operatorname{Symp}^{G}_{h}(M,\omega_{\lambda})$ such that
$\phi(C)=C$, that is, such that $\phi$ _stabilises_ $C$ but does not
necessarily act as the identity on $C$.

  * •

$\operatorname{Fix}^{G}(C)$ := The group of all
$\phi\in\operatorname{Symp}^{G}_{h}(M,\omega_{\lambda})$ such that
$\phi|_{C}=id$, that is, such that _fixes $C$ pointwise_.

  * •

$\operatorname{Fix}^{G}(N(C))$:= The group of all
$\phi\in\operatorname{Symp}^{G}_{h}(M,\omega_{\lambda})$ such that
$\phi|_{C}=\operatorname{id}$ and $d\phi|_{N(C)}:N(C)\to N(C)$ is the identity
on $N(C)$.

  * •

$\operatorname{Gauge}1^{G}(N(C))$:= The group of $G$-equivariant symplectic
gauge automorphisms of the symplectic normal bundle of $C$.

  * •

$\operatorname{Gauge}1^{G}(N(C\vee\overline{F}))$ := The group of
$G$-equivariant symplectic gauge automorphism of the symplectic normal bundle
of the crossing divisor $C\vee\overline{F}$ that are identity in a
neighbourhood of the wedge point.

  * •

$\mathcal{S}^{G}_{K}$ := The space of unparametrized $G$-invariant symplectic
embedded spheres in the homology class $K$.

  * •

$\mathcal{S}^{G}_{K,p_{0}}$:= The space of unparametrized $G$-invariant
symplectic embedded spheres in the homology class $K$ passing through $p_{0}$.

  * •

$\mathcal{J}_{\omega_{\lambda}}^{G}(C)$ := The space of $G$-equivariant
$\omega_{\lambda}$ compatible almost complex structures s.t the curve $C$ is
holomorphic.

  * •

$\operatorname{Symp}^{G}(C)$:= The space of all $G$-equivariant
symplectomorphisms of the curve C.

  * •

$\operatorname{Fix}^{G}(N(C\vee{\overline{F}}))$ := The space of all
$G$-equivariant symplectomorphisms that are the identity in the neighbourhood
of $C\vee{\overline{F}}$.

  * •

$\operatorname{Symp}^{S^{1}}({\overline{F}},N(p_{0}))$ := equivariant
symplectomorphism of the sphere $\overline{F}$ that are the identity in an
open set of $\overline{F}$ around $p_{0}$.

  * •

$\overline{\mathcal{S}^{G}_{F,p_{0}}}$:= The space of unparametrized
$G$-invariant symplectic spheres in the homology class $F$ that are equal to a
fixed curve ${\overline{F}}$ in a neighbourhood of $p_{0}$.

  * •

$\operatorname{Symp}^{G}_{p_{0},h}(M,\omega_{\lambda})$:= The group of all
$\phi\in\operatorname{Symp}^{G}_{h}(M,\omega_{\lambda})$ fixing $p_{0}$.

  * •

$\operatorname{Stab}^{G}_{p_{0}}(C)$:= The group of all
$\phi\in\operatorname{Stab}^{G}(C)$ such that $\phi(p_{0})=p_{0}$.

All the above spaces are equipped with the $C^{\infty}$ topology.

####  3.5.2 Case 1: $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$ action on $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{2s}$ with $s\neq 0$

Let $\lambda>1$ and consider a $S^{1}$-action $S^{1}(a,b;m)$ on $(S^{2}\times
S^{2},\omega_{\lambda})$ with $m=2k$. Let ${\mathcal{S}^{S^{1}}_{D_{2s}}}$
denote the space of all $S^{1}$ invariant symplectic embedded spheres in the
class $D_{2s}=B-sF$. We shall assume that ${\mathcal{S}^{S^{1}}_{D_{2s}}}$ is
non-empty which, by Theorems 3.24 and 3.25, means that $2s=m$ or $2s=|2b\pm
m|$ depending on $a$.

We first show that ${\mathcal{S}^{S^{1}}_{D_{2s}}}$ is a homogeneous space
under the natural action of $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$. To this end, we observe that among the two fixed
points on the curve in class $D_{2s}$, Table 2.1 shows that at least one fixed
point, say $p$, has weights $(w_{1},w_{2})$ with $w_{1}\neq w_{2}$. It follows
that if an invariant curve in the class of the fiber $F$ intersects a curve in
the class $D_{2s}$ at $p$, then the two curves intersect
$\omega_{\lambda}$-orthogonally. Let $\mathcal{C}(D_{2s}\vee F,p)^{S^{1}}$ be
the space of invariant configurations made of curves in classes $D_{2s}$ and
$F$ intersecting orthogonally at $p$.

######  Lemma 3.31.

The evaluation map at a standard configuration
$\overline{D}_{2s}\vee\overline{F}$ through $p$

$\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})\twoheadrightarrow\mathcal{C}(D_{2s}\vee F,p)^{S^{1}}$

is a serre fibration.

###### Proof.

We first show that the action is transitive. Given an invariant configuration
$C\vee A\in\overline{D}_{2s}\vee\overline{F}$, the equivariant symplectic
neighbourhood theorem implies that we can find an invariant neighbourhood V of
$C\vee A$, an invariant neighbourhood $V^{\prime}$ of
$\overline{D}\cup\overline{F}$, and an equivariant symplectomorphism
$\alpha:V\to V^{\prime}$. We claim that $\alpha$ can be extended to an ambient
diffeomorphism $\beta$ of $S^{2}\times S^{2}$. Assume this for the moment. By
construction, the pullback form $\omega_{\beta}:=\beta^{*}\omega_{\lambda}$ is
invariant under the the conjugate action $\beta^{-1}\rho\beta$.

We observe that the complement of the standard configuration
$\overline{D}\cup\overline{F}$ in $W_{2s}$ is symplectomorphic to
$\mathbb{C}^{2}$ with the symplectic form
$\omega_{f}:=\frac{1}{2\pi}\partial\bar{\partial}f$ where
$f=\log\left(\left(1+||w||^{2}\right)^{\lambda}\left(1+||w||^{4k}+||z||^{2}\right)\right)$,
see [Abr98, Lemma 3.5]. Under this identification, the standard
$\mathbb{T}^{2}_{2s}$ action on $W_{2s}$ become linear on $\mathbb{C}^{2}$. It
follows that near infinity, the form $\omega_{\beta}$ is equal to $\omega_{f}$
and that the action $\beta^{-1}\rho\beta$ is linear. By Proposition 3.27 we
get that there is an equivariant symplectomorphism $\gamma$ that is equal to
the identity near infinity, and that identifies
$(\mathbb{C}^{2},\omega_{f},\rho)$ with
$(\mathbb{C}^{2},\beta^{*}\omega_{\lambda},\beta^{-1}\rho\beta)$. By
construction, the equivariant symplectomorphism $\phi:=\gamma\circ\beta$ takes
the configuration $C\vee A$ to $\overline{D}\vee\overline{F}$.

It remains to see that the local diffeomorphism $\alpha$ can be extended to an
ambient diffeomorphism $\beta$ of $S^{2}\times S^{2}$. By the Isotopy
Extension Theorem (see [Hir94, Theorem 1.4, p.180]), it suffices to show that
any two configurations of embedded spheres in classes $\overline{D}\cup F$
intersecting transversely and positively are isotopic. In turns, this follows
from the fact that any two $F$-foliations corresponding to two almost-complex
structures $J$ and $J^{\prime}$ are diffeotopic, and that any two sections of
the product $S^{2}\times S^{2}$ are diffeotopic through sections iff they are
homotopic. This shows that $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$ acts transitively on $\mathcal{C}(D_{2s}\vee
F,p)^{S^{1}}$.

To prove the homotopy lifting property, consider any family of maps
$\gamma:D^{n}\times[0,1]\rightarrow\mathcal{C}(D_{2s}\vee F,p)^{S^{1}}$ from a
$n$ dimensional disk $D^{n}$ to $\mathcal{C}(D_{2s}\vee F,p)^{S^{1}}$, and
choose a lift
$\overline{\gamma_{0}}:D^{n}\rightarrow\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$ of $\gamma_{0}$. Since the complement of a
configuration is contractible, the equivariant version of Banyaga’s Extension
Theorem for families implies that there exists a lift
$\overline{\gamma}:D^{n}\times[0,1]\rightarrow\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$ extending $\overline{\gamma_{0}}$.

${D^{n}\times\\{0\\}}$${\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})}$${D^{n}\times[0,1]}$${\mathcal{C}(D_{2s}\vee
F,p)^{S^{1}}}$$\scriptstyle{\overline{\gamma_{0}}}$$\scriptstyle{\theta}$$\scriptstyle{\gamma}$$\scriptstyle{\exists~{}\overline{\gamma}}$

Alternatively, one can apply the equivariant Gromov-Auroux Lemma C.4 to show
the existence of the lift $\overline{\gamma}$. In both cases, this concludes
the proof. ∎

######  Corollary 3.32.

Fix the action $S^{1}(a,b;m)$. Then
$\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times S^{2},\omega_{\lambda})$ acts
transitively on ${\mathcal{S}^{S^{1}}_{D_{2s}}}$.∎

######  Lemma 3.33.

Let $\overline{D}$ be an invariant symplectic sphere in the class $B-sF$ for
which $\mathcal{S}^{S^{1}}_{D_{2s}}$ is nonempty. Then the evaluation map

$\displaystyle\theta:\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$
$\displaystyle\twoheadrightarrow{\mathcal{S}^{S^{1}}_{D_{2s}}}$
$\displaystyle\phi$ $\displaystyle\mapsto\phi(\overline{D})$

is a Serre fibration with fibre over $\overline{D}$ given by

$\operatorname{Stab}(\overline{D}):=\left\\{\phi\in\operatorname{Symp}^{S^{1}}(S^{2}\times
S^{2},\omega_{\lambda})~{}|~{}\phi(\overline{D})=\overline{D}\right\\}$

###### Proof.

The evaluation map is transitive by Corollary 3.32. The homotopy lifting
property follows from Lemma C.4 as in the proof of Lemma 3.31. Alternatively,
one can also note that the action map factors through the restriction map

$\mathcal{C}(D_{2s}\vee F,p)^{S^{1}}\to\mathcal{S}^{S^{1}}_{D_{2s}}$

which is itself a fibration. To see this, note that the restriction maps fits
into a commuting diagram

${\mathcal{J}^{S^{1}}_{\omega_{\lambda}}}$${\mathcal{C}(D_{2s}\vee
F,p)^{S^{1}}}$${\mathcal{S}^{S^{1}}_{D_{2s}}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$

where the maps $f_{1}$ and $f_{2}$ are fibrations. Observe that the map
$f_{1}$ is well defined because the weights at the chosen fixed point $p$ are
not equal. Hence, for any choice of invariant almost complex structure
$J\in\mathcal{J}^{S^{1}}_{\omega_{\lambda}}$, the unique invariant
$J$-holomorphic curve $C$ in class $D_{2s}$ intersects the unique invariant
$J$-holomorphic fiber through $\omega_{\lambda}$-orthogonally at $p$. ∎

######  Remark 3.34.

As both $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times S^{2},\omega_{\lambda})$
and $\mathcal{S}^{S^{1}}_{D_{2s}}$ can be shown to be CW-complexes, we see
from Theorem 1 in [SW84] (with proof corrected in [Cau92]), that a Serre
fibration in which the total space and base space are both CW complexes is
necessarily a Hurewicz fibration. Thus the map
$\theta:\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})\twoheadrightarrow{\mathcal{S}^{S^{1}}_{D_{2s}}}$ is in
fact a Hurewicz fibration and hence the fibre over any arbitrary
$D\in\mathcal{S}^{S^{1}}_{D_{2s}}$ is homotopy equivalent to
$\operatorname{Stab}(\overline{D})$.

The homotopy type of $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$ is related to the strata
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{2s}$ through the following
sequence of fibrations. We use the symbol “$\simeq$” to mean “weakly homotopy
equivalent” throughout the rest of the document. In the fibrations below, we
use the notation established in Section 3.5.1.

$\operatorname{Stab}^{S^{1}}(\overline{D})\longrightarrow\operatorname{Symp}^{S^{1}}_{h,p_{0}}(S^{2}\times
S^{2},\omega_{\lambda})\relbar\joinrel\twoheadrightarrow{\mathcal{S}^{S^{1}}_{D_{2s}}}\mathbin{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xrightarrow{\text{~{}~{}~{}$\simeq$~{}~{}~{}}}}}\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{2s}\rule{0.0pt}{20.00003pt}$

$\operatorname{Fix}^{S^{1}}(\overline{D})\longrightarrow\operatorname{Stab}^{S^{1}}(\overline{D})\relbar\joinrel\twoheadrightarrow\operatorname{Symp}^{S^{1}}(\overline{D})\mathbin{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xrightarrow{\text{~{}~{}~{}$\simeq$~{}~{}~{}}}}}S^{1}~{}\text{or}~{}\operatorname{SO}(3)\rule{0.0pt}{20.00003pt}$

$\operatorname{Fix}^{S^{1}}(N(\overline{D}))\longrightarrow\operatorname{Fix}^{S^{1}}(\overline{D})\relbar\joinrel\twoheadrightarrow\operatorname{Gauge}1^{S^{1}}(N(\overline{D}))\mathbin{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xrightarrow{\text{~{}~{}~{}$\simeq$~{}~{}~{}}}}}S^{1}\rule{0.0pt}{20.00003pt}$

$\operatorname{Stab}^{S^{1}}(\overline{F})\cap\operatorname{Fix}^{S^{1}}(N(\overline{D}))\longrightarrow\operatorname{Fix}^{S^{1}}(N(\overline{D}))\relbar\joinrel\twoheadrightarrow\overline{\mathcal{S}^{S^{1}}_{F,p_{0}}}\mathbin{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xrightarrow{\text{~{}~{}~{}$\simeq$~{}~{}~{}}}}}\mathcal{J}^{S^{1}}(\overline{D})\simeq\\{*\\}\rule{0.0pt}{20.00003pt}$

$\operatorname{Fix}^{S^{1}}(\overline{F})\longrightarrow\operatorname{Stab}^{S^{1}}(\overline{F})\cap\operatorname{Fix}^{S^{1}}(N(\overline{D}))\relbar\joinrel\twoheadrightarrow\operatorname{Symp}^{S^{1}}(\overline{F},N(p_{0}))\mathbin{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xrightarrow{\text{~{}~{}~{}$\simeq$~{}~{}~{}}}}}\left\\{*\right\\}\rule{0.0pt}{20.00003pt}$

$\left\\{*\right\\}\mathbin{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xleftarrow{\text{~{}~{}~{}$\simeq$~{}~{}~{}}}}}\operatorname{Fix}^{S^{1}}(N(\overline{D}\vee\overline{F}))\longrightarrow\operatorname{Fix}^{S^{1}}(\overline{F})\relbar\joinrel\twoheadrightarrow\operatorname{Gauge}1^{S^{1}}(N(\overline{D}\vee\overline{F}))\mathbin{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\xrightarrow{\text{~{}~{}~{}$\simeq$~{}~{}~{}}}}}\left\\{*\right\\}\rule{0.0pt}{20.00003pt}$

Here $\overline{F}$ and $\overline{D}$ are the $\omega_{\lambda}$-orthogonally
intersecting invariant curves in the $2s^{\,\text{th}}$-Hirzebruch surface
$W_{2s}$ and whose moment map images are depicted below in red. We denote by
$\overline{F}$ the unique curve in class $F$ intersecting the given curve
$\overline{D}\in{\mathcal{S}^{S^{1}}_{D_{2s}}}$
$\omega_{\lambda}$-orthogonally at $p_{0}$. In the second fibration, the group
$\operatorname{Symp}^{S^{1}}(\overline{D})$ is homotopy equivalent to
$\operatorname{SO}(3)$ when the $S^{1}$ action fixes the curve $\overline{D}$
pointwise. Otherwise, it is homotopy equivalent to $S^{1}$.

$\overline{D}$$\overline{F}$$p_{0}$ Figure 3.1: The isolated fixed point
$p_{0}$

Assuming the homotopy equivalence in the first fibration, we immediately get

######  Theorem 3.35.

Consider the $S^{1}(a,b;m)$ action on $(S^{2}\times S^{2},\omega_{\lambda})$
with $\lambda>1$. If $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{2s}\neq\phi$, then $\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})/\operatorname{Stab}^{S^{1}}(\overline{D})\simeq\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{2s}$.

Furthermore, tracking down the various homotopy equivalences in the other
fibrations, we will prove that the equivariant stabilizer of the curve
$\overline{D}$, namely $\operatorname{Stab}^{S^{1}}(\overline{D})$, is
homotopy equivalent to the equivariant stabilizer of the corresponding complex
structure under the natural action of
$\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times S^{2},\omega_{\lambda})$. More
precisely,

  * •

$\operatorname{Stab}^{S^{1}}(\overline{D})\simeq\mathbb{T}^{2}_{2s}$ when
$(a,b)\neq(0,\pm 1)$;

  * •

$\operatorname{Stab}^{S^{1}}(\overline{D})\simeq SO(3)\times S^{1}$ when
$(a,b)=(0,\pm 1)$.

We shall now justify each of the homotopy equivalences in the above
fibrations.

Now that we know that the action map

$\displaystyle\operatorname{Stab}^{S^{1}}(\overline{D})\to\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$
$\displaystyle\twoheadrightarrow{\mathcal{S}^{S^{1}}_{D_{2s}}}$

is a fibration, we show that ${\mathcal{S}^{S^{1}}_{D_{2s}}}$ is weakly
homotopic to $\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{2s}$.

######  Lemma 3.36.

The natural map $\alpha:\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{2s}\to{\mathcal{S}^{S^{1}}_{D_{2s}}}$ defined by sending an almost complex
structure $J\in\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{2s}$ to the
unique $J$-holomorphic curve in class $D_{2s}$ is a weak homotopic
equivalence.

###### Proof.

To show that $\alpha$ is a weak homotopy equivalence, we first show that the
map is Serre fibration. To do so, consider an arbitrary element
${D}\in{\mathcal{S}^{S^{1}}_{D_{2s}}}$. As in the proof of Lemma 3.33, it
suffices to show that given a family of map $\gamma_{t}$ from a n-dimensional
disk $D^{n}$, such that $\gamma_{0}(0)={D}$, and a lift
$g_{0}^{\prime}:D^{n}:\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{2s}$
lifting $\gamma_{0}$, then there exists a lift $\gamma_{t}^{\prime}$ to
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap U_{2s}$.

${D^{n}\times\\{0\\}}$${\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{2s}}$${D^{n}\times[0,1]}$${\mathcal{S}^{S^{1}}_{D_{2s}}}$$\scriptstyle{\gamma_{0}^{\prime}}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma}$$\scriptstyle{\exists~{}\gamma^{\prime}}$

As in the proof of Lemma 3.33, we have that there exists a lift
$\overline{\gamma}:D^{n}\times[0,1]\to\operatorname{Symp}^{S^{1}}_{h}(S^{2}\times
S^{2},\omega_{\lambda})$ of $\gamma$. Pick an element $J\in\alpha^{-1}({D})$
and define $\gamma^{\prime}(s):=\overline{\gamma}^{*}J$. This defines a lift
$\gamma^{\prime}$ of $\gamma$. Hence $\alpha$ is a fibration, with fibre begin
contractible. Thus we get the required result that
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}\cap
U_{2s}\simeq{\mathcal{S}^{S^{1}}_{D_{2s}}}$. ∎

######  Lemma 3.37.

The restriction map
$\operatorname{Stab}^{S^{1}}(\overline{D})\twoheadrightarrow\operatorname{Symp}^{S^{1}}(\overline{D})$
is a fibration.

###### Proof.

To show that the restriction map is a fibration we use Theorem C.5, in which
we set $X=\operatorname{Symp}^{S^{1}}(\overline{D})$,
$G=\operatorname{Stab}^{S^{1}}(\overline{D})$ and the action is given by

$\displaystyle G\times X$ $\displaystyle\to X$ $\displaystyle(\phi,\psi)$
$\displaystyle\to\phi|_{\overline{D}}\circ\psi$

Hence in order to show that the restriction map
$r:\operatorname{Stab}^{S^{1}}(\overline{D})\to\operatorname{Symp}^{S^{1}}(\overline{D})$
is a fibration, we only need to show that the action described above admits
local cross sections. Suppose we only show that a neighbourhood of identity
admits local cross sections and that
$\operatorname{Stab}^{S^{1}}(\overline{D})$ acts transitively on
$\operatorname{Symp}^{S^{1}}(\overline{D})$ this would suffice to show that
$r$ is a fibration as by Theorem C.5, its a local fibration near the identity
and the map $r$ is equivariant with respect to the action of
$\operatorname{Stab}^{S^{1}}(\overline{D})$, thus completing the proof.

Consider the identity
$\operatorname{id}\in\operatorname{Symp}^{S^{1}}(\overline{D})$. As Let
$\alpha:N(\overline{D})\to U$ be a equivariant diffeomorphism between the
symplectic normal bundle $N(\overline{D})$ and a neighbourhood $U$ of
$\overline{D}$. As $\operatorname{Symp}^{S^{1}}(\overline{D})$ is locally
contractible (this can be seen for example by noticing that the proof of Prop
3.3.14 in [MS17] can be made equivariant) we can find a neighbourhood V of
$\operatorname{id}$, and a fixed retraction $\beta_{t}$ of the neighbourhood
$V$ onto the identity. Hence given any
$\psi\in\operatorname{Symp}^{S^{1}}(\overline{D})$, we get a one parameter
family $\beta_{t}(\psi)$ of symplectomorphisms. As $\pi_{1}(\overline{D})=0$,
$\beta_{t}(\psi)$ is Hamiltonian and is generated by a function $H_{t}$. Let
$\pi:N(\overline{D})\to\overline{D}$ be the projection of the normal bundle.
Define $\tilde{H_{t}}:=\alpha\circ\pi^{*}H_{t}$. Thus $\tilde{H_{t}}$ defines
an invariant function on a U. Fix an invariant bump function $\rho$ with
support in $U$ and is 1 in a small neighbourhood around $\overline{D}$, then
$\rho\tilde{H_{t}}$ is an invariant function and the corresponding
symplectomorphism it generates $\tilde{\psi}$ belong to
$\operatorname{Stab}^{S^{1}}(\overline{D})$ and extends $\psi$. Note that if
we fix the neighbourhood $U$, the bump function and the retraction of the
neighbourhood in $\operatorname{Symp}^{S^{1}}(\overline{D})$ then this
procedure gives us a lift of $\psi$ near id in
$\operatorname{Symp}^{S^{1}}(\overline{D})$ to
$\operatorname{Stab}^{S^{1}}(\overline{D})$. By Theorem C.5 this shows $r$ is
fibration. ∎

The above proof also shows that the action is transitive. As given
$\gamma\in\operatorname{Symp}^{S^{1}}(\overline{D})$ we give a procedure to
construct an element
$\tilde{\gamma}\in\operatorname{Stab}^{S^{1}}(\overline{D})$ such that the
restriction to $\overline{D}$ is $\gamma$, thus showing the action is
transitive.

######  Lemma 3.38.

$\operatorname{Symp}^{S^{1}}(\overline{D})$ is homotopic to
$\operatorname{SO}(3)$ for the circle action $S^{1}(0,\pm 1,m)$ and
$\operatorname{Symp}^{S^{1}}(\overline{D})$ is homotopic to $S^{1}$ for all
other circle actions.

###### Proof.

Consider the circle action induced on $\overline{D}$. The action $S^{1}(0,\pm
1,m)$ fixes $\overline{D}$ pointwise. Hence
$\operatorname{Symp}^{S^{1}}(\overline{D})=\operatorname{Symp}(\overline{D})$.
By Smale’s theorem we know that $\operatorname{Symp}(\overline{D})$ is
homotopy equivalent to $\operatorname{SO}(3)$.

The restriction for all other actions not equal to $S^{1}(0,\pm 1,m)$, do not
point wise fix the curve $\overline{D}$. For all other actions, we have the
following two subcases. Assume that the action is effective. Let
$\mu:\overline{D}\to\mathbb{R}$ be it’s moment map. Then as explained in the
proof of Proposition 2.7 we have that
$\operatorname{Symp}^{S^{1}}(\overline{D})\simeq
C^{\infty}(\mu(\overline{D}),S^{1})$, where
$C^{\infty}(\mu(\overline{D}),S^{1})$ denotes the space of smooth maps from
the image of the moment map to $S^{1}$. As the image of the moment map is an
interval, and as the space of smooth maps from an interval to $S^{1}$ is
homotopy equivalent to $S^{1}$, we have the required result that
$\operatorname{Symp}^{S^{1}}(\overline{D})\simeq S^{1}$.

Finally if the induced symplectic $S^{1}$ action on $\overline{D}$ is not
effective and has $\mathbb{Z}_{k}$ stabilizer, the action of
$S^{1}/\mathbb{Z}_{k}\cong S^{1}$, is effective and the space of
symplectomorphisms equivariant with respect to this quotient effective action
is the same as space of symplectomorphisms equivariant with respect to the
non-effective $S^{1}$ action. Thus the homotopy type of
$\operatorname{Symp}^{S^{1}}(\overline{D})\simeq S^{1}$. ∎

######  Lemma 3.39.

The map

$\displaystyle\alpha:\operatorname{Fix}^{S^{1}}(\overline{D})$
$\displaystyle\twoheadrightarrow\operatorname{Gauge}1^{S^{1}}(N(\overline{D}))$
$\displaystyle\phi$ $\displaystyle\mapsto d\phi|_{N(\overline{D})}$

is a Serre fibration with fibre homotopic to
$\operatorname{Fix}^{S^{1}}(N(\overline{D}))$. The base space
$\operatorname{Gauge}1^{S^{1}}(N(\overline{D}))$ is homotopy equivalent to
$S^{1}$.

###### Proof.

The fact that $\operatorname{Gauge}1^{S^{1}}(N(\overline{D}))\simeq S^{1}$ is
explained in Appendix A. Thus we only need to prove that the restriction of
the derivative indeed is a fibration and the fibre is homotopic to
$\operatorname{Fix}^{S^{1}}(N(\overline{D}))$.

Consider the action

$\displaystyle\operatorname{Fix}^{S^{1}}(\overline{D})\times\operatorname{Gauge}1^{S^{1}}(N(\overline{D}))$
$\displaystyle\to\operatorname{Gauge}1^{S^{1}}(N(\overline{D}))$
$\displaystyle(\phi,\psi)$ $\displaystyle\to d\phi|_{N(D)}\circ\psi$

Again by Theorem C.5 it suffices to show that the there is a local section to
above action. Such a local section is produced by Lemma C.1.

The fibre is apriori given by all equivariant symplectomorphisms that act as
identity on the normal bundle of $\overline{D}$. The claim is that this is in
fact homotopy equivalent to the space
$\operatorname{Fix}^{S^{1}}(N(\overline{D}))$. This follows from lemma C.3. ∎

Let $\overline{\mathcal{S}^{S^{1}}_{F,p_{0}}}$ be the space of unparametrized
$S^{1}$-invariant symplectic spheres in the homology class $F$ that are equal
to a fixed invariant curve ${\overline{F}}$ in a neighbourhood of $p_{0}$.

######  Lemma 3.40.

The map

$\displaystyle\operatorname{Fix}^{S^{1}}(N(\overline{D}))$
$\displaystyle\to\overline{\mathcal{S}^{S^{1}}_{F,p_{0}}}$
$\displaystyle\phi\mapsto\phi(\overline{F})$

is a fibration and
$\overline{\mathcal{S}^{S^{1}}_{F,p_{0}}}\simeq\mathcal{J}^{S^{1}}_{\omega_{\lambda}}(\overline{D})\simeq\\{*\\}$

###### Proof.

The proof for this is exactly the same as the proof of Corollary 3.32. We only
note that if $F^{\prime}\in\overline{\mathcal{S}^{S^{1}}_{F,p_{0}}}$ then the
map $\phi$ constructed in the proof of Corollary 3.32 belongs to
$\operatorname{Fix}^{S^{1}}(N(\overline{D}))$. Thus we have that
$\operatorname{Fix}^{S^{1}}(N(\overline{D}))$ acts transitively on
$\overline{\mathcal{S}^{S^{1}}_{F,p_{0}}}$, and thus the action map induces a
fibration.

Proof that
$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}(\overline{D})\simeq\\{*\\}$: This
follows from the equivariant version of the standard proof of considering the
homeomorphic space of equivariant compatible metrics and noting that this
space of metrics is contractible.

Proof that
$\overline{\mathcal{S}^{S^{1}}_{F,p_{0}}}\simeq\mathcal{J}^{S^{1}}_{\omega_{\lambda}}(\overline{D})\simeq\\{*\\}$:
Let $\mathcal{S}^{\perp}_{F,p_{0}}$ denote the space of all $S^{1}$ invariant
symplectically embedded spheres S in class F such that
$S\cap\overline{D}=p_{0}$ and $S$ and $\overline{D}$ intersect
$\omega_{\lambda}$-orthogonally at $p_{0}$. By Lemma B.1 we see that for every
$S\in\mathcal{S}^{\perp}_{F,p_{0}}$ there exists a
$J\in\mathcal{J}^{S^{1}}_{\omega_{\lambda}}(\overline{D})$ such that the
configuration $S\vee\overline{D}$ is $J$-holomorphic. We now have the
following fibration

$\mathcal{J}^{S^{1}}_{\omega_{\lambda}}(\overline{D})\relbar\joinrel\twoheadrightarrow\mathcal{S}^{\perp}_{F,p_{0}}$

where the map
$\gamma:\mathcal{J}^{S^{1}}_{\omega_{\lambda}}(\overline{D})\to\mathcal{S}^{\perp}_{F,p_{0}}$
is just sending $J\in\mathcal{J}^{S^{1}}_{\omega_{\lambda}}(\overline{D})$ to
the corresponding curve in class $F$ passing through $p_{0}$. Note that the
above map is well defined because the fixed $p_{0}$ was chosen such that the
weights at $p_{0}$ were distinct. Hence any $S^{1}$ invariant $F$ curve
passing through $p_{0}$ must be $\omega_{\lambda}$-orthogonal to
$\overline{D}$. Now we show that $\gamma$ is a homotopy equivalence. To do