# Toric one-skeletons for complexity-one spaces

Isabelle Charton

###### Abstract

A complexity-one space is a compact symplectic manifold $(M,\omega)$ endowed
with an effective Hamiltonian action of a torus $T$ of dimension
$\frac{1}{2}\dim(M)-1$. In this note we prove that for a certain class of
complexity-one spaces the Poincaré dual of the Chern class $c_{n-1}$ can be
represented by a collection of $\frac{n}{2}\chi(M)$ symplectic embedded
$2$-spheres, where $\chi(M)$ is the Euler characteristic of $M$ and
$\dim(M)=2n$. We call such a collection a toric one-skeleton. The
classification of complexity-one spaces is an important subject in symplectic
geometry. A nice subcategory of those spaces are the ones which are monotone.
The existence of a toric one-skeleton is a useful tool to understand six-
dimensional monotone complexity-one spaces. In particular, we will show that
the existence of a toric one-skeleton for such a space implies that the second
Betti number of $M$ is at most seven. This is a simple application of results
by Sabatini-Sepe and Lindsay-Panov.

###### Contents

  1. 1 Introduction
  2. 2 Basic properties of Hamiltonian $T$-spaces
    1. 2.1 The restriction to a subcircle
  3. 3 Equivariant Cohomology
  4. 4 The moment map polytope
    1. 4.1 Conventions for the fixed point data
    2. 4.2 The pre-toric one-skeleton
  5. 5 Hamiltonian $T$-spaces of dimension $4$
    1. 5.1 Reduced toric one-skeletons of fat edges
  6. 6 Proof of Theorem 1.4 and Corollary 1.6

##  1 Introduction

1112010 Mathematics Subjects Classification. 57R91, 57S25, 37J10
Keywords and phrase. Complexity-one spaces, Equivariant Cohomology

A symplectic action of a torus $T=(S^{1})^{d}$ on a symplectic manifold is
called Hamiltonian, if it admits a moment map $\phi\colon
M\rightarrow(\text{Lie}(T))^{*}$, i.e. $\phi$ is a smooth map such that

$\displaystyle\text{d}\left\langle\phi,\xi\right\rangle=-\iota_{X_{\xi}}\omega$
(1)

for all $\xi\in\text{Lie}(T)$, where $X_{\xi}$ is the vector field on $M$
generated by $\xi$. We call the quadruple $(M,\omega,T,\phi)$ a Hamiltonian
$T$-space, if the Hamiltonian action is effective and $M$ is compact. Since
the action is assumed to be effective and Hamiltonian, we have
$\dim(T)\leq\frac{1}{2}\dim(M)$. The non-negative integer
$k=\frac{1}{2}\dim(M)-\dim(T)$ is called the complexity of
$(M,\omega,T,\phi)$. We also call $(M,\omega,T,\phi)$ a complexity-k space. A
complexity-zero space is a symplectic toric manifold. The classification of
Hamiltonian $T$-spaces is an important subject in symplectic geometry. The
Convexity-Theorem - due to Atiyah [2] and Guillemin-Sternberg [7] \- states
that the image of the moment map $\phi(M)$ is a convex polytope. Moreover,
Delzant [4] proved that symplectic toric manifolds are classified by their
moment map image. Also the (equivariant) cohomology ring of a symplectic toric
manifold can be easily recovered from its moment map image. This is not true
for Hamiltonian $T$-spaces with complexity greater then zero. In this paper we
study properties of the second homology groups of complexity-one spaces. In
particular, a symplectic manifold $(M,\omega)$ always admits an almost complex
structure $J\colon TM\rightarrow TM$ which is compatible with the symplectic
form $\omega$, i.e. $\omega(\cdot,J\cdot)$ is a Riemannian metric on $M$.
Since the space of such structures is contractible, we can define complex
invariants of the tangent bundle $TM$, for instance Chern classes.
It is known that for a symplectic toric manifold of dimension $2n$ the
Poincaré dual of the Chern class $c_{n-1}(M)\in H^{2n-2}(M;\mathbb{Z})$ can be
realised by a collection $\mathcal{S}$ of $\frac{n}{2}\chi(M)$ symplectic
embedded surfaces, where $\chi(M)$ is the Euler Characteristic of $X$. Namely,
the preimage of $\phi^{-1}(e)$ of each edge $e$ of the moment map polytope
$\phi(M)$ is a smoothly embedded, symplectic $2$-sphere and $\mathcal{S}$ is
the collection of all these $2$-spheres. We call $\mathcal{S}$ the toric one-
skeleton of the corresponding symplectic toric manifold. In [5] Godinho,
Sabatini and von Heymann generalized the concept of toric one-skeletons for
Hamiltonian $T$-spaces with only isolated fixed points. Similarly, by the next
definition we generalize the concept of toric one-skeletons for all
Hamiltonian $T$-spaces.

######  Definition 1.1.

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space of dimension $2n$ and let
$\chi(M)$ be the Euler Characteristic of $M$. A toric one-skeleton
$\mathcal{S}$ of $(M,\omega,T,\phi)$ is a collection of $\frac{n}{2}\chi(M)$
smoothly embedded, symplectic surfaces, such that the Poincaré dual to their
union in $H_{2}(M;\mathbb{Z})$ is the Chern class $c_{n-1}(M)$, i.e.

$\displaystyle\int_{M}\mu\cdot
c_{n-1}(M)\quad=\quad\sum_{S\in\mathcal{S}}\left(\int_{S}\mu\right)$ (2)

for all $\mu\in H^{2}(M;\mathbb{Z})$.

This definition leads to the following question.

######  Question 1.2.

When does a Hamiltonian $T$-space admit a toric one-skeleton?

We investigate the following reasons into this question. An important
subcategory of Hamiltonian $T$-spaces are the ones which are monotone222A
symplectic manifold $(M,\omega)$ is monotone if its first Chern class $c_{1}$
is a multiple of the class $[\omega]$ in $H^{2}(M;\mathbb{R})$.. The
classification of monotone Hamiltonian $T$-spaces is not solved -even not in
dimension six-. By the work of Sabatini-Sepe [15] and Lindsay-Panov [12] we
have that the existence of a toric one-skeleton for a monotone complexity-one
space $(M,\omega,T,\phi)$ of dimension six implies that the second Betti
number of $M$ is at most $7$ (see Lemma 6.1). Hence, the existence of a toric
one-skeleton is a useful tool to understand six-dimensional monotone
complexity-one spaces.

In this notes we prove that a certain class of complexity-one spaces admit a
toric one-skeleton. Now we introduce the class of complexity-one spaces that
we will consider. Let $(M,\omega,T,\phi)$ be a complexity-one space. Consider
the moment map polytope $\phi(M)$ and let

$\displaystyle
e=\left(\mathbb{R}\left\langle\alpha_{e}\right\rangle+\beta_{e}\right)\cap\phi(M)\
$

be an edge of $\phi(M)$. Then $F_{e}=\phi^{-1}(e)$ is a smoothly embedded,
symplectic, $T$-invariant submanifold with stabilizer
$T_{e}=\text{exp}(\text{ker}(\alpha_{e}))$, a codimension-1 subtorus of $T$.
The $S^{1}\cong T/T_{e}$-action on $F_{e}$ is effective and Hamiltonian.
Moreover, $F_{e}$ has either dimension $2$ or $4$. We call $e$ a fat edge, if
the dimension of $F_{e}$ is $4$ (see Section 4).

######  Definition 1.3.

Let $(M,\omega,T,\phi)$ be a complexity-one space. We say $(M,\omega,T,\phi)$
has the Extension-Property $(\mathcal{P}_{E})$ if the following holds :
Let $F_{e}$ be a $4$-dimensional isotropic submanifold of $M$ which is the
preimage of a fat edge $e$ of the moment map polytope $\phi(M)$. Then the
$S^{1}=T/T_{e}$-action on $F_{e}$ extends to an effective Hamiltonian
$T^{2}$-action.

So the goal of this notes is to prove the following theorem.

######  Theorem 1.4.

A complexity-one space $(M,\omega,T,\phi)$ that satisfies the Extension-
Property $(\mathcal{P}_{E})$ admits a toric one-skeleton.

######  Remark 1.5.

Let $(M,\omega,T,\phi)$ be a complexity-one space. By Karshon’s classification
of $4$-dimensional Hamiltonian $S^{1}$-spaces [10], we have the following
facts:
$\bullet$ If all connected components of $M^{T}$ are isolated points, then
$(M,\omega,T,\phi)$ satisfies the Extension-Property $(\mathcal{P}_{E})$.
$\bullet$ If all connected components of $M^{T}$ are $2$-spheres, then
$(M,\omega,T,\phi)$ satisfies the Extension-Property $(\mathcal{P}_{E})$.
On the other side, we have that;
$\bullet$ If $(M,\omega,T,\phi)$ satisfies the Extension-Property
$(\mathcal{P}_{E})$, then a connected component of $M^{T}$ is an isolated
point or a $2$-sphere.

As a consequence of Theorem 1.4 we obtain the following corollary.

######  Corollary 1.6.

Let $(M,\omega,T,\phi)$ be a monotone complexity-one space of dimension six,
which satisfies the Extension-Property $(\mathcal{P}_{E})$. Then the second
Betti number of $M$ is at most 7.

Now we explain the idea behind the proof of Theorem 1.4.
Sketch of the Proof of Theorem 1.4
Our main tool to prove Theorem 1.4 is the ABBV Localization formula in
equivariant cohomology. In particular, we analyze the ABBV Localization
formula for Hamiltonian $S^{1}$-spaces, whose fixed submanifolds have at most
dimension $2$ (see Lemma 3.3 and Corollary 3.4).
In section 4 we define the pre-toric one-skeleton $\mathcal{S}_{pre}$ for
Hamiltonian $T$-spaces. This is a collection of $T$-invariant, smoothly
embedded, symplectic $2$-spheres whose stabilizers are codimensional-$1$
subtori of $T$. Moreover, let $F_{e}$ be an isotropic submanifold that belongs
to a fat edge of $\phi(M)$. The Extension-Property $(\mathcal{P}_{E})$ implies
that $F_{e}$ itself admits a ’special’ toric one-skeleton
$\mathcal{S}_{F_{e}}$. In particular $\mathcal{S}_{F_{e}}$ is a collection of
$T$-invariant $2$-spheres. We call

$\displaystyle\mathcal{\bar{S}}_{F_{e}}=\left\\{S\in\mathcal{S}_{F_{e}}\mid
S\text{ is not fixed by the $T$-action}\right\\}$

a reduced toric one-skeleton of $F_{e}$ (see Section 5). By $M_{2}^{T}$ we
denote the set of all $2$-dimensional components of $M^{T}$.
Finally, we will use the ABBV formula, in particular Corollary 3.4, to show
that the union

$\displaystyle\mathcal{S}=\mathcal{S}_{pre}\bigcup
M_{2}^{T}\bigcup\left(\bigcup_{e\text{ is a fat
edge}}\mathcal{\bar{S}}_{F_{e}}\right)$

is a toric one-skeleton for $(M,\omega,T,\phi)$.

Acknowledgements. This work is part of the SFB/TRR 191 ‘Symplectic Structure
in Geometry, Algebra and Dynamics‘, funded by the DFG. I would like to thank
Silvia Sabatini and Daniele Sepe for very helpful discussions and comments.

##  2 Basic properties of Hamiltonian $T$-spaces

Let $T$ be a torus. We denote its Lie algebra by $\text{Lie}(T)$ and its
lattice by

$\displaystyle\ell_{T}=\ker(\exp\colon\text{Lie}(T)\rightarrow T).$

Moreover, the dual Lie algebra of $T$ is $(\text{Lie}(T))^{*}$ and the dual
lattice is $\ell_{T}^{*}$.
For a Hamiltonian $T$-space $(M,\omega,T,\phi)$, we denote the set of
$T$-fixed points by $M^{T}$ and for any subgroup $H$ of $T$ we denote the set
of $H$-fixed points by $M^{H}$.

######  Lemma 2.1.

[13, Lemma 5.53] Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space and let
$H$ be a closed subgroup of $T$. Then any connected component of $M^{H}$ is a
closed symplectic submanifold of $(M,\omega)$.

So we call the connected components of $M^{T}$ the fixed submanifolds.
Moreover if $H$ is a subtours of $T$, then the quotient torus $T/H$ acts on
the connected components of $M^{H}$ in Hamiltonian fashion.

Next, we recall the local normal form near fixed points. Therefore, we fix a
$T$-invariant almost complex structure $J\colon TM\rightarrow TM$ which is
compatible with $\omega$. Hence, for each $p\in M^{T}$ we have an
$\mathbb{C}$-linear $T$-representation on $(T_{p}M,J_{p})\cong\mathbb{C}^{n}$.
The weights $\alpha_{p,1},\dots,\alpha_{p,n}\in\ell_{T}^{*}$ of this
representation are called the isotropic weights at $p$. The following theorem
by Guillemin-Sternberg [8] states that small neighborhoods of fixed points are
determined by their isotropic weights up to $T$-invariant symplectomorphims.

######  Theorem 2.2.

(local normal forms near fixed points) Let $(M,\omega,T,\phi)$ be a
Hamiltonian $T$-space of dimension $2n$. Let $p\in M^{T}$ be a fixed point
with weights $\alpha_{p,1},\dots,\alpha_{p,n}\in\ell_{T}^{*}$. Then there
exists a neighborhood $U_{p}$ of $p$ with complex coordinates
$z_{1},\dots,z_{n}$ centered at $p$ such that

$\displaystyle\bullet\text{the symplectic form is
}\omega=\frac{i}{2}\sum_{j=1}^{n}\text{d}z_{j}\wedge\text{d}\bar{z_{j}},$
$\displaystyle\bullet\text{the $T$-action is
}\exp(\xi)\cdot(z_{1},\dots,z_{n})=(\text{e}^{2\pi
i\alpha_{p,1}(\xi)}z_{1},\dots,\text{e}^{2\pi i\alpha_{p,n}(\xi)}z_{n})\text{
for }\xi\in\text{Lie}(T),$ $\displaystyle\bullet\text{the moment map is
}\phi(z)=\frac{1}{2}\sum_{j=1}^{n}\alpha_{p,j}|z_{j}|^{2}+\phi(p).$

Note that for each fixed submanifold $F$ and any $p,q\in F$ the isotropic
weights at $p$ and $q$ are equal. So we call these weights the isotropic
weights at $F$ and we denote them by $\alpha_{F,1},\dots\alpha_{F,n}$.
Moreover, the dimension of $F$ is equal to twice the number of weights
$\alpha_{F,1},\dots\alpha_{F,n}$ which are equal to zero.

######  Lemma 2.3.

Let $(M,\omega,T,\phi)$ be a complexity-$k$ space. Then the dimension of each
fixed submanifold is at most $2k$.

###### Proof.

Let dim($M$)=$2n$, so the complexity of $(M,\omega,T,\phi)$ is $k=n-d$, where
dim($T$)=$d$. Now let $F$ be a fixed submanifold and let
$\alpha_{F,1},\dots,\alpha_{F,n}$ be the isotropic weights at $F$. Since the
$T$-action on $M$ is effective, we have that the $\mathbb{Z}$-span of
$\alpha_{F,1},\dots,\alpha_{F,n}$ is equal to
$\ell_{T}^{*}\cong\mathbb{Z}^{d}$. Hence, at most $k$ of isotropic weights
$\alpha_{F,1},\dots,\alpha_{F,n}$ at $F$ are equal to zero. Since the
dimension of $F$ is equal to twice the number of isotropic weights at $F$ that
are zero, we conclude $dim(F)\leq 2k$. ∎

###  2.1 The restriction to a subcircle

Let $T$ be a real torus and let $\bar{\xi}\in\text{Lie}(T)$, such that
$S^{1}=\exp(\mathbb{R}\bar{\xi})$ is a subcircle of $T$. Let $\ell_{T}^{*}$ be
the dual lattice of $T$ and $\ell_{S^{1}}^{*}\cong\mathbb{Z}$ be the dual
lattice of $\text{Lie}(S^{1})\subset\text{Lie}(T)$. Note that we have a
natural restriction map

$\displaystyle\Phi\colon\ell_{T}^{*}\longrightarrow\ell_{S^{1}}^{*}\cong\mathbb{Z},$
(3)

which is induced by the inclusion $\text{Lie}(S^{1})\rightarrow\text{Lie}(T)$.
In particular, let $V$ be a complex vector space of complex dimension $n$,
such that $T$ acts on $V$ by $\text{GL}(V)$ with weights
$\alpha_{1},\dots,\alpha_{n}\in\ell_{T}^{*}$. Then $S^{1}$ acts on $V$ with
weights $\Phi(\alpha_{1}),\dots,\Phi(\alpha_{n})\in\ell_{S^{1}}^{*}$.

######  Definition 2.4.

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space. We call
$\bar{\xi}\in\text{Lie}(T)$ generic, if $S^{1}=\exp(\mathbb{R}\bar{\xi})$ is a
subcircle of $T$ and $M^{S^{1}}=M^{T}$.

######  Remark 2.5.

Since we assume a Hamiltonian $T$-space to be compact, we have that for any
Hamiltonian $T$-space there is a generic $\bar{\xi}\in\text{Lie}(T)$.

##  3 Equivariant Cohomology

In this section we review some facts about $S^{1}$-equivariant cohomology.
(For a detailed introduction, see for example [1] and [9]). Moreover, we
analyze the ABBV Localization formula for Hamiltonian $S^{1}$-spaces, whose
fixed submanifolds have at most dimension $2$, which is the key ingredient for
the proof of Theorem 1.4.

Let $M$ be a manifold endowed with an $S^{1}$-action. In the Borel-model the
$S^{1}$-equivariant cohomology of $M$ is defined as follows. The diagonal
action of $S^{1}$ on $M\times S^{\infty}$ is free. By
$M\times_{S^{1}}S^{\infty}$ we denote the orbit space. The $S^{1}$-equivariant
cohomology ring of $M$ is

$\displaystyle H_{S^{1}}^{*}(M;R)\colon=H^{*}(M\times_{S^{1}}S^{\infty};R),$

where $R$ is the coefficient ring.
Let $M^{S^{1}}$ be the set of fixed points and assume it is not empty. Let $F$
be one of its connected components. The inclusion map $i_{F}\colon
F\rightarrow M$ is an $S^{1}$-equivariant map, so it induces a map

$\displaystyle i_{F}^{*}\colon H_{S^{1}}^{*}(M)\rightarrow H_{S^{1}}^{*}(F).$

Given $\mu^{S^{1}}\in H_{S^{1}}^{*}(M)$, we write
$\mu^{S^{1}}|_{F}=i_{F}^{*}(\mu^{S^{1}})$. Moreover if $P\in M^{S^{1}}$ is a
point, then $\mu^{S^{1}}(P)=i_{\left\\{P\right\\}}^{*}(\mu^{S^{1}})$. Note
that

$\displaystyle H_{S^{1}}^{*}(F)=H^{*}(F)\otimes
H^{*}(\mathbb{C}P^{\infty})=H^{*}(F)\left[x\right],$ (4)

where $x$ has degree $2$.

######  Remark 3.1.

Let $P,Q\in F$, where $F$ is a connected component of $M^{S^{1}}$, then
$\mu^{S^{1}}(P)=\mu^{S^{1}}(Q)$. In particular, let

$\displaystyle\pi_{F}\colon H_{S^{1}}^{*}(F)=H^{*}(F)\otimes
H^{*}(\mathbb{C}P^{\infty})\longrightarrow H^{*}(\mathbb{C}P^{\infty})$ (5)

be the natural projection, then $\pi_{F}(\mu^{S^{1}}|_{F})=\mu^{S^{1}}(P)$ for
all $P\in F$.

Moreover, the projection $\ M\times_{S^{1}}S^{\infty}\
\rightarrow\mathbb{C}P^{\infty}$ induces a push-forward map in equivariant
cohomology

$\displaystyle H_{S^{1}}^{*}(M)\rightarrow
H^{*-\operatorname{dim}(M)}(\mathbb{C}P^{\infty}),$ (6)

which can be seen as integration along the fibers. So we denote it by
$\int_{M}$. The following theorem, due to Atiyah-Bott and Berline-Vergne (see
[1], [3]) gives a formula for the map $\int_{M}$ in terms of fixed point set
data.

######  Theorem 3.2.

(ABBV Localization formula) Let $M$ be a compact oriented manifold endowed
with a smooth $S^{1}$-action. Given $\mu^{S^{1}}\in
H_{S^{1}}^{*}(M;\mathbb{Q})$

$\displaystyle\int_{M}\mu^{S^{1}}\,=\,\sum_{F\subset
M^{S^{1}}}\int_{F}\dfrac{\mu^{S^{1}}|_{F}}{\text{e}^{S^{1}}(N_{F})},$

where the sum runs over all connected components $F$ of $M^{S^{1}}$ and
$\text{e}^{S^{1}}(N_{F})$ is the equivariant Euler class of the normal bundle
$N_{F}$ to $F$.

In particular, if $F=\left\\{P\right\\}$ is an isolated fixed point, then

$\displaystyle\int_{F}\dfrac{\mu^{S^{1}}|_{F}}{\text{e}^{S^{1}}(N_{F})}=\dfrac{\mu^{S^{1}}(p)}{(w_{P,1}\cdot\dots\cdot
w_{P,n})x^{n}},$ (7)

where $w_{P,1},\dots,w_{P,n}$ are the weights333Note that the signs of the
individual weights are not well-defined, but the sign of their product is.  of
the $S^{1}$-representation at $T_{P}M$. In the next lemma we analyze the term
$\int_{F}\frac{(\mu^{S^{1}}c_{n-1}^{S^{1}}(M))|_{F}}{\text{e}^{S^{1}}(N_{F})}$
for the case that $F=\Sigma$ is a fixed surface.

######  Lemma 3.3.

Let $(M,\omega,S^{1},\phi)$ be a Hamiltonian $S^{1}$-space of dimension $2n$
and let $\Sigma$ be a fixed surface. Let $w_{1},\dots,w_{n-1}$ be the weights
of $S^{1}$-action on the normal bundle $N_{\Sigma}$ of $\Sigma$. Then

$\displaystyle\int_{\Sigma}\dfrac{(\mu^{S^{1}}\cdot
c_{n-1}^{S^{1}}(M))|_{\Sigma}}{\text{e}^{S^{1}}(N_{\Sigma})}\quad=\quad\int_{\Sigma}\mu^{S^{1}}\quad+\quad
2(1-g_{\Sigma})\frac{\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma})}{x}{\sum_{i=1}^{n-1}\frac{1}{w_{i}}},$

for $\mu^{S^{1}}\in H_{S^{1}}^{2}(M;\mathbb{Z})$, where $g_{\Sigma}$ is the
genus of $\Sigma$.

###### Proof.

We have
$H_{S^{1}}^{*}(\Sigma;\mathbb{Z})=H^{*}(\Sigma;\mathbb{Z})\left[x\right]$. We
denote by $u_{\Sigma}$ the positive444with respect to the orientation of
$\Sigma$ induced by the symplectic form $\omega$ generator of
$H^{2}(\Sigma;\mathbb{Z})$.

First, we compute the total equivariant Chern class $c^{S^{1}}(N_{\Sigma})$ of
the normal bundle $N_{\Sigma}$ of $\Sigma$ in terms of $u_{\Sigma}$ and the
weights $w_{1},\dots,w_{n-1}$ . The normal bundle $N_{\Sigma}$ is an
$S^{1}$-equivariant complex vector bundle over $\Sigma$ of rank $n-1$. Due to
the fact that each complex vector bundle over a compact surface splits into
the direct sum of complex line bundles and that the $S^{1}$-action on $\Sigma$
is trivial, we have that $N_{\Sigma}$ splits into the direct sum of
$S^{1}$-equivariant line bundles $L_{1},\dots,L_{n-1}$, such that for all
$j=1,\dots,n-1$ the first equivariant Chern class $c_{1}^{S^{1}}(L_{j})$ of
$L_{j}$ is of the form

$\displaystyle c_{1}^{S^{1}}(L_{j})=w_{j}x+A_{j}u_{\Sigma}$ (8)

for some $A_{j}\in\mathbb{Z}$. Hence, the total $S^{1}$-equivariant Chern
class of $N_{\Sigma}$ is

$\displaystyle
c^{S^{1}}(N_{\Sigma})=\prod_{i=1}^{n-1}\left(1+c_{1}^{S^{1}}(L_{i})\right)=\prod_{i=1}^{n-1}\left(1+w_{i}x+A_{i}u_{\Sigma}\right).$
(9)

Note that $u_{\Sigma}\cdot u_{\Sigma}=0$ in
$H_{S^{1}}^{*}(\Sigma;\mathbb{Z})$, so the equivariant Euler class of
$N_{\Sigma}$ is

$\displaystyle
e^{S^{1}}(N_{\Sigma})=c_{n-1}^{S^{1}}(N_{\Sigma})=\prod_{i=1}^{n-1}\left(w_{i}x+A_{i}u_{\Sigma}\right)=\prod_{i=1}^{n-1}\left(w_{i}x\right)\left(1+\sum_{i=1}^{n-1}\dfrac{A_{i}}{w_{i}x}u_{\Sigma}\right)$
(10)

and the formal inverse of $e^{S^{1}}(N_{\Sigma})$ is

$\displaystyle\dfrac{1}{e^{S^{1}}(N_{\Sigma})}=\dfrac{1}{\prod_{i=1}^{n-1}\left(w_{i}x\right)}\left(1-\sum_{i=1}^{n-1}\dfrac{A_{i}}{w_{i}x}u_{\Sigma}\right).$
(11)

Next, we have

$\displaystyle
c_{n-2}^{S^{1}}(N_{\Sigma})=\sum_{j=1}^{n-1}\left(\prod_{i=1,i\neq
j}^{n-1}\left(w_{i}x+A_{i}u_{\Sigma}\right)\right)=\sum_{j=1}^{n-1}\left(\prod_{i=1,i\neq
j}^{n-1}\left(w_{i}x\right)\left(1+\sum_{i=1,i\neq
j}^{n-1}\dfrac{A_{i}}{w_{i}x}u_{\Sigma}\right)\right).$ (12)

Moreover, since the $S^{1}$-action on $T\Sigma$ is trivial, the total
equivariant Chern class of $T\Sigma$ is
$c^{S^{1}}(\Sigma)=1+2(1-g_{\Sigma})u_{\Sigma}$. Now we use
$c^{S^{1}}(M)|_{\Sigma}=c^{S^{1}}(\Sigma)c^{S^{1}}(N_{\Sigma})$, so we obtain

$\displaystyle
c_{n-1}^{S^{1}}(M)|_{\Sigma}=e^{S^{1}}(N_{\Sigma})+2(1-g_{\Sigma})\left(\sum_{j=1}^{n-1}\prod_{i=1,i\neq
j}^{n-1}\left(w_{i}\right)\right)x^{n-2}u_{\Sigma}$ (13)

and

$\displaystyle\dfrac{c_{n-1}^{S^{1}}(M)|_{\Sigma}}{e^{S^{1}}(N_{\Sigma})}=1+\quad
2(1-g_{\Sigma})\dfrac{u_{\Sigma}}{x}{\sum_{i=1}^{n-1}\frac{1}{w_{i}}}.$ (14)

Now let $\mu^{S^{1}}\in H_{S^{1}}^{*}(M,\mathbb{Z})$, then

$\displaystyle\mu^{S^{1}}|_{\Sigma}=\mu|_{\Sigma}+\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma}),$
(15)

where $\mu\in H^{2}(M;\mathbb{Z})$ is image of $\mu^{S^{1}}$ under the
restriction map $H_{S^{1}}^{2}(M;\mathbb{Z})\rightarrow H^{2}(M;\mathbb{Z})$.
We conclude that

$\displaystyle\dfrac{\mu^{S^{1}}\cdot
c_{n-1}^{S^{1}}(M)|_{\Sigma}}{e^{S^{1}}(N_{\Sigma})}\quad=\quad\mu|_{\Sigma}+\quad
2(1-g_{\Sigma})\dfrac{\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma})}{x}\cdot
u_{\Sigma}\left(\sum_{i=1}^{n-1}\dfrac{1}{w_{i}}\right)\quad+\quad\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma}).$
(16)

So the claim follows.

∎

As a consequence of the ABBV Localization formula and of Lemma 3.3, we obtain
the following corollary.

######  Corollary 3.4.

Let $(M,\omega,S^{1},\phi)$ be a Hamiltonian $S^{1}$-space of dimension $2n$,
such that $M^{S^{1}}$ consists just of fixed $2$-spheres
$\Sigma_{1},\dots,\Sigma_{N}$ and isolated fixed points $P_{1},\dots,P_{K}$.
For all $i=1,\dots,N$, let $w_{\Sigma_{i},1},\dots,w_{\Sigma_{i},n-1}$ be the
weights of the $S^{1}$-action on the normal bundle of $\Sigma_{i}$. Moreover,
for all $j=1,\dots,K$, let $w_{P_{j},1},\dots,w_{P_{j},n}$ be the weights of
the $S^{1}$-action on $T_{P_{j}}M$. Then

$\displaystyle\int_{M}\mu^{S^{1}}\cdot c_{n-1}^{S^{1}}(M)=$
$\displaystyle\sum_{i=1}^{N}\left(\int_{\Sigma_{i}}\mu^{S^{1}}+2\dfrac{\pi_{\Sigma_{i}}(\mu^{S^{1}}|_{\Sigma_{i}})}{x}\left(\frac{1}{w_{\Sigma_{i},1}}+\dots+\frac{1}{w_{\Sigma_{i},n-1}}\right)\right)$
(17)
$\displaystyle+\sum_{j=1}^{K}\left(\dfrac{\mu^{S^{1}}(P_{j})}{x}\left(\frac{1}{w_{P_{j},1}}+\dots+\frac{1}{w_{P_{j},n}}\right)\right)$
(18)

for $\mu^{S^{1}}\in H_{S^{1}}^{2}(M;\mathbb{Z})$

###### Proof.

For an isolated fixed point $P_{j}$, we have
$e^{S^{1}}(N_{P_{j}})=\left(\prod_{s=1}^{n}w_{P_{j},s}\right)x^{n}$ and

$\displaystyle\mu^{S^{1}}\cdot
c_{n-1}^{S^{1}}(M)|_{P_{j}}=\mu^{S^{1}}(P_{j})\left(\sum_{h=1}^{n}\left(\prod_{k=1,k\neq
h}^{n}w_{P_{j},k}\right)\right)x^{n-1}.$

Hence,

$\displaystyle\frac{\mu^{S^{1}}\cdot
c_{n-1}^{S^{1}}(M)|_{P_{j}}}{e^{S^{1}}(N_{P_{j}})}=\dfrac{\mu^{S^{1}}(P_{j})}{x}\left(\frac{1}{w_{P_{j},1}}+\dots+\frac{1}{w_{P_{j},n}}\right).$

So the claim follows from the ABBV formula and Lemma 3.3.

∎

##  4 The moment map polytope

The aim of this section is to introduce conventions for the fixed point data
of a Hamiltonian $T$-space and to define the pre-toric one-skeleton.
Therefore, we recall facts about the moment map polytope (i.e. the image of
the moment map) for a Hamiltonian $T$-space. One of most the important
theorems for Hamiltonian $T$-spaces is the famous Convexity-Theorem proved by
Atiyah [2] and Guillemin-Sternberg [7].

######  Theorem 4.1.

(Convexity-Theorem) Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space, then
the fibers of $\phi$ are connected and the image of the moment map $\phi(M)$
is a convex polytope. In particular, $\phi(M)$ is the convex hull of the
images of the fixed submanifolds.

Another important property of the moment map is stated in the following
theorem.

######  Theorem 4.2.

(Sjamaar [16]) Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space. As a map to
$\phi(M)$ the moment map is open.

The moment map polytope contains informations about the corresponding
Hamiltonian $T$-space. In particular, by the Convexity-Theorem we have that
the preimage of a vertex of $\phi(M)$ is a connected component of $M^{T}$.
Moreover, a face of the moment map polytope gives us a symplectic submanifold.
Namely, let $\mathcal{F}$ be a $d$-dimensional face of the polytope $\phi(M)$,
then $\mathcal{F}$ is of the form

$\displaystyle\mathcal{F}=\left(\mathcal{H}_{\mathcal{F}}+\left\\{p_{\mathcal{F}}\right\\}\right)\bigcap\phi(M),$

where $p_{\mathcal{F}}$ is a point in $\mathcal{F}$ and
$\mathcal{H}_{\mathcal{F}}$ is a linear subspace of $(\text{Lie}(T))^{*}$ of
dimension $d$. We call $\mathcal{H}_{\mathcal{F}}$ the defining subspace for
$\mathcal{F}$.

######  Proposition 4.3.

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space and let $\mathcal{F}$ be a
$d$-dimensional face of $\phi(M)$ with defining subspace
$\mathcal{H}_{\mathcal{F}}$. Then

$\displaystyle
T_{\mathcal{F}}=\exp\left(\ker\left(\mathcal{H}_{\mathcal{F}}\right)\right)$
(19)

is a subtorus of $T$ of codimension $d$ and the preimage
$\phi^{-1}(\mathcal{F})$ is a connected component of $M^{T_{\mathcal{F}}}$.
Moreover, $\phi^{-1}(\mathcal{F})$ is a symplectic submanifold of $(M,\omega)$
and the quotient torus $T/T_{\mathcal{F}}$ acts effectively and in Hamiltonian
fashion on $\phi^{-1}(\mathcal{F})$.

A prove of Proposition 4.3 can be found in [14, Chapter 3]. The next corollary
gives an upper bound for the dimension of $\phi^{-1}(\mathcal{F})$.

######  Corollary 4.4.

Let $(M,\omega,T,\phi)$ be a complexity-$k$ space and let $\mathcal{F}$ be a
$d$-dimensional face of $\phi(M)$ . Then $\phi^{-1}(\mathcal{F})$ is a
symplectic submanifold of dimension at most $2(k+d)$.

###### Proof.

Let $\mathcal{H}_{\mathcal{F}}$ be the defining subspace of $\mathcal{F}$. By
Proposition 4.3, we have that
$T_{\mathcal{F}}=\exp\left(\ker\left(\mathcal{H}_{\mathcal{F}}\right)\right)$
is a subtorus of $T$ of codimension $d$. Moreover, $T_{\mathcal{F}}$ acts on
$(M,\omega)$ in Hamiltonian fashion with complexity

$\displaystyle k_{\mathcal{F}}$
$\displaystyle=\dfrac{1}{2}\dim(M)-\dim(T_{\mathcal{F}})$ (20)
$\displaystyle=\dfrac{1}{2}\dim(M)-\left(\dim(T)-d\right)$ (21)
$\displaystyle=k+d.$ (22)

Moreover, by Proposition 4.3 we also have that $\phi^{-1}(\mathcal{F})$ is a
connected component of $M^{T_{\mathcal{F}}}$. And so Lemma 2.3 implies that

$\displaystyle\dim(\phi^{-1}(\mathcal{F}))\leq 2k_{\mathcal{F}}=2(k+d).$ (23)

∎

###  4.1 Conventions for the fixed point data

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space of dimension $2n$ and $p\in
M^{T}$ an isolated fixed point with isotropic weights
$\alpha_{p,1},\dots,\alpha_{p,n}\in\ell_{T}^{*}\setminus\left\\{0\right\\}$.
For $i=1,\dots,n\quad$ $T_{\alpha_{p,i}}=\exp(\ker(a_{p,i}))$ is a
codimension-$1$ subtorus of $T$. We denote by $S_{\alpha_{p,i}}$ the connected
component of $M^{T_{\alpha_{p,i}}}$ that contains $p$. So $S_{\alpha_{p,i}}$
is a symplectic and $T$-invariant submanifold.

######  Definition 4.5.

We call $S_{\alpha_{p,i}}$ the submanifold that belongs to the weight
$\alpha_{p,i}$. We say the weight $\alpha_{p,i}$ is light if the dimension of
$S_{\alpha_{p,i}}$ is $2$ (or equivalent $\alpha_{p,i}$ and $\alpha_{p,j}$ are
linearly independent for all $j\in\left\\{1,\dots,\hat{i},\dots,n\right\\}$ ).
Otherwise the weight $\alpha_{p,i}$ is called heavy.

Moreover, we classify the isolated fixed points of a Hamiltonian $T$-space
into two categories as follows.

######  Definition 4.6.

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space. We denote the set of
isolated fixed points by $M_{0}^{T}$. An isolated fixed point $p\in M_{0}^{T}$
is called GKM if the isotropic weights at $p$ are pairwise linearly
independent (i.e. all isotropic weights are light), otherwise $p$ is called
non-GKM. We denote by $M_{GKM}^{T}$ the set of all GKM-points.

######  Remark 4.7.

A Hamiltonian $T$-space is called GKM if for each codimension-$1$ subtorus $H$
of $T$ any connected component of $M^{H}$ has at most dimension $2$ -or
equivalent the fixed point set $M^{T}$ contains just isolated fixed points,
which are all GKM-. Hamiltonian GKM-spaces are a subcategory of GKM spaces,
which were introduced by Goresky-Kottwitz-MacPherson in [6].

######  Lemma 4.8.

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$ and let
$p\in M^{T}$ be an isolated fixed point which is not GKM. Then exactly two
weights of $p$ are heavy. Moreover, we have

$\displaystyle\phi(p)\text{ is a vertex of }\phi(M)\text{ and
}\phi^{-1}(\phi(p))=p\quad\quad\quad\text{or}$ $\displaystyle\phi(p)\text{ is
not a vertex of }\phi(M),\text{ but }\phi(p)\text{ lies on an edge of
}\phi(M).$

###### Proof.

Since the $T$-action is effective, we have that
$\mathbb{Z}-\text{span}\left\\{\alpha_{p,1},\dots,\alpha_{p,n}\right\\}=l^{*}(\cong\mathbb{Z}^{n-1})$.
So w.l.o.g we assume that the weights are ordered by the following convention:

$\displaystyle\bullet\mathbb{R}-\text{span}\left\\{\alpha_{p,1},\dots,\alpha_{p,n-1}\right\\}=(\text{Lie}(T))^{*}$
(24) $\displaystyle\bullet\alpha_{p,n}=\lambda\cdot\alpha_{p,n-1}\text{ for
some }\lambda\in\mathbb{Q}\setminus\left\\{0\right\\}$ (25)

Hence, the weights $\alpha_{p,1},\dots,\alpha_{p,n-2}$ are light and the
weights $\alpha_{p,n-1},\alpha_{p,n}$ are heavy. Moreover, Theorems 4.1, 4.2
and 4.3 imply that

$\displaystyle\bullet\text{If }\lambda>0,\text{ then }\phi(p)\text{ is a
vertex of }\phi(M)\text{ and }\phi^{-1}(\phi(p))=p.$
$\displaystyle\bullet\text{If }\lambda<0,\text{ then }\phi(p)\text{ is not a
vertex of }\phi(M),\text{ but }\phi(p)\text{ lies on an edge of }\phi(M).$

∎

######  Definition 4.9.

Let $(M,\omega,T,\phi)$ be a complexity-one space. An edge $e$ of the moment
map polytope is called fat if $\phi^{-1}(e)$ has dimension $4$. We denote by
$E_{\Phi(M)}^{fat}$ the set of fat edges of the moment map polytope $\Phi(M)$.

######  Lemma 4.10.

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$.
$(i)$ If $p\in M^{T}$ is a GKM isolated fixed point, then $\phi(p)$ does not
meet a fat edge of $\phi(M)$.
$(ii)$ If $p\in M^{T}$ is a non-GKM isolated fixed point, then $\phi(p)$ meets
precisely one fat edge of $\phi(M)$. In particular, let
$a_{p,n-1},\alpha_{p,n}$ be the heavy weights of $p$, then this fat edge is
given by

$\displaystyle\left(\phi(p)+\mathbb{R}\left\langle\alpha_{p,n-1}\right\rangle\right)\cap\phi(M)=\left(\phi(p)+\mathbb{R}\left\langle\alpha_{p,n}\right\rangle\right)\cap\phi(M)$

$(iii)$ If $\Sigma\subset M^{T}$ is a fixed surface. Then all $n-1$ edges of
$\phi(M)$ that are meeting at the vertex $\phi(\Sigma)$ are fat. Moreover,
these edges are given by

$\displaystyle\left(\phi(\Sigma)+\mathbb{R}\left\langle\alpha_{\Sigma,j}\right\rangle\right)\cap\phi(M)$

for $i=1,\dots,n-1$, where $\alpha_{\Sigma,1},\dots,\alpha_{\Sigma,n-1}$ are
the weights of the normal bundle of $\Sigma$.

###  4.2 The pre-toric one-skeleton

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space of dimension $2n$. Let
$p\in M^{T}$ be an isolated fixed point and let $\alpha$ be a light weight at
$p$. The submanifold $S_{\alpha}$ that belongs to $\alpha$ is a symplectic and
$T$-invariant surface. Moreover, $S_{\alpha}$ is fixed by the codimension-$1$
subtours $T_{\alpha}=\exp(\ker\alpha)$ and the action of the quotient circle
$T/T_{\alpha}$ on $S_{\alpha}$ is effective and Hamiltonian. Since each
Hamiltonian $S^{1}$-space of dimension $2$ is a $2$-sphere with exactly $2$
fixed points, we have that $S_{\alpha}$ is a $2$-sphere and it contains
precisely two points of $M^{T}$, namely $p$ and the other one is denoted by
$q$. Moreover, since $T$ acts on $TpS_{\alpha}$ with weight $\alpha$, we have
that $T$ acts on $TqS_{\alpha}$ with weight -$\alpha$. In particular,
-$\alpha$ is a light weight at $q$.

######  Definition 4.11.

Let $(M,\omega,T,\phi)$ be a Hamiltonian $T$-space. The pre-toric one-skeleton
$\mathcal{S}_{pre}$ of $(M,\omega,T,\phi)$ is the set of all submanifolds
which belong to a light weight of an isolated fixed point.

######  Remark 4.12.

From the discussion above, we have that $\mathcal{S}_{pre}$ is a collection of
smoothly embedded and symplectic $2$-spheres. Moreover, let $p$ be an isolated
fixed point and let $k$ be the number of light weights at $p$. Then there are
precisely $k$ spheres of $\mathcal{S}_{pre}$ meeting at $p$.

Together with the discussion above, we conclude that the per-toric one-
skeleton of a complexity-$1$ space satisfies the following properties:

######  Lemma 4.13.

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$. Let
$M_{0}^{T}$ be the set of isolated fixed points and $M_{GKM}^{T}$ be the set
of GKM isolated fixed points.
$\bullet$ If $p\in M_{GKM}^{T}$, then there are precisely $n$ spheres of
$\mathcal{S}_{pre}$ meeting at $p$, namely the submanifolds that belong to the
weights of $p$.
$\bullet$ If $p\in M_{0}^{T}\setminus M_{GKM}^{T}$, then there are precisely
$(n-2)$ spheres of $\mathcal{S}_{pre}$ meeting at $p$, namely the submanifolds
that belong to the light weights of $p$.
Moreover, the cardinality of $\mathcal{S}_{pre}$ is

$\displaystyle|\mathcal{S}_{pre}|\quad=\quad\frac{1}{2}n|M_{GKM}^{T}|+\frac{1}{2}(n-2)|M_{0}^{T}\setminus
M_{GKM}^{T}|.$ (26)

######  Proposition 4.14.

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$. For an
isolated fixed point $p\in M_{0}^{T}$ let $\alpha_{p,1},\dots,\alpha_{p,n}$ be
the weights of $p$, where the weight ordered such that
$\alpha_{p,1},\dots,\alpha_{p,n-2}$ are the light weights of $p$ if $p$ is not
GKM. Let $\bar{\xi}\in\text{Lie}(T)$ generic and
$S^{1}=\exp(\mathbb{R}\bar{\xi})$, then

$\displaystyle\sum_{S\in\mathcal{S}_{pre}}\int_{S}\mu^{S^{1}}=$
$\displaystyle\sum_{p\in
M^{T}_{GKM}}\dfrac{\mu^{S^{1}}(p)}{x}\left(\frac{1}{\Phi(\alpha_{p,1})}+\dots+\frac{1}{\Phi(\alpha_{p,n})}\right)\quad+$
(27) $\displaystyle\sum_{p\in M^{T}_{0}\setminus
M^{T}_{GKM}}\dfrac{\mu^{S^{1}}(p)}{x}\left(\frac{1}{\Phi(\alpha_{p,1})}+\dots+\frac{1}{\Phi(\alpha_{p,n-2})}\right)$
(28)

for all $\mu^{S^{1}}\in H_{2}^{*}(M;\mathbb{Z})$, where
$\Phi\colon\ell_{T}^{*}\rightarrow\ell_{S^{1}}^{*}$ is the restriction.

###### Proof.

Let $S\in\mathcal{S}_{pre}$ with $S\cap M^{T}=\left\\{p,q\right\\}$, then $T$
acts on $TpS$ with weight $\alpha\in\ell^{*}\setminus\left\\{0\right\\}$,
where $\alpha$ is a light weight at $p$ and $T$ acts on $TqS$ with weight
-$\alpha$, which is also a light weight at $q$. So this implies that
$S^{1}\cong\exp(\mathbb{R}\bar{\xi})$ acts on $T_{p}S$ with weight
$\Phi(\alpha)$ and on $T_{q}S$ with weight $-\Phi(\alpha)=\Phi(-\alpha)$.
Now let $\mu^{S^{1}}\in H_{S^{1}}^{2}(M;\mathbb{Z})$, then ABBV formula gives
us

$\displaystyle\int_{S}\mu^{S^{1}}=\dfrac{\mu^{S^{1}}(p)}{\Phi(\alpha)\cdot
x}\quad+\quad\dfrac{\mu^{S^{1}}(q)}{\Phi(-\alpha)\cdot x}.$ (29)

The claim follows form this and Lemma 4.13. ∎

##  5 Hamiltonian $T$-spaces of dimension $4$

Let $(F^{4},\omega,T^{2},\psi)$ be a symplectic toric manifold of dimension
$4$. Then $\psi(F)$ is a Delzant polytope and $(F,\omega,T,\psi)$ is
determined by this polytope (up to $T^{2}$-equivariant symplectomorphims) (see
[4]). We fix a splitting $T^{2}=S^{1}\times S^{1}$. So we identify
$\text{Lie}(T)=\mathbb{R}^{2}$ and
$(\text{Lie}(T))^{*}=(\mathbb{R}^{2})^{*}=\mathbb{R}^{2}$. Let $E_{\psi(F)}$
be the set of edges of $\psi(F)$. Then

$\displaystyle S_{e}=\psi^{-1}(e)$ (30)

is a symplectic embedded $2$-sphere for all $e\in E_{\psi(F)}$. Moreover,
$\left\\{S_{e}\mid e\in E_{\psi(F)}\right\\}$ is a toric one-skeleton for
$(F^{4},\omega,T^{2},\psi)$.
If we restrict the action to the subcircle $S^{1}=\left\\{1\right\\}\times
S^{1}\subset T^{2}$, we get an $4$-dimensional Hamiltonian $S^{1}$-space
$(F,\omega,S^{1},\phi)$, where $\phi=\text{pr}_{2}\circ\psi$ and
$\text{pr}_{2}\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ is the projection
onto the second factor. Moreover, we have
$\bullet$ If $e\in E_{\psi(F)}$ is a horizontal edge, then $S_{e}$ is fixed by
the $S^{1}$-action.
$\bullet$ If $e\in E_{\psi(F)}$ is not a horizontal edge, then let $v_{1}$ and
$v_{2}$ be the vertices of $\psi(F)$ connected by this edge. W.l.o.g. we
assume that $\text{pr}_{2}(v_{1})<\text{pr}_{2}(v_{2})$. Let
$\frac{k_{e}}{b_{e}}$ the slope of this edge555
$k_{e},b_{e}\in\mathbb{Z}_{>0}$ with $gcd(k_{e},b_{e})=1$ if $e$ is not a
vertical edge. Otherwise the slope is $\infty$ and $k_{e}=1$.  and
$p_{i}=\phi^{-1}(v_{i})$ for $i=1,2$. Then $S^{1}$ acts on $S_{e}$ with fixed
points $p_{1}$ and $p_{2}$ and $k_{e}$ is the weight at $p_{1}$ and $-k_{e}$
is the weight at $p_{2}$. Moreover, let $\mu^{S^{1}}\in
H_{S^{1}}^{2}(F;\mathbb{Z})$ then the ABBV formula gives us

$\displaystyle\int_{S_{e}}\mu^{S^{1}}=\dfrac{\mu^{S^{1}}(p_{1})-\mu^{S^{1}}(p_{2})}{x\cdot
k_{e}}.$ (31)

######  Definition 5.1.

Let $(F,\omega,S^{1},\phi)$ be a Hamiltonian $S^{1}$-space of dimension $4$
such that the $S^{1}$-action extends to an effective Hamiltonian
$T^{2}$-action. Let $\psi\colon F\rightarrow(\text{Lie}(T^{2}))^{*}$ a moment
map for the $T^{2}$-action. We call

$\displaystyle\mathcal{\bar{S}}_{F}=\left\\{S=\psi^{-1}(e)\mid e\in
E_{\psi(F)}\right\\}\setminus F^{S^{1}}_{2}$

the reduced toric one-skeleton  of $(F,\omega,S^{1},\phi)$ (with respect to
this extension of the $S^{1}$-action), where $F_{2}^{S^{1}}$ is the set of
2-dimensional components of $F^{S^{1}}$.

######  Lemma 5.2.

Let $(F,\omega,S^{1},\phi)$ be a Hamiltonian $S^{1}$-space of dimension $4$
such that the $S^{1}$-action extends to an effective Hamiltonian
$T^{2}$-action and let $\mathcal{\bar{S}}_{F}$ be the reduced toric one-
skeleton. Moreover, let $F_{0}^{S^{1}}$ be the set of isolated fixed point and
$F_{2}^{S^{1}}$ be the set of fixed $2$-spheres. Then

$\displaystyle\sum_{S_{\in}\mathcal{\bar{S}}_{F}}\int_{S}\mu^{S^{1}}=2\sum_{\Sigma\in
F_{2}^{S^{1}}}\mathcal{E}_{\Sigma}\dfrac{\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma})}{x}\quad+\quad\sum_{p\in
F_{0}^{S^{1}}}\dfrac{\mu^{S^{1}}(p)}{x}\left(\frac{1}{\lambda_{p,1}}+\frac{1}{\lambda_{p,2}}\right),$

for $\mu^{S^{1}}\in H_{S^{1}}^{2}(M;\mathbb{Z})$, where $\lambda_{p1}$ and
$\lambda_{p2}$ are the weights of the $S^{1}$-action at $p\in F_{0}^{S^{1}}$
and

$\displaystyle\mathcal{E}_{\Sigma}\colon=\begin{cases}+1\text{ if is $\phi$
attains its minimum at }\Sigma,\\\ -1\text{ if is $\phi$ attains its maximum
at }\Sigma.\end{cases}$

###### Proof.

Let $p$ be an isolated fixed point. Then there are precisely two $2$-spheres
$S_{p1},S_{p,2}$ of $\mathcal{\bar{S}}_{F}$ meeting at $p$, each belongs to
one weight at $\lambda_{p,1},\lambda_{p,2}$. In particular, $S^{1}$ acts on
$T_{p_{i}}S_{p,1}$ with weight $\lambda_{p,i}$ for $i=1,2$.
For a fixed surface $\Sigma$ there are precisely two $2$-spheres
$S_{\Sigma,1},S_{\Sigma,2}$ of $\mathcal{\bar{S}}_{F}$ that intersect with
$\Sigma$. Moreover, $\Sigma\cap S_{\Sigma,i}$ contains one point
$p_{\Sigma,i}$ and $S^{1}$ acts on $T_{p_{\Sigma,i}}S_{\Sigma,i}$ with weight
$+1$ resp. $-1$ if $\phi$ attains its minimum reps. maximum at $\Sigma$ .
Now for $S\in\mathcal{\bar{S}}_{F}$ and $p\in S\cap F^{S^{1}}$ let
$\lambda_{S,p}$ be the weight of $S^{1}$-action $T_{p}S$, then the ABBV
formula gives us

$\displaystyle\int_{S}\mu^{S^{1}}=\sum_{p\in S\cap
F^{S^{1}}}\left(\dfrac{\mu^{S^{1}}(p)}{x}\dfrac{1}{\lambda_{S,p}}\right).$

We conclude

$\displaystyle\sum_{S_{\in}\mathcal{\bar{S}}_{F}}\int_{S}\mu^{S^{1}}$
$\displaystyle=\sum_{S_{\in}\mathcal{\bar{S}}_{F}}\left(\sum_{p\in S\cap
F^{S^{1}}}\left(\dfrac{\mu^{S^{1}}(p)}{x}\dfrac{1}{\lambda_{S,p}}\right)\right)$
$\displaystyle=2\sum_{\Sigma\in
F_{2}^{S^{1}}}\mathcal{E}_{\Sigma}\dfrac{\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma})}{x}\quad+\quad\sum_{p\in
F_{0}^{S^{1}}}\dfrac{\mu^{S^{1}}(p)}{x}\left(\frac{1}{\lambda_{p,1}}+\frac{1}{\lambda_{p,2}}\right).$

∎

###  5.1 Reduced toric one-skeletons of fat edges

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$ with
$\bar{\xi}\in\text{Lie}(T)$ generic. Let

$\displaystyle
e=\left\\{\mathbb{R}\left\langle\alpha_{e}\right\rangle+\beta_{e}\right\\}\cap\phi(M)$
(32)

be a fat edge of the moment map polytope and $F_{e}=\phi^{-1}(e)$ the
corresponding isotropic submanifold of dimension $4$. We have two circle
actions on $F_{e}$, namely the one of the subcircle
$\exp(\mathbb{R}\bar{\xi})$ and the one of the quotient circle
$T/\exp(\ker(\alpha_{e}))$. In order to distinguish them, we denote by $S^{1}$
the subcircle $\exp(\mathbb{R}\bar{\xi})$ and by $\mathcal{C}$ the quotient
circle $T/\exp(\ker(\alpha_{e}))$.

We assume that $\alpha_{e}\in\ell_{T}^{*}$ is primitive. If $p\in M^{T}\cap
F_{e}$, then $T$ acts on $T_{p}F_{e}$ with weights $\lambda_{p,1}\alpha_{e}$
and $\lambda_{p,2}\alpha_{e}$, where
$\lambda_{p,1},\lambda_{p,2}\in\mathbb{Z}$. Hence, the quotient $\mathcal{C}$
acts on $T_{p}F_{e}$ with weights $\lambda_{p,1},\lambda_{p,2}\in\mathbb{Z}$.
Moreover let $\Phi\colon\ell_{T}^{*}\rightarrow\ell_{S^{1}}^{*}$ the
restriction map, then $S^{1}$ acts on $T_{p}F_{e}$ with weights
$\Phi(\lambda_{p,1}\alpha_{e})$ and $\Phi(\lambda_{p,2}\alpha_{e})$.

######  Lemma 5.3.

Assume that the $\mathcal{C}$-action on $F_{e}$ extends to an effective
Hamiltonian $T^{2}$-action on $F_{e}$. Let $\mathcal{\bar{S}}_{F_{e}}$ the
corresponding reduced toric one-skeleton. For $p\in M^{T}\cap F_{e}$ let
$\alpha_{p,n-1},\alpha_{p,n}$ be the heavy weights of the $T-$action on
$T_{p}M$ and for $\Sigma\in M_{2}^{T}\cap F_{e}$ let $\alpha_{\Sigma,e}$ be
the weight of the $T-$action on the normal bundle of $\Sigma$ in $F_{e}$. Then

$\displaystyle\sum_{S_{\in}\mathcal{\bar{S}}_{F_{e}}}\int_{S}\mu^{S^{1}}=2\sum_{\Sigma\in
M_{2}^{S^{1}}\cap
F_{e}}\dfrac{\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma})}{x}\dfrac{1}{\Phi(\alpha_{\Sigma
e})}\quad+\quad\sum_{p\in M_{0}^{S^{1}}\cap
F_{e}}\dfrac{\mu^{S^{1}}(p)}{x}\left(\frac{1}{\Phi(\alpha_{p,n-1})}+\frac{1}{\Phi(\alpha_{p,n})}\right),$

for $\mu^{S^{1}}\in H_{S^{1}}^{2}(M;\mathbb{Z})$.

The proof of Lemma 5.2 is the same as those of Lemma 5.3. Next, we consider
what happen if we pick for each fat edge a reduced toric one-skeleton.

######  Proposition 5.4.

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$ which
satisfies the Extension-Property $(\mathcal{P}_{E})$. For each fat edge $e\in
E_{\phi(M)}^{fat}$, we pick one reduced toric one-skeleton
$\mathcal{\bar{S}}_{F_{e}}$ of $F_{e}$.
For a fixed $2$-sphere $\Sigma\in M_{2}^{T}$ let
$\alpha_{\Sigma,1},\dots,\alpha_{\Sigma,n-1}$ be the weights of the normal
bundle of $\Sigma$ and for a non-GKM isolated fixed point $p\in
M_{0}^{T}\setminus M_{GKM}^{T}$ let $\alpha_{p,n-1},\alpha_{p,n}$ be the heavy
weights of $p$. Then

$\displaystyle\sum_{e\in
E_{\phi(M)}^{fat}}\left(\sum_{S\in\hat{\mathcal{S}}_{F_{e}}}\int_{S}\mu^{S^{1}}\right)=$
$\displaystyle 2\sum_{\Sigma\in
M_{2}^{T}}\dfrac{\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma})}{x}\left(\frac{1}{\Phi(\alpha_{\Sigma,1})}+\dots+\frac{1}{\Phi(\alpha_{\Sigma,n-1})}\right)\quad+$
(33) $\displaystyle\sum_{p\in M^{T}_{0}\setminus
M^{T}_{GKM}}\dfrac{\mu^{S^{1}}(p)}{x}\left(\frac{1}{\Phi(\alpha_{p,n-1})}+\frac{1}{\Phi(\alpha_{p,n})}\right)$
(34)

for $\mu^{S^{1}}\in H_{S^{1}}^{2}(M;\mathbb{Z})$, where
$\bar{\xi}\in\text{Lie}(T)$ is generic, $S^{1}=\exp(\mathbb{R}\bar{\xi})$ and
$\Phi\colon\ell_{T}^{*}\rightarrow\ell_{S^{1}}^{*}$ is the restriction.

###### Proof.

For a fixed $2$-sphere $\Sigma$ there are precisely $(n-1)$ fat edges meeting
at the vertex $\phi(\Sigma)$, each belongs to a weight
$\alpha_{\Sigma,1},\dots,\alpha_{\Sigma,n-1}$ of the normal bundle of $\Sigma$
in $M$. In particular,

$\displaystyle\left\\{\alpha_{\Sigma,1},\dots,\alpha_{\Sigma,n-1}\right\\}=\left\\{\alpha_{\Sigma,e}\mid
e\in E_{\phi(M)}^{fat}\text{ such that $\phi(\Sigma)$ is a vertex of $e$
}\right\\},$ (35)

where $\alpha_{\Sigma,e}$ is the weight of the normal bundle of $\Sigma$ in
$F_{e}$.
For a non-GKM point $p\in M_{0}^{T}\setminus M_{GKM}^{T}$, we have $p\in
F_{e}$ for precisely one $e\in E_{\phi(M)}^{fat}$. In particular,

$\displaystyle M_{0}^{T}\setminus M_{GKM}^{T}=\bigcup_{e\in
E_{\phi(M)}^{fat}}\left(M_{0}^{T}\cap F_{e}\right)$

Together with Lemma 5.3 the claim follows. ∎

######  Lemma 5.5.

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$ that
satisfies the Extension-Property $(\mathcal{P}_{E})$. For a fat edge $e\in
E_{\phi(M)}^{fat}$, we pick one reduced toric one-skeleton
$\mathcal{\bar{S}}_{F_{e}}$ of $F_{e}$. Then the cardinality of $\cup_{e\in
E_{\phi(M)}^{fat}}\mathcal{\bar{S}}_{F_{e}}$ is

$\displaystyle|\cup_{e\in
E_{\phi(M)}^{fat}}\mathcal{\bar{S}}_{F_{e}}|=|M_{0}^{T}\setminus
M_{GKM}^{T}|+(n-1)|M_{2}^{T}|.$ (36)

##  6 Proof of Theorem 1.4 and Corollary 1.6

In this section we prove the Theorem 1.4. Moreover, in Lemma 6.1 we prove that
the existence of a toric one-skeleton for a monotone complexity-one space of
dimension six implies that the second Betti number is bounded by $7$. Finally,
we see that Corollary 1.6 is a simple consequence of Theorem 1.4 and Lemma
6.1. So let us prove Theorem 1.4.

###### Proof of Theorem 1.4 .

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension $2n$ that
satisfies the Extension-Property $(\mathcal{P}_{E})$. Hence, the connected
components of $M^{T}$ are isolated points or $2$-spheres.
$\bullet$ For each fat edge $e\in E^{fat}_{\phi(M)}$, we pick one reduced
toric one-skeleton $\mathcal{\bar{S}}_{F_{e}}$ of $F_{e}=\phi^{-1}(e)$.
$\bullet$ Let $\mathcal{S}_{pre}$ be the pre-toric one-skeleton of
$(M,\omega,T,\phi)$.
$\bullet$ Let $M_{2}^{T}$ be the set of fixed $2$-spheres of $M^{T}$.

Now we want to show that

$\displaystyle\mathcal{S}$ $\displaystyle=\mathcal{S}_{pre}\bigcup
M_{2}^{T}\bigcup\left(\bigcup_{e\in
E^{fat}_{\phi(M)}}\bar{\mathcal{S}}_{F_{e}}\right)$ (37)

is a toric one-skeleton of $(M,\omega,T,\phi)$. Therefore, let $\bar{\xi}\in
Lie(T)$ generic, $S^{1}\cong\exp(\mathbb{R}\bar{\xi})$ and let
$\Phi\colon\ell_{T}^{*}\rightarrow\ell_{S^{1}}^{*}$.

Then Proposition 4.14 and 5.4 imply

$\displaystyle\sum_{S\in\mathcal{S}}\int_{S}\mu^{S^{1}}=$
$\displaystyle\sum_{\Sigma\in
M_{2}^{T}}\left(\int_{\Sigma}\mu^{S^{2}}+2\dfrac{\pi_{\Sigma}(\mu^{S^{1}}|_{\Sigma})}{x}\left(\frac{1}{\Phi(\alpha_{\Sigma,1})}+\dots+\frac{1}{\Phi(\alpha_{\Sigma,n-1})}\right)\right)$
$\displaystyle+\sum_{p\in
M^{T}_{0}}\dfrac{\mu^{S^{1}}(p)}{x}\left(\frac{1}{\Phi(\alpha_{p,1})}+\dots+\frac{1}{\Phi(\alpha_{p,n})}\right)$

for all $\mu^{S^{1}}\in H_{S^{1}}^{*}(M;\mathbb{Z})$. By Corollary 3.4 , this
is equal to $\int_{M}\mu^{S^{1}}c_{n-1}^{S^{1}}$.

Moreover, since the fixed point set $M^{T}$ is torsion-free the restriction
map $H_{S^{1}}^{*}(M;\mathbb{Z})\rightarrow H^{*}(M;\mathbb{Z})$ is surjective
([11, Kirwan]). Let $\mu\in H^{2}(M;\mathbb{Z})$, then $\mu$ admits an
$S^{1}$-equivariant extension $\mu^{S^{1}}\in H_{S^{1}}^{2}(M;\mathbb{Z})$. We
obtain

$\displaystyle\sum_{S\in\mathcal{S}}\int_{S}\mu=\sum_{s\in\mathcal{S}}\int_{S}\mu^{S^{1}}=\int_{M}\mu^{S^{1}}c_{n-1}^{S^{1}}=\int_{M}\mu
c_{n-1}.$ (38)

Hence, the union of the $2$-spheres of $\mathcal{S}$ in $H_{2}(M;\mathbb{Z})$
is the Poincaré dual to the Chern class $c_{n-1}$.
Moreover, by Lemma 4.13 and 5.5 we have

$\displaystyle|\mathcal{S}|=\dfrac{1}{2}n\cdot\left(|M_{0}^{T}|+2|M_{2}^{T}|\right).$
(39)

So it is left to show that the Euler Characteristic $\chi(M)$ is equal to
$|M_{0}^{T}|+2|M_{2}^{T}|$. Therefore let $\phi^{\bar{\xi}}\colon
M\rightarrow\mathbb{R}$ be the $\bar{\xi}$-component of the moment map. Since
$\bar{\xi}$ is generic, we have that $\phi^{\bar{\xi}}$ is a perfect Morse-
Bott function whose critical submanifolds are precisely the connected
components of $M^{S^{1}}=M^{T}$ (Kirwan [11]). Hence,

$\displaystyle H^{*}(M;\mathbb{R})=\bigoplus_{F\subset
M^{T}}H^{*-2d_{F}}(F;\mathbb{R}),$

where $d_{F}$ is the number of negative weights of the normal bundle of $F$.
Since all connected components of $M^{T}$ are isolated points or $2$-spheres,
we have

$\displaystyle\chi(M)=|M_{0}^{T}|+2|M_{2}^{T}|.$

∎

######  Lemma 6.1.

Let $(M,\omega,T,\phi)$ be a complexity-one space of dimension six, which is
monotone and admits a toric one-skeleton $\mathcal{S}$. Then the second Betti
number of $M$ is at most $7$.

###### Proof.

By a result of Sabatini-Sepe [15], the odd Betti numbers of $M$ are equal to
zero. So let $b_{0},\dots,b_{6}$ be the Betti numbers of $M$. Since $M$ is
connected we have $b_{0}=1$. Moreover, by the Poincaré duality we have
$b_{0}=b_{6}$ and $b_{2}=b_{4}$. Hence, the Euler Characteristic of $M$ is

$\displaystyle\chi(M)=b_{0}-b_{1}+b_{2}-b_{3}+b_{4}-b_{5}+b_{6}=2(1+b_{2}).$
(40)

Since each element $S\in\mathcal{S}$ is a smoothly embedded, symplectic
surface of $(M,\omega)$ we have

$\displaystyle\int_{S}\omega>0\quad\text{for all }S\in\mathcal{S}.$ (41)

Now let $c_{1}\in H^{2}(M;\mathbb{Z})$ be the first Chern class of
$(M,\omega)$. Since $(M,\omega)$ is monotone, the integer $\int_{S}c_{1}$ is
positive for all $S\in\mathcal{S}$. This implies

$\displaystyle\sum_{S\in\mathcal{S}}\int_{S}c_{1}\geq|\mathcal{S}|=\frac{3\cdot\chi(M)}{2}=3(1+b_{2}).$

Since $\mathcal{S}$ is the Poincaré dual of the second Chern class $c_{2}\in
H^{4}(M;\mathbb{Z})$, we have

$\displaystyle\int_{M}c_{1}c_{2}=\sum_{S\in\mathcal{S}}c_{1}\geq 3(1+b_{2})$
(42)

Moreover, by the work of Lindsay-Panov [12] we have that
$\int_{M}c_{1}c_{2}=24$. Hence,

$\displaystyle 24=\int_{M}c_{1}c_{2}\geq 3(1+b_{2}),$ (43)

and this implies $b_{2}\leq 7$.

∎

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