# Equivariant cohomology of odd–dimensional complex quadrics from a
combinatorial point of view

Shintarô Kuroki  Department of Applied Mathematics Faculty of Science, Okayama
University of Science, 1-1 Ridai-Cho Kita-Ku Okayama-shi Okayama 700-0005,
Okayama, Japan<EMAIL_ADDRESS>and  Bidhan Paul  Chennai Mathematical
Institute, SIPCOT IT Park, Siruseri, Kelambakkam, 603103, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper aims to determine the ring structure of the torus equivariant
cohomology of odd-dimensional complex quadrics by computing the graph
equivariant cohomology of their corresponding GKM graphs. We show that its
graph equivariant cohomology is generated by three types of subgraphs in the
GKM graph, which are subject to four different types of relations.
Furthermore, we consider the relationship between the two graph equivariant
cohomology rings induced by odd- and even-dimensional complex quadrics.

###### Key words and phrases:

complex quadrics, GKM manifolds, GKM graph, equivariant cohomology

###### 1991 Mathematics Subject Classification:

55N91, 05C90

##  1\. Introduction

A complex quadric $Q_{N}$ is defined by the quadratic equation in the complex
projective space $\mathbb{CP}^{N+1}$:

$\displaystyle
Q_{N}:=\Big{\\{}[z_{1}:\cdots:z_{N+2}]\in\mathbb{CP}^{N+1}\leavevmode\nobreak\
|\leavevmode\nobreak\ \sum_{i=1}^{N+2}z_{i}^{2}=0\Big{\\}}.$

If $N$ is odd (resp. even), then $Q_{N}$ is called an odd-dimensional (resp.
even-dimensional) complex quadric. The standard $SO(N+2)$-action on
$\mathbb{CP}^{N+1}$ preserves this quadratic equation, leading to a maximal
torus $T$-action on $Q_{N}$.

The cohomology ring of $Q_{N}$ has been studied. Over the integer
coefficients, it is isomorphic to one of the following rings for $n\geq 1$
(see [4, Excercise 68.3] or [16] if $N$ is even):

$\displaystyle
H^{*}(Q_{N})\simeq\left\\{\begin{array}[]{lll}\mathbb{Z}[c,x]/\langle
c^{n}-2x,x^{2}\rangle&{\rm if}\ N=2n-1,&{\rm where}\ \deg c=2,\ \deg x=2n\\\
\mathbb{Z}[c,x]/\langle c^{2n}-2cx,x^{2}\rangle&{\rm if}\ N=4n-2,&{\rm where}\
\deg c=2,\ \deg x=4n-2\\\ \mathbb{Z}[c,x]/\langle
c^{2n+1}-2cx,x^{2}-c^{2n}x\rangle&{\rm if}\ N=4n,&{\rm where}\ \deg c=2,\ \deg
x=4n\end{array}\right.$

In [15], the first author computed the $T$-equivariant cohomology ring of the
even-dimensional complex quadric using GKM theory (also see [18]). This
provides a unified formula for the cohomology rings $H^{*}(Q_{4n-2})$ and
$H^{*}(Q_{4n})$ using GKM graphs. The approach also explains why two distinct
relations appear in these rings through combinatorial interpretations. Thus,
GKM theory proves to be a powerful tool for studying (equivariant) cohomology
rings of $T$-spaces.

We briefly recall GKM theory here (see also the introduction of [15]). A GKM
manifold is an equivariantly formal manifold $M^{2k}$ (i.e., $H^{odd}(M)=0$)
endowed with a compact torus $T:=(S^{1})^{n}$ action. Goresky, Kottwitz, and
MacPherson introduced this concept in [8], where the action ensures that the
set of $0$\- and $1$-dimensional orbits forms a graph structure. Motivated by
their work, Guillemin and Zara introduced the concept of an _abstract GKM
graph_ in [10], an abstract graph with labeled edges (see Section 2 for
details). An abstract GKM graphs, which extend beyond their geometric
motivations, have enriched the field known as _GKM theory_. In this paper, we
study the $T^{n}$-equivariant cohomology of the odd-dimensional complex
quadric $Q_{2n-1}$ from a GKM theoretical perspective.

It is well-known that the $Q_{2n-1}$ is $T^{n}$-equivariantly diffeomorphic to
the zero locus of a quadratic equation in $\mathbb{CP}^{2n}$ (see e.g. [19,
Chapter V.1, 1.1 Theorem]):

$\displaystyle
Q_{2n-1}:=\Big{\\{}[z_{1}:\cdots:z_{2n+1}]\in\mathbb{CP}^{2n}\leavevmode\nobreak\
|\leavevmode\nobreak\ \sum_{i=1}^{n}z_{i}z_{2n+1-i}+z_{2n+1}^{2}=0\Big{\\}}.$

In this paper, we consider $Q_{2n-1}$ as this manifold. From this definition,
the natural $T^{n}$-action on $Q_{2n-1}$ is defined by:

$[z_{1}:\cdots:z_{2n+1}]\cdot(t_{1},\ldots,t_{n}):=[z_{1}t_{1}:\cdots:z_{n}t_{n}:t_{n}^{-1}z_{n+1}:\cdots:t_{1}^{-1}z_{2n}:z_{2n+1}],$
(1.1)

where $(t_{1},\ldots,t_{n})\in T^{n}$. Since $Q_{2n-1}$ is diffeomorphic to
the oriented Grassmannian $SO(2n+1)/(SO(2n-1)\times SO(2))$ –the set of
oriented $2$-planes through the origin in $\mathbb{R}^{2n+1}$– this action is
equivariantly diffeomorphic to the maximal torus $T^{n}$-action induced by
restricting the transitive $SO(2n+1)$-action. Furthermore, as $SO(2n-1)\times
SO(2)\subset SO(2n+1)$ is a maximal rank subgroup of $SO(2n+1)$, the set of
$0$\- and $1$-dimensional orbits of $T^{n}$-action forms a graph structure
(see [9]). Therefore, the GKM graph of $Q_{2n-1}$ with $T^{n}$-action (1.1)
can be determined by labeling the edges with tangential representations.

By the formula for $H^{*}(Q_{2n-1})$ as above, we conclude that
$H^{odd}(Q_{2n-1})=0$; hence, $Q_{2n-1}$ is an equivariantly formal GKM
manifold. Consequently, the equivariant cohomology $H_{T^{n}}^{*}(Q_{2n-1})$
can be computed using the graph equivariant cohomology of its GKM graph,
denoted by $\mathcal{GQ}_{2n-1}$. The main theorem of this paper, precisely
presented in Section 5, is as follows:

######  Theorem 1.1 (Theorem 5.1).

There exist the following isomorphisms as a ring:

$\displaystyle H^{*}_{T^{n}}(Q_{2n-1})\simeq
H^{*}(\mathcal{GQ}_{2n-1})\simeq\mathbb{Z}[\mathcal{GQ}_{2n-1}].$

On the other hand, there is a non-effective $T^{n}$-action on $Q_{2n-2}$,
which is restricted from that on $Q_{2n-1}$. In the second part of this paper
(Section 7), we study the homomorphism $H^{*}_{T}(Q_{2n-1})\to
H_{T}^{*}(Q_{2n-2})$ using GKM theory. We also compute the equivariant
cohomology of the non-effective $T^{n}$-action on $Q_{2n-2}$, obtaining the
following result by combining Lemma 7.2 and Theorem 7.8 (see Section 7 for
details):

######  Theorem 1.2.

There exist the following isomorphisms as a ring:

$\displaystyle H^{*}_{T^{n}}(Q_{2n-2})\simeq
H^{*}(\mathcal{GQ}_{2n-2})\simeq\mathbb{Z}[\mathcal{M},\mathcal{D},X]/\mathcal{I}.$

Note that the generators $\mathcal{M}$ and $\mathcal{D}$ in Theorem 1.2 also
appear in the equivariant cohomology of $Q_{2n-2}$ with the effective
$T^{n}$-action as the equivariant Thom classes, as shown in [15]. Theorem 1.2
reveals that in the non-effective case, we also need an additional generator
$X$, which is not induced from the equivariant Thom classes. Motivated by this
phenomenon, in the final part of this paper (Section 8), we further study the
non-effective $T^{1}$-actions on $\mathbb{CP}^{1}$, and obtain the following
result:

######  Theorem 1.3.

Let $\varphi_{n}$ be the $n$-times rotated $T^{1}$-action on $\mathbb{CP}^{1}$
for $n\geq 0$, and $H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})$ be its equivariant
cohomology. Then, there is the following ring isomorphism:

$\displaystyle
H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})\simeq\mathbb{Z}[\tau_{p},\tau_{q},\alpha]/\langle\tau_{p}\tau_{q},n\alpha-\tau_{p}+\tau_{q}\rangle,$

where $\tau_{p}$ (resp. $\tau_{q}$) is the equivariant Thom class associated
with the fixed points $p=[1:0]$ (resp. $q=[0:1]$), and $\alpha$ is the pull-
back of the generator of $H^{*}(BT^{1})\simeq\mathbb{Z}[\alpha]$.

Comparing the effective ($n=1$) and the non-effective ($n\not=1$) cases, we
also observe that there is an additional generator for the non-effective
$T^{1}$-actions on $\mathbb{CP}^{1}$, which may not be induced from the
equivariant Thom classes.

This article is structured as follows: In Section 2, we compute the GKM graph
$\mathcal{GQ}_{2n-1}$ of the effective $T^{n}$-action on $Q_{2n-1}$. In
Section 3, we introduce the graph equivariant cohomology
$H^{*}(\mathcal{GQ}_{2n-1})$ and define the generators $M_{v},$ $\Delta_{K}$
and $Q$, studying their properties. In Section 4, we present the four
relations among $M_{v},\leavevmode\nobreak\ \Delta_{K}$ and $Q$. The main
theorem (Theorem 5.1) is proved in Section 5. In Section 6, the ordinary
cohomology ring of $Q_{2n-1}$ is studied from a GKM theoretical perspective.
The comparison of two graph equivariant cohomologies induced from even- and
odd- dimensional complex quadrics is studied in Section 7. Finally, in Section
8, we provide a GKM description for non-effective $T^{1}$-actions on
$Q_{1}\simeq\mathbb{CP}^{1}$.

##  2\. GKM graphs of odd-dimensional complex quadrics $Q_{2n-1}$

We use the symbol $(Q_{2n-1},T^{n})$ to denote the $T^{n}$-action on
$Q_{2n-1}$ defined by (1.1). We first describe the GKM graph of
$(Q_{2n-1},T^{n})$. For the basics of GKM manifolds and GKM graphs, see the
paper [10]. In this article, we identify the cohomology ring $H^{*}(BT^{n})$
as the following polynomial ring generated by degree 2 generators
$\alpha_{1},\ldots,\alpha_{n}$:

$H^{*}(BT^{n})\simeq\mathbb{Z}[\alpha_{1},\ldots,\alpha_{n}].$ (2.2)

Here, one may consider the generator $\alpha_{j}$, for $j=1,\ldots,n$, as the
$j$-th coordinate projection $pr_{j}:T^{n}\to S^{1}$. Namely, we often use the
following identifications:

$H^{2}(BT^{n})\simeq\mbox{Hom}(T^{n},S^{1})\simeq(\mathfrak{t}^{n}_{\mathbb{Z}})^{*}\simeq\mathbb{Z}^{n},$
(2.3)

where $\mathfrak{t}^{n}_{\mathbb{Z}}$ is the lattice of the Lie algebra of
$T^{n}$ and $(\mathfrak{t}^{n}_{\mathbb{Z}})^{*}$ is its dual.

###  2.1. The GKM graph of the $T^{n}$-action on $Q_{2n-1}$

We now compute the GKM graph of $(Q_{2n-1},T^{n})$. It is easy to check that
the $T^{n}$-fixed points of $Q_{2n-1}$ are

$Q_{2n-1}^{T}=\\{[e_{i}]\,:\,1\leq i\leq 2n\\},$

where $[e_{i}]=[0:\cdots:0:1:0:\cdots:0]\in\mathbb{CP}^{2n}$ is the $i$-th
coordinate with $1$ and $0$ otherwise. Furthermore, the $T^{n}$-invariant
$2$-spheres of $Q_{2n-1}$ are of the following two types:

$\begin{split}&[z_{i}:z_{j}]:=[0:\cdots:0:z_{i}:0:\cdots:0:z_{j}:0:\cdots:0]\in
Q_{2n-1},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{if $i+j\neq 2n+1$};\\\
&[z_{i}:z_{j}]:=[0:\cdots:0:z_{i}:0:\cdots:0:z_{j}:0:\cdots:z_{2n+1}]\in
Q_{2n-1},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{if $i+j=2n+1$}.\end{split}$

Note that the second type as above satisfies the equation
$z_{i}z_{j}+z_{2n+1}^{2}=0$. Therefore, every pair of $[e_{i}],[e_{j}]\in
Q_{2n-1}^{T}$ becomes the fixed points of a $T^{n}$-invariant $2$-sphere.
Hence, we associate the following graph
$\Gamma_{2n-1}:=(\mathcal{V}_{2n-1},\leavevmode\nobreak\ \mathcal{E}_{2n-1})$
with the pair $(Q_{2n-1},T^{n})$:

Vertices:

The set of vertices $\mathcal{V}_{2n-1}$ consists of $Q_{2n-1}^{T}$, where we
may identify the vertices $Q_{2n-1}^{T}=\\{[e_{i}]\ |\ 1\leq i\leq 2n\\}$ with
the set of numbers $[2n]:=\\{1,\ldots,2n\\}$;

Edges:

The set of edges $\mathcal{E}_{2n-1}$ consists of $ij$ for every
$i,j\in\mathcal{V}_{2n-1}$.

Combinatorially, $\Gamma_{2n-1}$ is just the complete graph with $2n$
vertices.

543261 Figure 1. The above graph shows $\Gamma_{5}$ for $n=3$ induced from
$(Q_{5},T^{3})$.

######  Remark 2.1.

For simplicity, we often write $j$ as $\overline{i}$ when $i+j=2n+1$. Hence
the set of vertices and edges in $\Gamma_{2n-1}$ can be rewritten as follows:

$\displaystyle\mathcal{V}_{2n-1}$
$\displaystyle:=\\{1,\ldots,n,\overline{n},\overline{n-1},\ldots,\overline{1}\\};$
(2.4) $\displaystyle\mathcal{E}_{2n-1}$
$\displaystyle:=\\{ij\,:\,i,j\in\mathcal{V}_{2n-1}\text{ such that }j\neq
i\\}.$ (2.5)

We next define an _axial function_ $\alpha:{\mathcal{E}}_{2n-1}\to
H^{2}(BT^{n})$ on the edges by computing the tangential representations around
the fixed points. Since the computation for the other vertices follows a
similar method, we will compute the tangential representation only around the
fixed point $[e_{1}]=[1:0:\cdots:0]$. Around $[e_{1}]\in Q_{2n-1}^{T}$, the
tangential representation is given by

$[1:z_{2}:\cdots:z_{2n}:z_{2n+1}]\mapsto[1:t_{1}^{-1}t_{2}z_{2}:\cdots:t_{1}^{-1}t_{n}z_{n}:t_{1}^{-1}t_{n}^{-1}z_{n+1}:\cdots:t_{1}^{-2}z_{2n}:t_{1}^{-1}z_{2n+1}].$

We notice that the quadric equation for the $z_{2n}$ coordinate satisfies the
following relation:

$\Big{(}z_{2n}=-\sum_{i=2}^{n}z_{i}z_{2n+1-i}-z_{2n+1}^{2}\Big{)}\mapsto\Big{(}t_{1}^{-2}z_{2n}=-\sum_{i=2}^{n}(t_{1}^{-1}t_{i})z_{i}(t_{1}^{-1}t_{i}^{-1})z_{2n+1-i}-(t_{1}^{-1}z_{2n+1})^{2}\Big{)}.$

This shows that the representation on the $z_{2n}$ coordinate is automatically
determined by the representations on the other corrdinates. Therefore, the
tangential representation around the fixed point $[e_{1}]$ can be obtained
from the representations on the coordinates except $z_{1}$ and $z_{2n}$. More
precisely, it splits into the following complex $1$-dimensional irreducible
representations:

$T_{[e_{1}]}Q_{2n-1}\simeq\bigoplus_{i=2}^{2n-1}V(-\alpha_{1}+\alpha_{i})\oplus
V(-\alpha_{1}),$

where $V(\beta)$ is the complex $1$-dimensional representation defined by the
homomorphism $\beta:T^{n}\to S^{1}$. Recall that
$\alpha_{i}\in\mbox{Hom}(T^{n},S^{1})$ for $i\in[n]$ is the representation
corresponding to the $i$-th coordinate projection $pr_{i}:T^{n}\to S^{1}$ (see
(2.2) and (2.3)). For $i\in\\{n+1,\ldots,2n\\}$, we regard

$\alpha_{i}:=-\alpha_{\overline{i}}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{ for $i\in\\{n+1,\ldots,2n\\}$ and $\overline{i}=2n+1-i$.}$ (2.6)

Similarly, each tangential representation around a fixed point decomposes into
complex $1$-dimensional irreducible representations. Furthermore, each
$1$-dimensional irreducible representation corresponds to the tangential
representation on the fixed point of the invariant $2$-sphere. Therefore, we
can define the following axial function on the edges:

$\alpha:{\mathcal{E}}_{2n-1}\to H^{2}(BT^{n})$ (2.7)

which takes

$\displaystyle\alpha(ij)=-\alpha_{i}+\alpha_{j}\text{ for
$j\not=i,\overline{i}$},$ $\displaystyle\alpha(i\overline{i})=-\alpha_{i}.$

More precisely, using the notation (2.6), we have the following assignments:

  * •

$\alpha(ij)=-\alpha_{i}+\alpha_{j}$ for $1\leq i\neq j\leq n$;

  * •

$\alpha(i\overline{j})=-\alpha_{i}+\alpha_{\overline{j}}=-\alpha_{i}-\alpha_{j}$
for $1\leq i\neq j\leq n$;

  * •

$\alpha(\overline{i}j)=-\alpha_{\overline{i}}+\alpha_{j}=+\alpha_{i}+\alpha_{j}$
for $1\leq i\neq j\leq n$;

  * •

$\alpha(\overline{i}\overline{j})=-\alpha_{\overline{i}}+\alpha_{\overline{j}}=\alpha_{i}-\alpha_{j}$
for $1\leq i\neq j\leq n$;

  * •

$\alpha(i\overline{i})=-\alpha_{i}$ for $1\leq i\leq n$,

  * •

$\alpha(\overline{i}i)=-\alpha_{\overline{i}}=\alpha_{i}$ for $1\leq i\leq n$.

Note that the relation $\alpha(pq)=-\alpha(qp)$ holds for all
$pq\in\mathcal{E}_{2n-1}$.

######  Notation 2.2.

Henceforth, the symbol $\mathcal{GQ}_{2n-1}$ represents the GKM graph
$(\Gamma_{2n-1},\alpha)$, where
$\Gamma_{2n-1}=({\mathcal{V}}_{2n-1},{\mathcal{E}}_{2n-1})$ in Remark 2.1, and
the axial function $\alpha:\mathcal{E}_{2n-1}\to H^{2}(BT^{n})$ is defined by
(2.7).

######  Remark 2.3.

Note that the $T^{n}$-action on $Q_{2n-1}\subset\mathbb{CP}^{2n}$ is a
Hamiltonian torus action obtained by restricting the $T^{n}$-action on
$\mathbb{CP}^{2n}$. It is easy to check that its moment-map image is the
$n$-dimensional crossed polytope, i.e., ${\rm Conv}\\{\pm e_{i}\ |\
i=1,\ldots,n\\}\subset(\mathfrak{t}^{n})^{*}.$ However, the GKM graph
$\Gamma_{2n-1}$ of $Q_{2n-1}$ is the complete graph with $2n$ vertices, not
the one-skeleton of the crossed polytope. On the other hand, the restricted
(non-effective) $T^{n}$-action on $Q_{2n-2}(\subset Q_{2n-1})$ is also a
Hamiltonian torus action whose moment-map image is the $n$-dimensional crossed
polytope. By Section 7.1 (also see [15]), we see that the GKM graph of
$Q_{2n-2}$ is indeed the one-skeleton of the crossed polytope.

###  2.2. Examples of $\mathcal{GQ}_{2n-1}$ for $n=2,3$.

In this subsection, we present two low-dimensional examples of
$\mathcal{GQ}_{2n-1}$. Figure 2 dipicts the labeled graph on the edges of
$\Gamma_{3}$ using axial functions, i.e., the GKM graph $\mathcal{GQ}_{3}$.

$1$$2$$4=\overline{1}$$3=\overline{2}$$-\alpha_{1}+\alpha_{2}$$-\alpha_{1}$$\alpha_{1}-\alpha_{2}$$\alpha_{1}+\alpha_{2}$$-\alpha_{2}$$-\alpha_{1}-\alpha_{2}$
Figure 2. The above figure shows the GKM graph $\mathcal{GQ}_{3}$. This
satisfies $\alpha(pq)=-\alpha(qp)$; therefore, we omit the labels in the
opposite directions.

On the other hand, the following table represents the axial functions on the
edges of $\Gamma_{5}$. For $p,q\in\\{1,\ldots,6\\}$, the $pq$ box corresponds
to the edge $pq\in\mathcal{E}_{5}$ of $\Gamma_{5}$ in Figure 1 and its value
indicates $\alpha(pq)$. From this table, we can immediately reconstruct the
labeled graph, i.e., the GKM graph $\mathcal{GQ}_{5}$, from this table.
Therefore, this table also represents the GKM graph.

| 1 | 2 | 3 | $4=\overline{3}$ | $5=\overline{2}$ | $6=\overline{1}$
---|---|---|---|---|---|---
$1$ | N/A | $-\alpha_{1}+\alpha_{2}$ | $-\alpha_{1}+\alpha_{3}$ | $-\alpha_{1}-\alpha_{3}$ | $-\alpha_{1}-\alpha_{2}$ | $-\alpha_{1}$
$2$ | $\alpha_{1}-\alpha_{2}$ | N/A | $-\alpha_{2}+\alpha_{3}$ | $-\alpha_{2}-\alpha_{3}$ | $-\alpha_{2}$ | $-\alpha_{1}-\alpha_{2}$
$3$ | $\alpha_{1}-\alpha_{3}$ | $\alpha_{2}-\alpha_{3}$ | N/A | $-\alpha_{3}$ | $-\alpha_{2}-\alpha_{3}$ | $-\alpha_{1}-\alpha_{3}$
$\overline{3}$ | $\alpha_{1}+\alpha_{3}$ | $\alpha_{2}+\alpha_{3}$ | $\alpha_{3}$ | N/A | $-\alpha_{2}+\alpha_{3}$ | $-\alpha_{1}+\alpha_{3}$
$\overline{2}$ | $\alpha_{1}+\alpha_{2}$ | $\alpha_{2}$ | $\alpha_{2}+\alpha_{3}$ | $\alpha_{2}-\alpha_{3}$ | N/A | $-\alpha_{1}+\alpha_{2}$
$\overline{1}$ | $\alpha_{1}$ | $\alpha_{1}+\alpha_{2}$ | $\alpha_{1}+\alpha_{3}$ | $\alpha_{1}-\alpha_{3}$ | $\alpha_{1}-\alpha_{2}$ | $N/A$
Table 1. This table shows the axial functions on the edges of the graph
$\Gamma_{5}$ in Figure 1, where N/A indicates that there are no edges
connecting $pp$ for $p\in\mathcal{V}_{5}$. This information also provides the
same details of a GKM graph, as illustrated in Figure 2.

###  2.3. GKM subgraphs

In this subsection, we recall the notion of GKM subgraphs of
$\mathcal{GQ}_{2n-1}$. We will use the following symbols:

  * •

$({\mathcal{E}}_{2n-1})_{p}$ is the set of outgoing edges in $\Gamma_{2n-1}$
from the vertex $p\in\mathcal{V}_{2n-1}$;

  * •

for a subgraph $\Gamma^{\prime}\subset\Gamma_{2n-1}$,
${\mathcal{V}}(\Gamma^{\prime})$ (resp. ${\mathcal{E}}(\Gamma^{\prime})$) is
the set of vertices (resp. edges) of $\Gamma^{\prime}$;

  * •

${\mathcal{E}}_{p}(\Gamma^{\prime})$ is the set of outgoing edges in
${\mathcal{E}}(\Gamma^{\prime})$ from the vertex
$p\in{\mathcal{V}}(\Gamma^{\prime})$.

Let $e=pq\in{\mathcal{E}}_{2n-1}$ be an edge. We define a connection along $e$
by the bijection

$\nabla_{e}:({\mathcal{E}}_{2n-1})_{p}\to({\mathcal{E}}_{2n-1})_{q}$ (2.8)

which takes

$pi\mapsto\begin{cases}qp&\text{if $i=q$};\\\ q\overline{q}&\text{if
$i=\overline{p}$};\\\ qi&\text{if $i\not=q,\overline{p}$}.\end{cases}$

It is easy to verify that this satisfies the following congruence relation:

$\displaystyle\alpha(pi)-\alpha(\nabla_{e}(pi))\equiv 0\mod\alpha(e).$

Therefore, we define a (canonical) connection on $\Gamma_{2n-1}$ by
$\nabla:=\\{\nabla_{e}:e\in{\mathcal{E}}_{2n-1}\\}$ (see, [10, Section 1.1]
for further details). In this case, $(\Gamma_{2n-1},\alpha,\nabla)$ forms an
(integral) GKM graph in the usual sense (see [10]), which we also denote by
$\mathcal{GQ}_{2n-1}$ for short.

######  Definition 2.4 (GKM subgraph of $\mathcal{GQ}_{2n-1}$).

Let $\Gamma^{\prime}$ be a subgraph of $\Gamma_{2n-1}$ and $\alpha^{\prime}$
be the restricted axial function on its edges, i.e.,
$\alpha^{\prime}:=\alpha|_{\mathcal{E}(\Gamma^{\prime})}$. Let $\nabla$ be the
connection on $\mathcal{GQ}_{2n-1}$. We call
$(\Gamma^{\prime},\alpha^{\prime})$ a GKM subgraph if it is closed under
$\nabla$. More precisely, for all $e\in{\mathcal{E}}(\Gamma^{\prime})$ with
$i(e)=p$ and $t(e)=q\in{\mathcal{V}}(\Gamma^{\prime})$, the restricted
bijection

$(\nabla_{e})|_{{\mathcal{E}}_{p}(\Gamma^{\prime})}:{\mathcal{E}}_{p}(\Gamma^{\prime})\to{\mathcal{E}}_{q}(\Gamma^{\prime})$

is well-defined, In this case, $(\Gamma^{\prime},\alpha^{\prime})$ where
$\alpha^{\prime}:=\alpha|_{\Gamma^{\prime}}$ is again a GKM graph.

##  3\. Generators of $H^{*}(\mathcal{GQ}_{2n-1})$

The _graph equivariant cohomology_ of the GKM graph $\mathcal{GQ}_{2n-1}$ is
defined as follows:

$H^{*}(\mathcal{GQ}_{2n-1})=\\{f:{\mathcal{V}}_{2n-1}\to
H^{*}(BT^{n})\leavevmode\nobreak\ :\leavevmode\nobreak\ f(i)-f(j)\equiv
0\leavevmode\nobreak\ \mbox{mod}\leavevmode\nobreak\
\alpha(ij)\leavevmode\nobreak\ \mbox{for}\leavevmode\nobreak\
ij\in{\mathcal{E}}_{2n-1}\\}.$ (3.9)

The equation $f(i)-f(j)\equiv 0\leavevmode\nobreak\
\mbox{mod}\leavevmode\nobreak\ \alpha(ij)$ in (3.9) is often referred to as a
_congruence relation_. Note that $H^{*}(\mathcal{GQ}_{2n-1})$ has a graded
$H^{*}(BT^{n})$-algebra structure induced by the graded algebra structure of
$\bigoplus_{k\geq
0}\big{(}\bigoplus_{v\in{\mathcal{V}}_{2n-1}}H^{k}(BT^{n})\big{)}$. This
algebraic structure is also induced by the injective homomorphism

$\vartheta:H^{*}(BT^{n})\to H^{*}(\mathcal{GQ}_{2n-1})$ (3.10)

such that the image of $x\in H^{*}(BT^{n})$ (i.e.,
$\vartheta(x):{\mathcal{V}}_{2n-1}\to H^{*}(BT^{n})$) is defined by the
function

$\vartheta(x)(v)=x\text{ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ for all
$v\in{\mathcal{V}}_{2n-1}$}.$

Furthermore, $H^{*}(\mathcal{GQ}_{2n-1})$ also acquires an
$H^{*}(BT^{n})$-module structure induced by the above injective homomorphism
$\vartheta$.

######  Lemma 3.1.

For the $T^{n}$-action on $Q_{2n-1}$, we have the following graded
$H^{*}(BT^{n})$-algebra isomorphism:

$H^{*}_{T^{n}}(Q_{2n-1})\simeq H^{*}(\mathcal{GQ}_{2n-1}).$ (3.11)

###### Proof.

It is straightforward to verify that all isotropy subgroups are connected for
the effective $T^{n}$-action on $Q_{2n-1}$. Furthermore, since the odd-
cohomologies of $Q_{2n-1}$ vanish (see Section 6), by [6], this statement
holds. ∎

Therefore, to compute the $T^{n}$-equivariant cohomology ring
$H^{*}_{T^{n}}(Q_{2n-1})$ of $Q_{2n-1}$, it sufficies to compute the graph
equivariant cohomology $H^{*}(\mathcal{GQ}_{2n-1})$. In this section, we
introduce three types of elements of $H^{*}(\mathcal{GQ}_{2n-1})$, denoted
$M_{v}$, $Q$ and $\Delta_{K}$ which serves as its generators (see Section 5),
and describe the ring structure of $H^{*}(\mathcal{GQ}_{2n-1})$ in terms of
these generators and relations based on the combinatorial data of
$\mathcal{GQ}_{2n-1}$.

###  3.1. Degree $2$ generators

In this subsection, we will define two types of degree two generators, called
$M_{v}$ for $v\in\mathcal{V}_{2n-1}$ and $Q$.

######  Definition 3.2.

Let $v\in{\mathcal{V}}_{2n-1}$ be a vertex. We define the function
$M_{v}:{\mathcal{V}}_{2n-1}\to H^{2}(BT^{n})$ by

$M_{v}(j)=\begin{cases}0&\text{ if}\leavevmode\nobreak\ \leavevmode\nobreak\
j=v;\\\ 2\alpha_{v}&\text{ if}\leavevmode\nobreak\ \leavevmode\nobreak\
j=\overline{v};\\\ \alpha(jv)=-\alpha_{j}+\alpha_{v}&\text{
if}\leavevmode\nobreak\ \leavevmode\nobreak\ j\not=v,\overline{v};\end{cases}$
(3.12)

where $\alpha_{i}=-\alpha_{\overline{i}}$ for $i\in\\{n+1,\ldots,2n\\}$ (see
(2.6)).

######  Proposition 3.3.

For every $v\in{\mathcal{V}}_{2n-1}$, $M_{v}\in H^{2}(\mathcal{GQ}_{2n-1})$.

###### Proof.

To show this, it is enough to check that $M_{v}$ satisfies the congruence
relations (3.9) for every edge $ij\in{\mathcal{E}}_{2n-1}$. We verify the
above condition on a case-by-case basis:

  1. (a)

If $i=v$ and $j=\overline{v}$: Since $\alpha(v\overline{v})=-\alpha_{v}$, we
have

$M_{v}(i)-M_{v}(j)=-2\alpha_{v}\equiv 0\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{mod}\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha(ij).$

  2. (b)

If $i=v$ and $j\not=v,\overline{v}$: We have
$M_{v}(i)-M_{v}(j)=-\alpha(jv)\equiv 0\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{mod}\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha(ij)$.

  3. (c)

If $i=\overline{v}$ and $j\not=v,\overline{v}$: We have that

$M_{v}(i)-M_{v}(j)=2\alpha_{v}-\alpha(jv)=\alpha_{v}+\alpha_{j}\equiv
0\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{mod}\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha(ij).$

  4. (d)

If $i\not=v,\overline{v}$ and $j\not=v,\overline{v}$: We have

$M_{v}(i)-M_{v}(j)=(-\alpha_{i}+\alpha_{v})-(-\alpha_{j}+\alpha_{v})=-\alpha_{i}+\alpha_{j}\equiv
0\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{mod}\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha(ij).$

Therefore, $M_{v}\in H^{2}(\mathcal{GQ}_{2n-1})$ for all
$v\in{\mathcal{V}}_{2n-1}$. ∎

######  Example 3.4.

For $n=2$ (Figure 2), the following figure shows the element $M_{4}\in
H^{2}(\mathcal{GQ}_{3})$. This may be regarded as the full subgraph spanned by
the vertices $1,2,3\in\mathcal{V}_{3}$ in $\Gamma_{3}$. Note that $M_{v}(j)$
for $j\not=v,\overline{v}$ coincides with the normal axial function of this
full subgraph. Moreover, we can easily check that $M_{v}(\overline{v})$ is the
unique element that satisfies the congruence relations with the other
$M_{v}(j)$’s (cf. [15, Proposition 3.5]).

$M_{4}(1)=-2\alpha_{1}$$M_{4}(2)=-\alpha_{2}+\alpha_{4}=-\alpha_{2}-\alpha_{1}$$M_{4}(4)=0$$M_{4}(3)=-\alpha_{3}+\alpha_{4}=\alpha_{2}-\alpha_{1}$
Figure 3. The element $M_{4}\in H^{2}(\mathcal{GQ}_{3})$.

######  Example 3.5.

By using the table, we can represent all $M_{v}$’s at once. For example, for
$n=3$, the following table represents $M_{v}$ for all $v\in{\mathcal{V}}_{5}$.

| 1 | 2 | 3 | $4=\overline{3}$ | $5=\overline{2}$ | $6=\overline{1}$
---|---|---|---|---|---|---
$M_{1}$ | 0 | $\alpha_{1}-\alpha_{2}$ | $\alpha_{1}-\alpha_{3}$ | $\alpha_{1}+\alpha_{3}$ | $\alpha_{1}+\alpha_{2}$ | $2\alpha_{1}$
$M_{2}$ | $\alpha_{2}-\alpha_{1}$ | 0 | $\alpha_{2}-\alpha_{3}$ | $\alpha_{2}+\alpha_{3}$ | $2\alpha_{2}$ | $\alpha_{2}+\alpha_{1}$
$M_{3}$ | $\alpha_{3}-\alpha_{1}$ | $\alpha_{3}-\alpha_{2}$ | 0 | $2\alpha_{3}$ | $\alpha_{3}+\alpha_{2}$ | $\alpha_{3}+\alpha_{1}$
$M_{\overline{3}}$ | $-\alpha_{3}-\alpha_{1}$ | $-\alpha_{3}-\alpha_{2}$ | $-2\alpha_{3}$ | 0 | $-\alpha_{3}+\alpha_{2}$ | $-\alpha_{3}+\alpha_{1}$
$M_{\overline{2}}$ | $-\alpha_{2}-\alpha_{1}$ | $-2\alpha_{2}$ | $-\alpha_{2}-\alpha_{3}$ | $-\alpha_{2}+\alpha_{3}$ | 0 | $-\alpha_{2}+\alpha_{1}$
$M_{\overline{1}}$ | $-2\alpha_{1}$ | $-\alpha_{1}-\alpha_{2}$ | $-\alpha_{1}-\alpha_{3}$ | $-\alpha_{1}+\alpha_{3}$ | $-\alpha_{1}+\alpha_{2}$ | $0$

Furthermore, we define another degree 2 generator.

######  Definition 3.6.

We define the function $Q:{\mathcal{V}}_{2n-1}\to H^{2}(BT^{n})$ by

$Q(j)=\alpha(j\overline{j})=-\alpha_{j}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
j\in{\mathcal{V}}_{2n-1}.$ (3.13)

######  Proposition 3.7.

The function $Q$ is an element of $H^{2}(\mathcal{GQ}_{2n-1})$.

###### Proof.

The result follows since, for any edge $ij\in{\mathcal{E}}_{2n-1}$, we have
the following:

$Q(i)-Q(j)=-\alpha_{i}-(-\alpha_{j})\equiv 0\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{mod}\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha(ij).$

∎

######  Remark 3.8.

Let $\mathcal{GQ}_{2n-2}$ be the GKM subgraph of $\mathcal{GQ}_{2n-1}$ that
has the same set of vertices ${\mathcal{V}}_{2n-1}=[2n]$ and includes all
edges of $\mathcal{GQ}_{2n-1}$ except

$\\{v\overline{v}\in{\mathcal{E}}_{2n-1}\leavevmode\nobreak\
:\leavevmode\nobreak\ v\in{\mathcal{V}}_{2n-1}\\}.$

Combinatorially, $Q$ is the _equivariant Thom class_ of $\mathcal{GQ}_{2n-2}$,
i.e., the normal axial functions of the GKM subgraph (see [17, Section 4] for
more details). Geometrically, this is nothing but the equivariant Thom class
of the $T^{n}$-invariant submanifold $Q_{2n-2}\subset Q_{2n-1}$ defined by
$z_{2n+1}=0$. In Section 7, we will study $\mathcal{GQ}_{2n-2}$ more
precisely.

######  Example 3.9.

For $n=2$, the following example represents $Q:{\mathcal{V}}_{3}\to
H^{2}(BT^{2})$. This function $Q$ is defined by the normal axial functions of
the GKM subgraph $\mathcal{GQ}_{2}$ of $\mathcal{GQ}_{3}$, i.e., the
equivariant Thom class of the GKM subgraph $\mathcal{GQ}_{2}$.

$Q(1)=-\alpha_{1}$$Q(2)=-\alpha_{2}$$Q(4)=\alpha_{2}$$Q(3)=\alpha_{2}$ Figure
4. The element $Q\in H^{2}(\mathcal{GQ}_{3})$.

######  Example 3.10.

For $n=3$, the following table shows $Q:{\mathcal{V}}_{5}\to H^{2}(BT^{3})$.
This also represents the Thom class of the GKM subgraph $\mathcal{GQ}_{4}$ of
$\mathcal{GQ}_{5}$ (see Figure 5).

| 1 | 2 | 3 | $4=\overline{3}$ | $5=\overline{2}$ | $6=\overline{1}$
---|---|---|---|---|---|---
$Q$ | $-\alpha_{1}$ | $-\alpha_{2}$ | $-\alpha_{3}$ | $\alpha_{3}$ | $\alpha_{2}$ | $\alpha_{1}$

543261 Figure 5. The subgraph as above represents the GKM subgraph
$\mathcal{GQ}_{4}\subset\mathcal{GQ}_{5}$.

###  3.2. Some properties for degree 2 generators $M_{v}$ and $Q$

In this subsection, we will prove Proposition 3.13. To do this, we will
prepare two lemmas.

######  Lemma 3.11.

For any $v\in{\mathcal{V}}_{2n-1}$, the following equality holds:

$M_{v}+M_{\overline{v}}=2Q.$ (3.14)

###### Proof.

Let $w\in{\mathcal{V}}_{2n-1}$ be a vertex of $\mathcal{GQ}_{2n-1}$. We now
verify the result by evaluating (3.14) for each $w\in{\mathcal{V}}_{2n-1}$.

  1. (a)

If $w=v$:
$\big{(}M_{v}+M_{\overline{v}}\big{)}(w)=0+2\alpha_{\overline{v}}=-2\alpha_{v}=2Q(w).$

  2. (b)

If $w=\overline{v}$:
$\big{(}M_{v}+M_{\overline{v}}\big{)}(w)=2\alpha_{v}+0=-2\alpha_{\overline{v}}=2Q(w).$

  3. (c)

If $w\not=v,\overline{v}$:
$\big{(}M_{v}+M_{\overline{v}}\big{)}(w)=-\alpha_{w}+\alpha_{v}+(-\alpha_{w}+\alpha_{\overline{v}})=-2\alpha_{w}=2Q(w)$.

∎

######  Lemma 3.12.

For any $v\in{\mathcal{V}}_{2n-1}$, the following equality holds:

$M_{v}-M_{\overline{v}}=2\alpha_{v}.$ (3.15)

###### Proof.

We verify the result by evaluating (3.15) for each $w\in{\mathcal{V}}_{2n-1}$.

  1. (1)

If $w=v$:
$\big{(}M_{v}-M_{\overline{v}}\big{)}(w)=-2\alpha_{\overline{v}}=2\alpha_{v}.$

  2. (2)

If $w=\overline{v}$: $\big{(}M_{v}-M_{\overline{v}}\big{)}(w)=2\alpha_{v}.$

  3. (3)

If $w\not=v,\overline{v}$:
$\big{(}M_{v}-M_{\overline{v}}\big{)}(w)=-\alpha_{w}+\alpha_{v}-(-\alpha_{w}+\alpha_{\overline{v}})=2\alpha_{v}.$

∎

######  Proposition 3.13.

The generator $\alpha_{i}\in H^{*}(BT^{n})$, for $i=1,\ldots,n$, is obtained
by the following equality:

$\alpha_{i}=M_{i}-Q.$ (3.16)

###### Proof.

The proposition follows from (3.14) and (3.15). ∎

###  3.3. Higher degree generators

In this subsection, we define the degree $2m$ element $\Delta_{K}\in
H^{2m}(\mathcal{GQ}_{2n-1})$ for some $K\subset{\mathcal{V}}_{2n-1}$ such that
$|K|=2n-m$.

######  Definition 3.14.

Let $K\subset{\mathcal{V}}_{2n-1}=[2n]$ be a non-empty subset such that
$\\{i,\overline{i}\\}\not\subset K$ for every $i\in\mathcal{V}_{2n-1}$. We
define the function $\Delta_{K}:{\mathcal{V}}_{2n-1}\to H^{4n-2|K|}(BT^{n})$
by

$\Delta_{K}(j)=\begin{cases}\prod_{k\notin K}\alpha(jk)&\leavevmode\nobreak\
\leavevmode\nobreak\ \text{if}\leavevmode\nobreak\ \leavevmode\nobreak\ j\in
K,\\\ 0&\leavevmode\nobreak\ \leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\ j\notin K.\end{cases}$
(3.17)

Note that, by definition, $\Delta_{\emptyset}=0$.

######  Lemma 3.15.

If $K\subset{\mathcal{V}}_{2n-1}$ is a non-empty subset such that
$\\{i,\overline{i}\\}\not\subset K$ for every $i\in\mathcal{V}_{2n-1}$, then
$\Delta_{K}\in H^{4n-2|K|}(\mathcal{GQ}_{2n-1})$.

###### Proof.

Let $\Gamma_{K}$ be the full subgraph spanned by $K$ such that
$\\{i,\overline{i}\\}\not\subset K$ for every $i\in\mathcal{V}_{2n-1}$, and
let $\mathcal{GK}$ be the pair consisting of $\Gamma_{K}$ and the restricted
axial function $\alpha_{K}:=\alpha|_{\mathcal{E}(\Gamma_{K})}$ on
$\mathcal{E}(\Gamma_{K})$. Since $\mathcal{GK}$ is closed under the connection
$\nabla$ (see (2.8)), $\mathcal{GK}$ is a GKM subgraph of
$\mathcal{GQ}_{2n-1}$ (see Definition 2.4). Moreover, $\Delta_{K}$ is the
_equivariant Thom class_ of the GKM subgraph $\mathcal{GK}$ (see [17, Section
4]). Therefore, by similar arguments as in [17, Lemma 4.1], the lemma follows.
∎

######  Example 3.16.

For the GKM graph $\mathcal{GQ}_{5}$, the set of vertices $P=\\{4,5,6\\}$ (see
Figure 6 below) satisfies the condition given in Definition 3.14.

543261 Figure 6.  $\Gamma_{P}$ : a subgraph consisting of the vertices
$P=\\{4,5,6\\}\subset{{\mathcal{V}}_{5}}$.

Therefore, according to Table 1 in Section 2.2, the function $\Delta_{P}$ for
the subgraph $\Gamma_{P}$ is defined by

$\Delta_{P}(i)=\begin{cases}0\leavevmode\nobreak\ \leavevmode\nobreak\
&\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\ i=1,2,3,\\\
\alpha(41)\alpha(42)\alpha(43)=\alpha_{3}(\alpha_{1}+\alpha_{3})(\alpha_{2}+\alpha_{3})\leavevmode\nobreak\
\leavevmode\nobreak\ &\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\
i=4,\\\
\alpha(51)\alpha(52)\alpha(53)=\alpha_{2}(\alpha_{1}+\alpha_{2})(\alpha_{2}+\alpha_{3})\leavevmode\nobreak\
\leavevmode\nobreak\ &\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\
i=5,\\\
\alpha(61)\alpha(62)\alpha(63)=\alpha_{1}(\alpha_{1}+\alpha_{2})(\alpha_{1}+\alpha_{3})\leavevmode\nobreak\
\leavevmode\nobreak\ &\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\
i=6.\end{cases}$

######  Remark 3.17.

For $n=1$, i.e., $\mathcal{GQ}_{1}$, its vertices are defined by
$\mathcal{V}_{1}:=\\{1,\overline{1}\\}$. In this case, from the definitions of
$M_{a}$ and $\Delta_{a}$ for $a\in\mathcal{V}_{1}:=\\{1,\overline{1}\\}$, we
obtain the following equality:

$\displaystyle M_{\overline{a}}=2\Delta_{a}.$

Moreover, we have

$\displaystyle Q=\Delta_{1}+\Delta_{\overline{1}}.$

Therefore, for $n=1$, the generators can be reduced to
$\Delta_{1},\Delta_{\overline{1}}\in H^{2}(\mathcal{GQ}_{1})$. Furthermore,
since $\mathcal{GQ}_{1}$ is the same GKM graph induced from the standard
$T^{1}$-action on $\mathbb{CP}^{1}(\simeq Q_{1})$, it follows from a well-
known formula that we have

$\displaystyle H^{*}_{T^{1}}(Q_{1})\simeq
H^{*}(\mathcal{GQ}_{1})\simeq\mathbb{Z}[\Delta_{1},\Delta_{\overline{1}}]/\langle\Delta_{1}\Delta_{\overline{1}}\rangle\simeq
H^{*}_{T^{1}}(\mathbb{CP}^{1}).$

##  4\. Four types of relations among $M_{v}$, $Q$ and $\Delta_{K}$’s

This section introduces the four types of relations among
$M_{v},\leavevmode\nobreak\ Q$ and $\Delta_{K}$’s.

We use the following notation for $J\subset{\mathcal{V}}_{2n-1}$ with $n\geq
2$:

$G_{J}=\begin{cases}M_{v}&\text{if
$J={\mathcal{V}}_{2n-1}\setminus\\{v\\}$}\\\ \Delta_{J}&\text{if $J$ satisfies
the property that $\\{i,\overline{i}\\}\not\subset J$ for all
$i\in{\mathcal{V}}_{2n-1}$.}\end{cases}$ (4.18)

Then we can write the first multiplicative relation as follows.

######  Lemma 4.1 (Relation 1).

For $n\geq 2$, there is the following relation:

$\prod_{\cap J=\emptyset}G_{J}=0.$ (4.19)

For $n=1$, i.e., $\mathcal{V}_{1}=\\{1,\overline{1}\\}$, there are the
following relations:

$M_{1}M_{\overline{1}}=M_{1}\Delta_{1}=M_{\overline{1}}\Delta_{\overline{1}}=\Delta_{1}\Delta_{\overline{1}}=0.$

###### Proof.

The lemma follows from the Definitions 3.2 and 3.14. ∎

The next relation follows straightforwardly from Lemma 3.11.

######  Lemma 4.2 (Relation 2).

For every distinct $i,j\in{\mathcal{V}}_{2n-1}$, the following relation holds:

$M_{i}+M_{\overline{i}}=M_{j}+M_{\overline{j}}\leavevmode\nobreak\ =2Q.$

###### Proof.

The lemma follows from (3.14). ∎

The next relation follows straightforwardly from Definition 3.2 and Definition
3.14.

######  Lemma 4.3 (Relation 3).

If $K\subset\mathcal{V}_{2n-1}$ satisfies that $|K|=n$ and
$\\{i,\overline{i}\\}\not\subset K$ for every $i\in\mathcal{V}_{2n-1}$, then
the following relation holds:

$2\Delta_{K}=\prod_{{i\in K^{c}}}M_{i}.$ (4.20)

###### Proof.

The lemma follows from (3.12) and (3.17). ∎

######  Remark 4.4.

For $n=1$, i.e., $\mathcal{V}_{1}=\\{1,\overline{1}\\}$, this relation yields
the following three relations:

$2\Delta_{1}=M_{\overline{1}},\quad 2\Delta_{\overline{1}}=M_{1},\quad
2\Delta_{\emptyset}=0=M_{1}M_{\overline{1}}.$

Moreover, together with Lemma 4.2 (Relation 2), we have

$\Delta_{1}+\Delta_{\overline{1}}=Q.$

######  Example 4.5.

For $n=2$ and $K=\\{1,2\\}\subset{\mathcal{V}}_{3}$, we have

$2\Delta_{K}=M_{\overline{1}}M_{\overline{2}}\in H^{4}(\mathcal{GQ}_{3}).$

We also have the following multiplicative relation for two generators,
$\Delta_{K}$ and $M_{i}$:

######  Lemma 4.6 (Relation 4).

Fix $i\in{\mathcal{V}}_{2n-1}$. If $K\subset{\mathcal{V}}_{2n-1}$ satisfies
$\\{i\\}\subsetneq K$ and $\\{v,\overline{v}\\}\not\subset K$ for every
$v\in{\mathcal{V}}_{2n-1}$, then the following equality holds:

$\Delta_{K}\cdot M_{i}=\Delta_{K\setminus\\{i\\}}.$

###### Proof.

Note that $\Delta_{K}\cdot M_{i}$ is non-zero only on
$K\cap({\mathcal{V}}_{2n-1}\setminus\\{i\\})=K\setminus\\{i\\}$. By the
assumption of $K$, we have $\overline{i}\not\in K\setminus\\{i\\}$. This
implies that we can define the element $\Delta_{K\setminus\\{i\\}}$ by
Definition 3.14. On the other hand, since $\overline{i}\not\in
K\setminus\\{i\\}$, and using the definitions of $\Delta_{K}$ and $M_{i}$ (see
Definition 3.14 and Definition 3.2), we have the following equality:

$\big{(}\Delta_{K}\cdot
M_{i}\big{)}(w)=\begin{cases}\displaystyle\prod_{k\notin
K\setminus\\{i\\}}\alpha(wk)\leavevmode\nobreak\ \leavevmode\nobreak\
&\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\ w\in K\setminus\\{i\\}\\\
0\leavevmode\nobreak\ \leavevmode\nobreak\ &\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ w\notin K\setminus\\{i\\}.\end{cases}$ (4.21)

Hence, from (3.17), the lemma follows. ∎

######  Remark 4.7.

For $n=1$, i.e., $\mathcal{V}_{1}=\\{1,\overline{1}\\}$, this relation yields
the following two relations:

$\Delta_{1}M_{1}=\Delta_{\overline{1}}M_{\overline{1}}=\Delta_{\emptyset}=0.$

######  Example 4.8.

For $n=3$, $i=1$ and $K=\\{1,3,5\\}$, we have

$\Delta_{\\{1,3,5\\}}\cdot M_{1}=\Delta_{\\{3,5\\}}.$

##  5\. Main theorem and its proof

This section aims to prove this paper’s main theorem (Theorem 5.1). To do
that, we first prepare some notations. Then, in Lemma 5.2, we show that
$M_{v}$, $Q$ and $\Delta_{K}$ as defined in Section 3, generate
$H^{*}(\mathcal{GQ}_{2n-1})$. In the final step, we will prove that the
relations defined in Section 4 are enough relations to describe the ring
structure of $H^{*}(\mathcal{GQ}_{2n-1})$. As a consequence, we present the
equivariant cohomology of $Q_{2n-1}$ (i.e., $H^{*}_{T^{n}}(Q_{2n-1})$) in
terms of generators and relations.

We first prepare some notations. Let $\mathbf{M}$ denote the following set of
cohomology classes in $H^{*}(\mathcal{GQ}_{2n-1})$:

$\\{M_{v}\leavevmode\nobreak\ :\leavevmode\nobreak\
v\in{\mathcal{V}}_{2n-1}\\},$

and, let $\mathbf{D}$ denote the set of cohomology classes

$\\{\Delta_{P}\leavevmode\nobreak\ :\leavevmode\nobreak\
\text{$P(\not=\emptyset)\subset{\mathcal{V}}_{2n-1},\leavevmode\nobreak\
\\{i,\overline{i}\\}\not\subset P$ for all $i\in{\mathcal{V}}_{2n-1}$\\}}.$

Let $\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$ be the polynomial ring generated by
all elements in $\mathbf{M},\mathbf{D}$ and $Q$.

Let $\mathfrak{I}\lhd\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$ be an ideal
generated by the following types of elements:

  1. (i)

$\displaystyle\prod_{\cap J=\emptyset}G_{J}$ for $G_{J}$ as in (4.18) if
$n\geq 2$. If $n=1$, $M_{1}M_{\overline{1}},\leavevmode\nobreak\
\Delta_{1}M_{1},\leavevmode\nobreak\ \Delta_{\overline{1}}M_{\overline{1}}$
and $\Delta_{1}\Delta_{\overline{1}}$;

  2. (ii)

$M_{v}+M_{\overline{v}}-2Q$ for every $v\in{\mathcal{V}}_{2n-1}$;

  3. (iii)

$\displaystyle 2\Delta_{K}-\prod_{i\in K^{c}}M_{i}$ for every
$K\subset\mathcal{V}_{2n-1}$ such that $|K|=n$ and
$\\{i,\overline{i}\\}\not\subset K$ for all $i\in\mathcal{V}_{2n-1}$;

  4. (iv)

$\Delta_{P}\cdot M_{i}-\Delta_{P\setminus\\{i\\}}$ for every subset
$P\subset{\mathcal{V}}_{2n-1}$ such that $\\{i\\}\subset P$ and
$\\{j,\overline{j}\\}\not\subset P$ for all $j\in{\mathcal{V}}_{2n-1}$.

We define
$\mathbb{Z}[\mathcal{GQ}_{2n-1}]:=\mathbb{Z}[\mathbf{M},\mathbf{D},Q]/\mathfrak{I}$.
Let $\widetilde{\varphi}:\mathbb{Z}[\mathbf{M},\mathbf{D},Q]\to
H^{*}(\mathcal{Q}_{2n-1})$ be a ring homomorphism, and let

${\varphi}:\mathbb{Z}[\mathcal{GQ}_{2n-1}]\to H^{*}(\mathcal{GQ}_{2n-1})$
(5.22)

be the homomorphism induced from $\widetilde{\varphi}$. Note that we can
immediately prove the well-definedness of $\varphi$ by lemmas in Section 4.

In other words, the following diagram commutes:

${\mathbb{Z}[\mathbf{M},\mathbf{D},Q]}$${\mathbb{Z}[\mathcal{GQ}_{2n-1}]}$${H^{*}(\mathcal{GQ}_{2n-1})}$$\scriptstyle{\widetilde{\varphi}}$$\scriptstyle{\varphi}$
(5.23)

where the vertical map is the natural projection. Now we state the main
theorem of this paper.

######  Theorem 5.1.

The homomorphism $\varphi$ is an isomorphism of rings, i.e.,

$\varphi:\mathbb{Z}[\mathcal{GQ}_{2n-1}]\xrightarrow{\cong}H^{*}(\mathcal{GQ}_{2n-1}).$
(5.24)

In particular, for the effective $T^{n}$-action on $Q_{2n-1}$ defined by
(1.1), the following isomoprhism holds:

$H^{*}_{T^{n}}(Q_{2n-1})\cong
H^{*}(\mathcal{GQ}_{2n-1})\cong\mathbb{Z}[\mathcal{GQ}_{2n-1}].$ (5.25)

In the proofs below, for simplicity, we often abuse the notations $M_{i}$ and
$\Delta_{P}$ to mean $\widetilde{\varphi}(M_{i})$ and
$\widetilde{\varphi}(\Delta_{P})$, respectively. We first prove the
surjectivity of $\varphi$.

######  Lemma 5.2.

The homomorphism $\varphi:\mathbb{Z}[\mathcal{GQ}_{2n-1}]\to
H^{*}(\mathcal{GQ}_{2n-1})$ is surjective.

###### Proof.

This is enough to prove that
$\widetilde{\varphi}:\mathbb{Z}[\mathbf{M},\mathbf{D},Q]\to
H^{*}(\mathcal{GQ}_{2n-1})$ is surjective. Consider an element $f\in
H^{*}(\mathcal{GQ}_{2n-1})$. For the vertex $1\in{\mathcal{V}}_{2n-1}$, one
can write $f(1)\in H^{*}(BT^{n})\cong\mathbb{Z}[\alpha_{1},\ldots,\alpha_{n}]$
(see (3.9)) as

$f(1)=\sum_{\mathbf{j}}c_{\mathbf{j}}\alpha_{1}^{j_{1}}\cdots\alpha_{n}^{j_{n}}=h_{1}$

where $c_{\mathbf{j}}\in\mathbb{Z}$ and
$\mathbf{j}=(j_{1},\ldots,j_{n})\in(\mathbb{N}\cup\\{0\\})^{n}$. Since
$\alpha_{i}=M_{i}-Q$ (see (3.16)) for each $i$, it follows that

$f(1)=\sum_{\mathbf{j}}c_{\mathbf{j}}(M_{1}-Q)^{j_{1}}\cdots(M_{n}-Q)^{j_{n}}(1)=h_{1}.$

This implies that there exists an element in
$\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$ whose image under $\widetilde{\varphi}$
coincides with $f(1)$ on the vertex $1\in{\mathcal{V}}_{2n-1}$.

We next put $f_{2}=f-h_{1}$ and hence $f_{2}(1)=0$. By using the congruence
relation (3.9) on the edge $21\in{\mathcal{E}}_{2n-1}$, we have

$f_{2}(2)-f_{2}(1)\equiv 0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{mod}\leavevmode\nobreak\ \alpha(21)=M_{1}(2).$

Therefore, $f_{2}(2)=h_{2}M_{1}(2)$ for some $h_{2}\in H^{*}(BT^{n})$.

Furthermore, from (3.16), we have that $h_{2}M_{1}$ is in the image of
$\widetilde{\varphi}$. Define

$f_{3}=f_{2}-h_{2}M_{1}\leavevmode\nobreak\ (=f-h_{1}-h_{2}M_{1}).$

Since $f_{2}(1)=0$, $M_{1}(1)=0$, and $f_{2}(2)=h_{2}M_{1}(2)$, we have

$f_{3}(1)=0=f_{3}(2).$

Thus, by the congruence relations on the edges $31$ and
$32\in{\mathcal{E}}_{2n-1}$, we can write

$f_{3}(3)=h_{3}M_{1}M_{2}(3)$

for some $h_{3}\in H^{*}(BT^{n})\subset\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$.

Similarly, we can also verify that $f_{4}:=f_{3}-h_{3}M_{1}M_{2}$ satisfies
$f_{4}(1)=f_{4}(2)=f_{4}(3)=0$. By iterating this procedure $n$ times, we
obtain an element

$f_{n}:=f_{n-1}-h_{n-1}M_{1}\cdots M_{n-2}$

for some $h_{n-1}\in H^{*}(BT^{n})\subset\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$
and $f_{n}\in H^{*}(\mathcal{GQ}_{2n-1})$, which satisfies

$f_{n}(i)=0\text{ for $1\leq i\leq n-1$}$

as $f_{n-1}(i)=0$ for $1\leq i\leq n-2$ and $f_{n-1}(n-1)=h_{n-1}M_{1}\cdots
M_{n-2}(n-1)$.

Furthermore, applying the congruence relations for the edges
$ni\in{\mathcal{E}}_{2n-1}$ for $1\leq i\leq n-1$, we have

$f_{n}(n)=h_{n}M_{1}\cdots M_{n-1}(n)$

for some $h_{n}\in H^{*}(BT^{n})$. Then, we have

$\begin{split}f_{n+1}&=f_{n}-h_{n}M_{1}\cdots M_{n-1}\\\
&=f_{n-1}-h_{n-1}M_{1}\cdots M_{n-2}-h_{n}M_{1}\cdots M_{n-1}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \vdots\\\
&=f-h_{1}-h_{2}M_{1}-\cdots-h_{n-1}M_{1}\cdots M_{n-2}-h_{n}M_{1}\cdots
M_{n-1}\end{split}$ (5.26)

which statisfies $f_{n+1}(i)=0$ for all $1\leq i\leq n$. Therefore by using
the congruence relation (3.9) for the edge
$\overline{n}i\in{\mathcal{E}}_{2n-1}$ for every $1\leq i\leq n$ (where
$\overline{n}=n+1$), one can verify that

$f_{n+1}(n+1)\equiv 0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{mod}\leavevmode\nobreak\ \alpha(\overline{n}i)$

for all $1\leq i\leq n$. In particular, from the definition of
$\Delta_{\\{n+1,\ldots,2n\\}}$ (see Definition 3.14), one can write

$f_{n+1}(n+1)=h_{n+1}\Delta_{\\{n+1,\ldots,2n\\}}(n+1),$

for some $h_{n+1}\in H^{*}(BT^{n})\subset\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$.

Next, we define

$f_{n+2}=f_{n+1}-h_{n+1}\Delta_{\\{n+1,\ldots,2n\\}}$

which satisfies $f_{n+2}(1)=\cdots=f_{n+2}(n+1)=0$ since
$\Delta_{\\{n+1,\ldots,2n\\}}(i)=0$ for all $1\leq i\leq n$.

Similarly, for $2\leq j\leq n$, there exists $h_{n+j}\in
H^{*}(BT^{n})\subset\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$ such that

$f_{n+j+1}:=f_{n+j}-h_{n+j}\Delta_{\\{n+j,\ldots,2n\\}}$ (5.27)

satisfies $f_{n+j+1}(1)=\cdots=f_{n+j+1}(n+j)=0$. Notice that, when $j=n$,

$f_{2n+1}:=f_{2n}-h_{2n}\Delta_{\\{2n\\}},$

which satisfies $f_{2n+1}(1)=\cdots=f_{2n+1}(2n)=0$, i.e., $f_{2n+1}\equiv 0$.
Therefore, $f_{2n}=h_{2n}\Delta_{\\{2n\\}}$. By using (5.27) repeatedly, we
have the following:

$\begin{split}f_{2n-1}&=h_{2n-1}\Delta_{\\{2n-1,2n\\}}+h_{2n}\Delta_{\\{2n\\}};\\\
f_{2n-2}&=h_{2n-2}\Delta_{\\{2n-2,2n-1,2n\\}}+h_{2n-1}\Delta_{\\{2n-1,2n\\}}+h_{2n}\Delta_{\\{2n\\}};\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \vdots\\\
f_{n+1}&=h_{n+1}\Delta_{\\{n+1,\ldots,2n\\}}+\cdots+h_{2n-2}\Delta_{\\{2n-2,2n-1,2n\\}}+h_{2n-1}\Delta_{\\{2n-1,2n\\}}+h_{2n}\Delta_{\\{2n\\}}.\end{split}$
(5.28)

Therefore, from (5.26) and (5.28), we have

$\begin{split}f=h_{1}+h_{2}M_{1}&+\cdots+h_{n}M_{1}\cdots M_{n-1}\\\
&+h_{n+1}\Delta_{\\{n+1,\ldots,2n\\}}+\cdots+h_{2n-2}\Delta_{\\{2n-2,2n-1,2n\\}}+h_{2n-1}\Delta_{\\{2n-1,2n\\}}+h_{2n}\Delta_{\\{2n\\}}\end{split}$
(5.29)

where $h_{i}\in H^{*}(BT^{n})\subset\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$ for
each $1\leq i\leq 2n$. Hence, the lemma follows. ∎

######  Remark 5.3.

For any $g\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]=\mathbb{Z}[{\bf
M,\leavevmode\nobreak\ D},\leavevmode\nobreak\ Q]/\mathfrak{I}$, one can
choose $f\in\mathbb{Z}[{\bf M},{\bf D},Q]$ of the form give by (5.29) such
that $g=f+\mathfrak{I}$.

We will establish some facts to prove the injectivity of $\varphi$ in Lemma
5.6.

Let $v\in{\mathcal{V}}_{2n-1}=[2n]$. For $n\geq 2$, we define
$I_{v}\subset[n]\subset{\mathcal{V}}_{2n-1}$ by

$I_{v}=\begin{cases}[n]\setminus\\{v\\}&\text{ if }1\leq v\leq n\\\
[n]\setminus\\{\overline{v}\\}&\text{ if }n+1\leq v\leq 2n.\\\ \end{cases}$

######  Lemma 5.4.

Let $v\in{\mathcal{V}}_{2n-1}$. For $n\geq 2$, let $\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ v\notin J\rangle$ be an ideal
in $\mathbb{Z}[\mathcal{GQ}_{2n-1}]$ generated by $G_{J}$ (see (4.18)) with
$v\notin J$. Then, we have the following isomorphism:

$\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle G_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ v\notin
J\rangle\simeq\mathbb{Z}[M_{i},Q\leavevmode\nobreak\ |\leavevmode\nobreak\
i\in I_{v}]\simeq H^{*}(BT^{n}).$ (5.30)

For $n=1$, the following isomoprhism holds:

$\mathbb{Z}[\mathcal{GQ}_{1}]/\langle
M_{v},\Delta_{\overline{v}}\rangle\simeq\mathbb{Z}[\Delta_{v}]\simeq
H^{*}(BT^{1}),$ (5.31)

where $v\in\mathcal{V}_{1}=\\{1,\overline{1}\\}.$

###### Proof.

For $n=1$, by using Relations 1–4, we have that

$\mathbb{Z}[\mathcal{GQ}_{1}]/\langle
M_{v},\Delta_{\overline{v}}\rangle\simeq\mathbb{Z}[M_{\overline{v}},\Delta_{v},Q]/\langle
M_{\overline{v}}-2Q,2\Delta_{v}-M_{\overline{v}},\rangle\simeq\mathbb{Z}[\Delta_{v}].$

This establishes the statement for $n=1$.

We next assume that $n\geq 2$, We prove the statement only for the vertex
$v=1\in{\mathcal{V}}_{2n-1}$ because the proof for other vertices will follow
similarly. If $v=1\in{\mathcal{V}}_{2n-1}$, the isomorphism (5.30) to be
proved is stated as follows:

$\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle G_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\notin
J\rangle\simeq\mathbb{Z}[M_{2},\ldots,M_{n},Q]\simeq H^{*}(BT^{n}).$ (5.32)

Note that the following elements generate
$\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle G_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\notin J\rangle$:

$\\{\overline{Q}\\}\cup\\{\overline{M_{v}}:v\in{\mathcal{V}}_{2n-1}\\}\cup\\{\overline{\Delta_{P}}:P\subset{\mathcal{V}}_{2n-1},\leavevmode\nobreak\
\\{i,\overline{i}\\}\not\subset P\text{ for all
$i\in{\mathcal{V}}_{2n-1}$}\\}=:\\{\overline{Q}\\}\cup\overline{\mathbf{M}}\cup\overline{\mathbf{D}},$
(5.33)

where

$\displaystyle\overline{M_{v}}:=M_{v}+(\mathfrak{I}+\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle),$
$\displaystyle\overline{Q}:=Q+(\mathfrak{I}+\langle G_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\notin J\rangle),$
$\displaystyle\overline{\Delta_{P}}:=\Delta_{P}+(\mathfrak{I}+\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle).$

Let $L\subset{\mathcal{V}}_{2n-1}$ such that $\\{i,\overline{i}\\}\not\subset
L$ for every $1\leq i\leq n$. If $1\notin L$, then $\Delta_{L}=G_{L}\in\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$. Therefore,
$\overline{\Delta_{L}}=0$ in $\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$. On the
other hand, if $1\in L$ and $|L|=l<n$, then there exist vertices
$v_{i}(\not=1)\in\mathcal{V}_{2n-1}$ $(i=1,\ldots,n-l)$ such that
$v_{i}\not\in L$ and $v_{i}+v_{j}\not=2n+1$ for $1\leq i<j\leq n-l$. By
repeatedly using Lemma 4.6 (Relation 4), we have the following equality:

$\displaystyle\Delta_{L}$ $\displaystyle=\Delta_{L\cup\\{v_{1}\\}}\cdot
M_{v_{1}}$ $\displaystyle=\Delta_{L\cup\\{v_{1},v_{2}\\}}\cdot M_{v_{1}}\cdot
M_{v_{2}}$ $\displaystyle\quad\vdots$
$\displaystyle=\Delta_{L\cup\\{v_{1},v_{2},\ldots,v_{n-l}\\}}\cdot
M_{v_{1}}\cdots M_{v_{n-l}}.$

This shows that every generators presented by
$\overline{\Delta_{P}}\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$ can be
written in terms of some $\overline{M_{v}}$’s and $\overline{\Delta_{L}}$’s
with $1\in L$ and $|L|=n$. Furthermore, for such an $L$, we can put
$L=\\{1,i_{1},\ldots,i_{n-1}\\}$. Then, by Lemma 4.3 (Relation 3), we have
that

$\displaystyle 2\Delta_{L}=\prod_{i\in L^{c}}M_{i}.$

On the other hand, for $L^{\prime}=\\{2n,i_{1},\ldots,i_{n-1}\\}$, we have

$\displaystyle 2(\Delta_{L}+\Delta_{L^{\prime}})=\prod_{i\in
L^{c}}M_{i}+\prod_{i\in(L^{\prime})^{c}}M_{i}=(M_{1}+M_{2n})\prod_{i\in
L^{c}\setminus\\{2n\\}}M_{i}=2Q\prod_{i\in L^{c}\setminus\\{2n\\}}M_{i}.$

The last equality follows from Relation 2. Since $1\not\in L^{\prime}$, we
have

$\displaystyle\overline{\Delta_{L}}=\overline{Q}\prod_{i\in
L^{c}\setminus\\{2n\\}}\overline{M_{i}}.$

This concludes that every element in $\overline{{\bf D}}$ can be expressed in
terms of $\overline{Q}$ and the elements in $\overline{{\bf M}}$. Namely, we
can reduce the generators of (5.33) to $\overline{Q}$ and $\overline{{\bf
M}}$.

Next, from (4.18), we have $M_{1}=G_{{\mathcal{V}}_{2n-1}\setminus\\{1\\}}$,
i.e., $\overline{M_{1}}=0\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$. Therefore,
together with Lemma 4.2 (Relation 2), we have

$2\overline{Q}=\overline{M_{\overline{1}}}=\overline{M_{2}}+\overline{M_{\overline{2}}}=\cdots=\overline{M_{n}}+\overline{M_{\overline{n}}}\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle.$

Consequently,

$2\overline{Q}=\overline{M_{\overline{1}}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{M_{\overline{k}}}=2\overline{Q}-\overline{M_{k}}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for }k=2,\ldots,n.$

This implies that all generators in (5.33) can be reduced to the following:

$\overline{Q}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\\{\overline{M_{i}}\leavevmode\nobreak\ |\leavevmode\nobreak\ 2\leq i\leq
n\\}.$

Hence, the natural projection
$p:\mathbb{Z}[Q,M_{2},\ldots,M_{n}]\to\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\langle
G_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$ (see (5.23))
is surjective. Note that

$\mathbb{Z}[Q,M_{2},\ldots,M_{n}]=\mathbb{Z}[-Q,M_{2}-Q,\ldots,M_{n}-Q].$

We finally consider the following composition homomorphism

$\mathbb{Z}[-Q,M_{2}-Q,\ldots,M_{n}-Q]\xrightarrow{p}\frac{\mathbb{Z}[\mathcal{GQ}_{2n-1}]}{\langle
G_{J}|\leavevmode\nobreak\ 1\notin J\rangle}\xrightarrow{i_{1}}H^{*}(BT^{n}),$

where $i_{1}$ is induced from the composition
$\mathbb{Z}[\mathcal{GQ}_{2n-1}]\xrightarrow{\varphi}H^{*}(\mathcal{GQ}_{2n-1})\to
H^{*}(BT^{n})$, and is defined by $f\mapsto\varphi(f)(1)$ for
$f\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]$. Namely, $i_{1}$ is the restriction map
onto the vertex $1\in\mathcal{V}_{2n-1}$. Using (3.12) and (3.13), we have

$i_{1}\circ p(-Q)=\alpha_{1},\leavevmode\nobreak\ \text{ and,
}\leavevmode\nobreak\ i_{1}\circ p(M_{k}-Q)=\alpha_{k}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for }k=2,\ldots,n.$

Hence, $i_{1}\circ p$ is an isomorphism, which shows that $p$ is injective.
Consequently, $p$ is an isomorphism.

Hence,

$\frac{\mathbb{Z}[\mathcal{GQ}_{2n-1}]}{\langle G_{J}|\leavevmode\nobreak\
1\notin J\rangle}\cong\mathbb{Z}[Q,M_{2},\ldots,M_{n}]\cong
H^{*}(BT^{n})=\mathbb{Z}[\alpha_{1},\ldots,\alpha_{n}].$ (5.34)

This establishes (5.32). ∎

######  Corollary 5.5.

We have the following injective homomorphisms:

$\displaystyle\displaystyle
H^{*}(\mathcal{GQ}_{2n-1})\hookrightarrow\bigoplus_{v\in{\mathcal{V}}_{2n-1}}H^{*}(BT^{n})\simeq\bigoplus_{v\in{\mathcal{V}}_{2n-1}}\mathbb{Z}[Q,M_{i}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}]\ {\text{for $n\geq 2$};}$ (5.35)
$\displaystyle\displaystyle{H^{*}(\mathcal{GQ}_{1})\hookrightarrow\bigoplus_{v\in{\mathcal{V}}_{1}}H^{*}(BT^{1})\simeq\mathbb{Z}[\Delta_{1}]\oplus\mathbb{Z}[\Delta_{\overline{1}}].}$
(5.36)

###### Proof.

The first inclusion directly follows from the definition of graph equivariant
cohomology of GKM graphs, while the the second part follows from Lemma 5.4. ∎

######  Lemma 5.6.

The homomorphism $\varphi:\mathbb{Z}[\mathcal{GQ}_{2n-1}]\to
H^{*}(\mathcal{GQ}_{2n-1})$ is injective.

###### Proof.

We first assume that $n=1$. Then, it follows from Relations 1–4 that

$\displaystyle\mathbb{Z}[\mathcal{GQ}_{1}]\simeq\mathbb{Z}[\Delta_{1},\Delta_{\overline{1}}]/\langle\Delta_{1}\Delta_{\overline{1}}\rangle.$

Therefore, every element in $\mathbb{Z}[\mathcal{GQ}_{1}]$ can be written as

$\displaystyle k+f(\Delta_{1})+g(\Delta_{\overline{1}})$ (5.37)

for some constant term $k\in\mathbb{Z}$ and two polynomials
$f(\Delta_{1})\in\mathbb{Z}[\Delta_{1}]$ and
$g(\Delta_{\overline{1}})\in\mathbb{Z}[\Delta_{\overline{1}}]$, where
$f(\Delta_{1})$ and $g(\Delta_{\overline{1}})$ do not have constant terms.
Therefore, together with Corollary 5.5, we have that the following
homomorphism is injective:

$\Phi:\displaystyle\mathbb{Z}[\mathcal{GQ}_{1}]\xrightarrow{\varphi}H^{*}(\mathcal{GQ}_{1})\hookrightarrow\bigoplus_{v\in{\mathcal{V}}_{1}}H^{*}(BT^{1})\simeq\mathbb{Z}[\Delta_{1}]\oplus\mathbb{Z}[\Delta_{\overline{1}}],$

where $k$ in (5.37) maps to the diagonal element. This establishes that
$\varphi$ is injective.

We next assume that $n\geq 2$. Similarly, we claim that the following map
$\Phi$ is injective:

$\Phi:\displaystyle\mathbb{Z}[\mathcal{GQ}_{2n-1}]\xrightarrow{\varphi}H^{*}(\mathcal{GQ}_{2n-1})\hookrightarrow\bigoplus_{v\in{\mathcal{V}}_{2n-1}}H^{*}(BT^{n})\simeq\bigoplus_{v\in{\mathcal{V}}_{2n-1}}\mathbb{Z}[Q,M_{i}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}].$

Let

$r_{u}:\displaystyle\bigoplus_{v\in{\mathcal{V}}_{2n-1}}\mathbb{Z}[Q,M_{i}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}]\to\mathbb{Z}[Q,M_{i}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{u}]$

be the restriction map at $u\in{\mathcal{V}}_{2n-1}$. For any
$f\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]$, let $f(u)$ denote the image of $f$ by
$r_{u}\circ\Phi$. Assume that $\Phi(f)=0$ for an element
$f\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]$. Therefore,

$r_{v}\circ\Phi(f)=f(v)=0\in\mathbb{Z}[Q,M_{i}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all }v\in{\mathcal{V}}_{2n-1}.$ (5.38)

Here, the element $f\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]$ can be written as
follows (see (5.29) and Remark 5.3):

$\begin{split}f=h_{1}+h_{2}M_{1}&+\cdots+h_{n}M_{1}\cdots M_{n-1}\\\
&+h_{n+1}\Delta_{\\{n+1,\ldots,2n\\}}+\cdots+h_{2n}\Delta_{\\{2n\\}}+\mathfrak{I},\end{split}$
(5.39)

where $h_{i}\in H^{*}(BT^{n})\subset\mathbb{Z}[{\bf M},\leavevmode\nobreak\
{\bf D},\leavevmode\nobreak\ Q]$ for all $i=1,\ldots,2n$. Note that, for all
$i=1,\ldots,2n$, $\varphi(h_{i})\in\mathbb{Z}[\alpha_{1},\ldots,\alpha_{n}]$
(see (2.2)). Consequently, if there exists a vertex $u\in\mathcal{V}_{2n-1}$
such that $h_{i}(u)=0$, then $h_{i}$ must be identically zero, as
$\varphi(h_{i})$ is a constant in $H^{*}(\mathcal{GQ}_{2n-1})$.

Since

$\Delta_{\\{n+1,\ldots,2n\\}}(i)=\cdots=\Delta_{\\{2n\\}}(i)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
i=1,\ldots,n,$ (5.40)

and $M_{1}(1)=0$, using /(5.38) and (5.39), we have

$0=f(1)=h_{1}(1)+0+\cdots+0.$

Hence, $h_{1}=0$. Furthermore, by plugging $h_{1}=0$ into (5.39), and using
$M_{2}(2)=0$, we have

$0=f(2)=h_{2}M_{1}(2)+0+\cdots+0.$

Notice that
${h_{2}M_{1}(2)=h_{2}(2)M_{1}(2)}\in\mathbb{Z}[Q,\leavevmode\nobreak\
M_{i}\leavevmode\nobreak\ |\leavevmode\nobreak\ i\in I_{2}]$ (see (5.35)) and
$M_{1}(2)\not=0$. Hence $h_{2}(2)=0$ since
$\mathbb{Z}[{Q},M_{i}\leavevmode\nobreak\ |\leavevmode\nobreak\ i\in I_{2}]$
is an integral domain. Therefore, $h_{2}=0$. By following similar arguments
for $i=3,\ldots,n$, one can have

$h_{3}=\cdots=h_{n}=0.$

Therefore, by (5.39),

$\begin{split}f=h_{n+1}\Delta_{\\{n+1,\ldots,2n\\}}+\cdots+h_{2n-1}\Delta_{\\{2n-1,2n\\}}+h_{2n}\Delta_{\\{2n\\}}+\mathfrak{I}.\end{split}$
(5.41)

From (5.41), we next obtain the following equality for the vertex
$n+1\in{\mathcal{V}}_{2n-1}$:

$0=f(n+1)=h_{n+1}\Delta_{\\{n+1,\ldots,2n\\}}(n+1)+h_{n+2}\Delta_{\\{n+2,\ldots,2n\\}}(n+1)+\cdots+h_{2n}\Delta_{\\{2n\\}}(n+1).$

Since

$\begin{split}\Delta_{\\{n+2,\ldots,2n\\}}&(n+1)=\cdots=\Delta_{\\{2n\\}}(n+1)=0,\\\
&\Delta_{\\{n+1,\ldots,2n\\}}(n+1)\not=0,\end{split}$

we have $h_{n+1}=0$, by the same reason as above. Iterating similar arguments
for the other vertices $n+2,\ldots,2n\in{\mathcal{V}}_{2n-1}$, we have
$h_{i}=0$ for all $n+2\leq i\leq 2n$.

Therefore $f=0$ in $\mathbb{Z}[\mathcal{GQ}_{2n-1}]=\mathbb{Z}[{\bf
M,\leavevmode\nobreak\ D},\leavevmode\nobreak\ Q]/\mathfrak{I}$, which shows
the injectivity of $\Phi$. Hence, the lemma follows. ∎

Proof of Theorem 5.1.  The result (5.24) follows from Lemma 5.2 and Lemma 5.6.
Moreover, since the all isotropy subgroups of $(Q_{2n-1},T^{n})$ are
connected, it follows from [6] that we have the isomorphism (5.25). ∎

##  6\. Ordinary cohomology of odd dimensional complex quadrics

In the paper [15], we compare the ordinary cohomologies of $H^{*}(Q_{4n})$ and
$H^{*}(Q_{4n-2})$ by using the graph equivariant cohomology. In this section,
we compute the ordinary cohomology $H^{*}(Q_{2n-1})$ from the graph
equivariant cohomology $H^{*}(\mathcal{GQ}_{2n-1})$.

Recall the ordinary cohomology ring formulas of $Q_{m}$. According to [3,
Appendix C.3.4], we have

$\displaystyle H^{*}(Q_{2n})\simeq\mathbb{Z}[c,x]/\langle
c^{n+1}-2cx,x^{2}-\delta(n)c^{n}x\rangle,$ $\displaystyle
H^{*}(Q_{2n-1})\simeq\mathbb{Z}[c,x]/\langle c^{n}-2x,x^{2}\rangle,$

where $\deg c=2$, $\deg x=2n$ and

$\displaystyle\delta(n)=\left\\{\begin{array}[]{ll}0&n\equiv 1\mod 2\\\
1&n\equiv 0\mod 2\end{array}\right.$

Since $H^{odd}(Q_{2n-1})=0$, the odd-dimensional complex quadric $Q_{2n-1}$ is
the equivariantly formal GKM manifold (see [8]). Therefore, its ordinary
cohomology is isomorphic to the quotient of $H_{T}^{*}(Q_{2n-1})$ by the ideal
generated by $\pi^{*}(\alpha_{1}),\ldots,\pi^{*}(\alpha_{n})$, where
$\alpha_{1},\ldots,\alpha_{n}\in H^{2}(BT^{n})$ are generators and
$\pi^{*}:H^{*}(BT^{n})\to H_{T}^{*}(Q_{2n-1})$ is the induced (injective)
homomorphism from the projection $\pi:ET^{n}\times_{T^{n}}Q_{2n-1}\to BT^{n}$.
Recall that the equivariant cohomology $H_{T}^{*}(Q_{2n-1})$ is defined by the
ordinary cohomology of the Borel construction $ET^{n}\times_{T^{n}}Q_{2n-1}$.
Thus, using Theorem 5.1 and Proposition 3.13, we also have the ordinary
cohomology of $Q_{2n-1}$ as follows.

######  Corollary 6.1.

The ordinary cohomology $H^{*}(Q_{2n-1})$ is isomorphic to
$\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\mathcal{J}$, where $\mathcal{J}$ is
generated by

$\displaystyle M_{i}-Q$

for $i=1,\ldots,n$.

To reduce the relations of $\mathbb{Z}[\mathbf{M},\mathbf{D},Q]$, we first
show the following lemma.

######  Lemma 6.2.

If $K,H\subset\mathcal{V}_{2n-1}$ are the subsets such that $|K|=|H|=n$ and
$\\{i,\overline{i}\\}\not\not\subset K,H$ for every $i\in\mathcal{V}_{2n-1}$,
then, there is the following formula in
$\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\mathcal{J}$:

$\Delta_{K}=\Delta_{H}.$ (6.42)

Furthermore, if $L\subset\mathcal{V}_{2n-1}$ satisfies $|L|=n-l$ for some
$1\leq l\leq n$ and $\\{i,\overline{i}\\}\not\not\subset L$ for every
$i\in\mathcal{V}_{2n-1}$, then

$\displaystyle\Delta_{L}=\Delta_{K}Q^{l}.$

###### Proof.

By the definition of $\mathcal{J}$ and Lemma 4.2 (Relation 2), in
$\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\mathcal{J}$, we have

$\displaystyle Q=M_{1}=M_{2}=\cdots=M_{n}=M_{n+1}=\cdots=M_{2n}.$ (6.43)

Moreover, by using Lemma 4.3 (Relation 3), we have that

$\displaystyle 2\Delta_{K}=\prod_{j\in K^{c}}M_{j}=Q^{n}=\prod_{j\in
H^{c}}M_{j}=2\Delta_{H}.$

This establishes the 1st statement.

The 2nd statement follows by using (6.42), (6.43) and Relation 4 repeatedly. ∎

By Remark 5.3, Lemma 6.2 and (6.43), every element
$f\in\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\mathcal{J}$ can be written as

$\displaystyle
f=k_{0}+k_{1}Q+\cdots+k_{n-1}Q^{n-1}+k_{n}\Delta_{K}+k_{n+1}\Delta_{K}Q+\cdots+k_{2n-1}\Delta_{K}Q^{n-1},$

for some unique $k_{0},\ldots,k_{2n-1}\in\mathbb{Z}$. Therefore, there is the
following isomorphism as a $\mathbb{Z}$-module:

$\displaystyle\mathbb{Z}[\mathcal{GQ}_{2n-1}]/\mathcal{J}\simeq\mathbb{Z}\oplus\mathbb{Z}Q+\cdots\oplus\mathbb{Z}Q^{n-1}\oplus\mathbb{Z}\Delta_{K}\oplus\mathbb{Z}\Delta_{K}Q\oplus\cdots\oplus\mathbb{Z}\Delta_{K}Q^{n-1}$
$\displaystyle\simeq
H^{*}(Q_{2n-1})\simeq\mathbb{Z}\oplus\mathbb{Z}c+\cdots\oplus\mathbb{Z}c^{n-1}\oplus\mathbb{Z}x\oplus\mathbb{Z}xc\oplus\cdots\oplus\mathbb{Z}xc^{n-1}.$

By Lemma 4.3 (Relation 3), we have that

$\displaystyle Q^{n}=\prod_{i\in K^{c}}M_{i}^{n}=2\Delta_{K}.$

Together with Lemma 4.1 (Relation 1), we also have that

$\displaystyle
0=\prod_{i\in\mathcal{V}_{2n-1}}M_{i}=Q^{2n}=4\Delta_{K}\Delta_{K^{c}}=4\Delta_{K}^{2}.$

This shows that $\Delta_{K}^{2}=0$. Consequently we have the ordinary
cohomology.

######  Proposition 6.3.

There is the following isomorphism :

$\displaystyle H^{*}(Q_{2n-1})\simeq\mathbb{Z}[Q,\Delta_{K}]/\langle
Q^{n}-2\Delta_{K},\Delta_{K}^{2}\rangle,$

where $\deg Q=2$ and $\deg\Delta_{K}=2n$.

##  7\. GKM graphs of even- and odd-dimensional complex quadrics

By Remark 3.8, there is a GKM subgraph $\mathcal{GQ}_{2n-2}$ of
$\mathcal{GQ}_{2n-1}$. This induces a homomorphism (see Section 7.2 for
details)

$\displaystyle H^{*}(\mathcal{GQ}_{2n-1})\to H^{*}(\mathcal{GQ}_{2n-2}).$

In this section, we compare the graph equivariant cohomologies induced by
even- and odd-dimensional complex quadrics.

We first recall the restricted $T^{n}$-action of (1.1) on the even-dimensional
complex quadric

$\displaystyle
Q_{2n-2}:=\Big{\\{}[z_{1}:\cdots:z_{2n}:0]\in\mathbb{CP}^{2n}\leavevmode\nobreak\
|\leavevmode\nobreak\ \sum_{i=1}^{n}z_{i}z_{2n+1-i}=0\Big{\\}}\subset
Q_{2n-1}.$

By restricting (1.1) to $Q_{2n-2}$ we have the following action

$[z_{1}:\cdots:z_{2n}:0]\cdot(t_{1},\ldots,t_{n}):=[z_{1}t_{1}:\cdots:z_{n}t_{n}:t_{n}^{-1}z_{n+1}:\cdots:t_{1}^{-1}z_{2n}:0].$
(7.44)

We use the notation $(Q_{2n-2},T^{n})$ to denote the $T^{n}$-action on
$Q_{2n-2}$ defined by (7.44). It is easy to check that the kernel of
$(Q_{2n-2},T^{n})$, i.e., the intersection of isotropy subgroups of all
elements in $Q_{2n-2}$, is
$\Delta(\mathbb{Z}_{2})=\\{(1,\ldots,1),(-1,\ldots,-1)\\}\subset T^{n}$.
Therefore, this action is not effective.

###  7.1. The GKM graph $\mathcal{GQ}_{2n-2}$ induced from the non-effective
$T^{n}$-action on $Q_{2n-2}$

In this subsection, we compute the GKM graph of $(Q_{2n-2},T^{n})$ (cf. the
GKM graph of $Q_{2n-2}$ with the effective torus $T^{n}$-actions in [15,
Section 2.1]). Let $\Gamma_{2n-2}\subset\Gamma_{2n-1}$ be the subgraph
consisting of:

Vertices:

$\mathcal{V}_{2n-2}:=\mathcal{V}_{2n-1}$;

Edges:

$\mathcal{E}_{2n-2}:=\mathcal{E}_{2n-1}\setminus\\{i\overline{i}\ |\ 1\leq
i\leq n\\}$.

The GKM graph $\mathcal{GQ}_{2n-2}:=(\Gamma_{2n-2},\alpha^{\prime})$ is
defined by the restricted axial function

$\displaystyle\alpha^{\prime}:=\alpha|_{\mathcal{E}_{2n-2}}:\mathcal{E}_{2n-2}\to
H^{2}(BT^{n}),$

where $\alpha:\mathcal{E}_{2n-1}\to H^{2}(BT^{n})$ is the axial function of
$\mathcal{GQ}_{2n-1}:=(\Gamma_{2n-1},\alpha)$ (see Section 2.1). More
precisely, it is defined by the following equations:

  * •

$\alpha^{\prime}(ij)=\alpha(ij)=-\alpha_{i}+\alpha_{j}$ for $1\leq i\neq j\leq
n$;

  * •

$\alpha^{\prime}(i\overline{j})=\alpha(i\overline{j})=-\alpha_{i}-\alpha_{j}$
for $1\leq i\neq j\leq n$;

  * •

$\alpha^{\prime}(\overline{i}j)=\alpha(\overline{i}j)=\alpha_{i}+\alpha_{j}$
for $1\leq i\neq j\leq n$;

  * •

$\alpha^{\prime}(\overline{i}\overline{j})=\alpha(\overline{i}\overline{j})=\alpha_{i}-\alpha_{j}$
for $1\leq i\neq j\leq n$.

Since $(\Gamma_{2n-2},\alpha^{\prime})$ is defined by the restriction of
$(\Gamma_{2n-1},\alpha)$, there is an inclusion from $\mathcal{GQ}_{2n-2}$ to
$\mathcal{GQ}_{2n-1}$. We denote it by

$\displaystyle\iota:\mathcal{GQ}_{2n-2}\hookrightarrow\mathcal{GQ}_{2n-1}.$

$1$$2$$\overline{1}$$\overline{2}$$-\alpha_{1}+\alpha_{2}$$\alpha_{1}-\alpha_{2}$$\alpha_{1}+\alpha_{2}$$-\alpha_{1}-\alpha_{2}$$\stackrel{{\scriptstyle\iota}}{{\hookrightarrow}}$$1$$2$$\overline{1}$$\overline{2}$$-\alpha_{1}+\alpha_{2}$$-\alpha_{1}$$\alpha_{1}-\alpha_{2}$$\alpha_{1}+\alpha_{2}$$-\alpha_{2}$$-\alpha_{1}-\alpha_{2}$
Figure 7. The inclusion of GKM graphs
$\iota:\mathcal{GQ}_{2}\hookrightarrow\mathcal{GQ}_{3}$ for $n=2$, where
$\alpha_{1}$ and $\alpha_{2}$ are generators in $H^{2}(BT^{2})$.

###  7.2. The equivariant cohomology of $H^{*}_{T}(Q_{2n-2})$

In [15], the equivariant cohomology of $Q_{2n-2}$ with effective
$T^{n}$-actions is computed using the graph equivariant cohomology of its GKM
graph. In this subsection, in Lemma 7.2, we prove that the equivariant
cohomology of the non-effective $(Q_{2n-2},T^{n})$ is also computed using the
graph equivariant cohomology of $\mathcal{GQ}_{2n-2}$.

The following fact from [7, Corollary 2.2] is the essential result to prove
it.

######  Theorem 7.1 (Franz-Puppe).

Let $M$ be a $T$-space such that $H^{*}_{T}(X)$ is free over $H^{*}(BT)$. If
the isotropy group of each $x\not\in X_{1}:=\\{x\in X\ |\ \dim T(x)\leq 1\\}$
is contained in a proper subtorus of $T$, then the following sequence is
exact:

$\displaystyle 0\longrightarrow H_{T}^{*}(X;\mathbb{Z})\stackrel{{\scriptstyle
i^{*}}}{{\longrightarrow}}H_{T}^{*}(X^{T};\mathbb{Z})\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}H_{T}^{*+1}(X_{1},X^{T};\mathbb{Z}),$

where the homomorphism $i^{*}$ is induced by the inclusion $i:X^{T}\to X$, and
the homomorphism $\delta$ is the connecting homomorphism of the long exact
sequence of cohomology of the pair $(X_{1},X^{T})$.

By using this theorem, we obtain the following lemma.

######  Lemma 7.2.

The equivariant cohomology $H_{T^{n}}^{*}(Q_{2n-2})$ is isomorphic to the
graph equivariant cohomology $H^{*}(\mathcal{GQ}_{2n-2})$, i.e.,

$\displaystyle H_{T^{n}}^{*}(Q_{2n-2})$ $\displaystyle\simeq
H^{*}(\mathcal{GQ}_{2n-2})$ $\displaystyle:=\\{f:\mathcal{V}_{2n-2}\to
H^{*}(BT)\ |\ f(p)-f(q)\equiv 0\mod\alpha(pq),\ \text{for all
$pq\in\mathcal{E}_{2n-2}$}\\}.$ (7.45)

###### Proof.

Since $H^{odd}(Q_{2n-2})=0$ (see Section 6), it follows from the spectral
sequence argument that $H^{*}_{T}(Q_{2n-2})$ is free over $H^{*}(BT)$.
Moreover, it is easy to check that an element $x\in Q_{2n-2}$ with its orbit
$T(x)\simeq T^{n-k}$ ($k\geq n-2$) has the isotropy group $T_{x}\simeq
T^{k}\times\mathbb{Z}_{2}$. Therefore, there is the proper subtorus $K\subset
T^{n}$ which is isomorphic to $T^{k+1}$ such that $T_{x}\subset K$. For
example, the point $x=[z_{1}:\ldots:z_{n}:0:\ldots:0]$ with $z_{i}\not=0$ for
$i=1,\ldots,n$ has the isotropy subgroup $\Delta(\mathbb{Z}_{2})$, and
$\Delta(\mathbb{Z}_{2})$ is contained in the proper subtorus
$K=\\{(t,\ldots,t)\ |\ t\in S^{1}\\}\subset T^{n}$, where $K\simeq T^{1}$.
This shows that $(Q_{2n-2},T^{n})$ satisfies the assumptions of Theorem 7.1.
Therefore, we have

$\displaystyle H^{*}_{T^{n}}(Q_{2n-2})\simeq{\rm Im}\,i^{*}=\ker\delta.$

Because $\delta$ is the connecting homomorphism of the long exact sequence of
$(X_{1},X^{T})$ for $X=Q_{2n-2}$, we have

$\displaystyle H^{*}_{T^{n}}(Q_{2n-2})\simeq\ker\delta={\rm Im}\,j^{*},$

where $j:X^{T}\to X_{1}$ is the inclusion from the fixed points $X^{T}$ to
$X_{1}$ and $j^{*}:H_{T}^{*}(X_{1})\to H_{T}^{*}(X^{T})$ is its induced
homomorphism. Using the method with Mayer-Vietoris sequence demonstrated in
e.g. [1, 5] (also see the proof of [2, Theorem 2.9]), we have that

$\displaystyle{\rm Im}\,j^{*}\simeq\\{f:\mathcal{V}_{2n-2}\to H^{*}(BT)\ |\
f(p)-f(q)\equiv 0\mod\alpha(pq),\ \text{for all
$pq\in\mathcal{E}_{2n-2}$}\\}.$

This completes the proof. ∎

###  7.3. The graph equivariant cohomology of $\mathcal{GQ}_{2n-2}$ and the
induced homomorphism $\iota^{*}$

By (3.9) and (7.45), the inclusion
$\iota:\mathcal{GQ}_{2n-2}\hookrightarrow\mathcal{GQ}_{2n-1}$ induces the
following homomorphism:

$\displaystyle\iota^{*}:H^{*}(\mathcal{GQ}_{2n-1})\to
H^{*}(\mathcal{GQ}_{2n-2})$

such that

$\displaystyle\iota^{*}(f):=f^{\prime},$

where $f^{\prime}(p):=f(p)$ for all
$p\in\mathcal{V}_{2n-2}=\mathcal{V}_{2n-1}$. In fact, since $f$ satisfies the
congruence relations for every edge in $\mathcal{E}_{2n-1}$, its restriction
$f^{\prime}$ also satisfies the congruence relations for every edge in
$\mathcal{E}_{2n-2}$; therefore, $f^{\prime}\in H^{*}(\mathcal{GQ}_{2n-2})$.
We call $\iota^{*}$ an induced homomorphism from $\iota$.

Note that the comcepts described above are also defined for any GKM graph
$\mathcal{G}$ and its GKM subgraph $\mathcal{G}^{\prime}$. That is, if there
is an inclusion of a GKM subgraph
$\iota:\mathcal{G}^{\prime}\hookrightarrow\mathcal{G}$ then there is the
induced homomorphism $\iota^{*}:H^{*}(\mathcal{G})\to
H^{*}(\mathcal{G}^{\prime})$. We have the following lemma:

######  Lemma 7.3.

Let $\mathcal{G}$ be a GKM graph. Assume that there is a GKM subgraph
$\mathcal{G}^{\prime}$ of $\mathcal{G}$ such that the set of vertices of
$\mathcal{G}^{\prime}$ coincides with that of $\mathcal{G}$. Then the induced
homomorphism $\iota^{*}:H^{*}(\mathcal{G})\to H^{*}(\mathcal{G}^{\prime})$ is
injective.

###### Proof.

Let $\mathcal{V}$ be the set of vertices of $\mathcal{G}$ and
$\mathcal{G}^{\prime}$. By the definition of graph equivariant cohomology (see
e.g. [10]), the restriction to the vertices

$\displaystyle
H^{*}(\mathcal{G})\stackrel{{\scriptstyle\iota^{*}}}{{\to}}H^{*}(\mathcal{G}^{\prime})\to\bigoplus_{v\in\mathcal{V}}H^{*}(BT)$

is injective. Therefore, the homomorphism $\iota^{*}$ is injective. ∎

Since $\mathcal{V}_{2n-2}=\mathcal{V}_{2n-1}$ for
$\iota:\mathcal{GQ}_{2n-2}\hookrightarrow\mathcal{GQ}_{2n-1}$, we have the
following corollary:

######  Corollary 7.4.

The induced homomorphism $\iota^{*}:H^{*}(\mathcal{GQ}_{2n-1})\to
H^{*}(\mathcal{GQ}_{2n-2})$ is injective.

###  7.4. The ring structure of $H^{*}(\mathcal{GQ}_{2n-2})$ and the image
${\rm Im}\leavevmode\nobreak\ \iota^{*}$

In this subsection, we compute $H^{*}(\mathcal{GQ}_{2n-2})$, and then
determine the image of $\iota^{*}:H^{*}(\mathcal{GQ}_{2n-1})\to
H^{*}(\mathcal{GQ}_{2n-2})$.

For $H^{*}(\mathcal{GQ}_{2n-2})$, the following elements will serve as
generators. Some of them can be defined by using elements in
$H^{*}(\mathcal{GQ}_{2n-1})$ (i.e., the functions
$\mathcal{V}_{2n-1}(=\mathcal{V}_{2n-2})\to H^{*}(BT^{n})$ that satisfies
(3.9)).

Generator 1:

$\mathcal{M}:=\\{M^{\prime}_{v}:\mathcal{V}_{2n-2}\to H^{2}(BT^{n})\ |\
v\in\mathcal{V}_{2n-2}\\}$ where $M_{v}^{\prime}:=\iota^{*}(M_{v})$. More
precisely, by (3.12),

$M^{\prime}_{v}(j)=\begin{cases}0&\text{if $j=v$}\\\
\alpha(jv)=-\alpha_{j}+\alpha_{v}&\text{if $j\not=v,\overline{v}$}\\\
2\alpha_{v}&\text{if $j=\overline{v}$}\end{cases}$

Generator 2:

$\mathcal{D}:=\\{\Delta^{\prime}_{K}:\mathcal{V}_{2n-2}\to
H^{4n-2|K|-2}(BT^{n})\ |\ K\subset{\mathcal{V}}_{2n-2},\ \text{
$\\{i,\overline{i}\\}\not\subset K$ $\forall i\in{\mathcal{V}}_{2n-2}$}\\}$
where

$\Delta^{\prime}_{K}(j)=\begin{cases}\displaystyle\prod_{k\notin
K\cup\\{\overline{j}\\}}\alpha(jk)&\text{if $j\in K$},\\\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 0&\text{if
$j\notin K$}.\end{cases}$ (7.46)

Generator 3:

$X:=\iota^{*}(Q):\mathcal{V}_{2n-2}\to H^{2}(BT^{n})$ such that
$X(i):=Q(i)=-\alpha_{i}$ for $1\leq i\leq 2n$.

######  Remark 7.5.

The element $X$ also appears in the graph equivariant cohomology of the GKM
graph in [15, Section 4], which is induced from the effective $T^{n}$-action
on $Q_{2n-2}$. In [15], $X$ is not needed as a generator because
$M_{v}+M_{\overline{v}}=X$ holds (see Relation 2 of [15]). However, in the GKM
graph induced from the non-effective action, for any
$v\in{\mathcal{V}}_{2n-2}$, the following equality holds:

$M^{\prime}_{v}+M^{\prime}_{\overline{v}}=2X.$ (7.47)

Therefore, $X$ is needed as a generator in the non-effective case because of
(7.47).

######  Remark 7.6.

Let $K\subset{\mathcal{V}}_{2n-2}$ be a subset such that
$\\{i,\overline{i}\\}\not\subset K$ for any $i\in{\mathcal{V}}_{2n-2}$. We
note that $\Delta_{K}^{\prime}\in H^{4n-2|K|-2}(\mathcal{GQ}_{2n-2})$ is not
induced from $\Delta_{K}\in H^{4n-2|K|}(\mathcal{GQ}_{2n-1})$ (see Definition
3.14), i.e., $\iota^{*}(\Delta_{K})\not=\Delta^{\prime}_{K}$, because their
degrees are different (see Figure 8).

Let $\mathbb{Z}[\mathcal{M},\mathcal{D},X]$ be the polynomial ring generated
by all elements in $\mathcal{M}$, $\mathcal{D}$ and $X$. Let $\mathcal{I}$ be
the ideal in $\mathbb{Z}[\mathcal{M},\mathcal{D},X]$ generated by the
following four types of relations:

Relation 1:

$\prod_{\cap J=\emptyset}G_{J}$ for $G_{J}\in\mathcal{M}\sqcup\mathcal{D}$,
where $G_{J}$ is defined similarly to (4.18);

Relation 2:

$M^{\prime}_{v}+M^{\prime}_{\overline{v}}-2X$ for any
$v\in{\mathcal{V}}_{2n-1}$;

Relation 3:

$\prod_{i\in
I}M^{\prime}_{i}-(\Delta^{\prime}_{(I\cup\\{a\\})^{c}}+\Delta^{\prime}_{(I\cup\\{\overline{a}\\})^{c}})$
for every subset $I\subset\mathcal{V}_{2n-2}$ such that $|I|=n-1$ and there
exists a unique pair $\\{a,\overline{a}\\}\subset I^{c}$;

Relation 4:

$\Delta^{\prime}_{K}\cdot M^{\prime}_{i}-\Delta^{\prime}_{K\setminus\\{i\\}}$
for $\\{i\\}\subsetneq K$ and $\\{j,\overline{j}\\}\not\subset K$ for all
$j\in{\mathcal{V}}_{2n-2}$.

######  Remark 7.7.

Relations 1, 3 and 4 as descrived above coincide with those appearing in the
graph equivariant cohomology of the GKM graph induced from the effective torus
action in [15]. The only difference is Relation 2.

Using a similar proof to that of Theorem 5.1 (or the main theorem of [15]), we
have the following theorem:

######  Theorem 7.8.

For the GKM graph $\mathcal{GQ}_{2n-2}$, the following isomorphism holds:

$\displaystyle
H^{*}(\mathcal{GQ}_{2n-2})\simeq\mathbb{Z}[\mathcal{M},\mathcal{D},X]/\mathcal{I}.$

Moreover, the following proposition holds:

######  Proposition 7.9.

The injective homomorphism $\iota^{*}:H^{*}(\mathcal{GQ}_{2n-1})\to
H^{*}(\mathcal{GQ}_{2n-2})$ is induced by the following correspondence of
generators:

$\displaystyle\iota^{*}:M_{v}\mapsto M^{\prime}_{v};$
$\displaystyle\iota^{*}:Q\mapsto X;$ $\displaystyle\iota^{*}:\Delta_{J}\mapsto
X\cdot\Delta^{\prime}_{J}.$

###### Proof.

The induced homomorphism $\iota^{*}$ is injective by Corollary 7.4.

The first and second correspondences in the statement follow directly from the
definitions of generators and $\iota^{*}$. The third correspondence is
established by comparing the generators from Theorem 5.1 and Theorem 7.8. ∎

######  Example 7.10.

In Figure 8, we compare two elements: $\Delta^{\prime}_{1,2,3}\in
H^{4}(\mathcal{GQ}_{4})$ and $\Delta_{1,2,3}\in H^{6}(\mathcal{GQ}_{5})$. More
precisely: The left element $\Delta^{\prime}_{1,2,3}\in
H^{4}(\mathcal{GQ}_{4})$ is defined by:

$\displaystyle\Delta^{\prime}_{1,2,3}(1)=\alpha(14)\alpha(15),\Delta^{\prime}_{1,2,3}(2)=\alpha(24)\alpha(26),\Delta^{\prime}_{1,2,3}(3)=\alpha(35)\alpha(36),$
$\displaystyle\Delta^{\prime}_{1,2,3}(4)=\Delta^{\prime}_{1,2,3}(5)=\Delta^{\prime}_{1,2,3}(6)=0.$

The right element $\Delta_{1,2,3}\in H^{6}(\mathcal{GQ}_{5})$ is defined by:

$\displaystyle\Delta_{1,2,3}(1)=\alpha(14)\alpha(15)\alpha(16),\Delta_{1,2,3}(2)=\alpha(24)\alpha(25)\alpha(26),\Delta_{1,2,3}(3)=\alpha(34)\alpha(35)\alpha(36),$
$\displaystyle\Delta_{1,2,3}(4)=\Delta_{1,2,3}(5)=\Delta_{1,2,3}(6)=0.$

It is straightforward to verify that
$\iota^{*}(\Delta_{1,2,3})=X\Delta^{\prime}_{1,2,3}$ (see the third
correspondence in Proposition 7.9).

$6$$3$$2$$1$$4$$5$$\stackrel{{\scriptstyle\iota^{*}}}{{\longleftarrow}}$$6$$3$$2$$1$$4$$5$
Figure 8. $\Delta^{\prime}_{1,2,3}\in H^{4}(\mathcal{GQ}_{4})$ (left), and
$\Delta_{1,2,3}\in H^{6}(\mathcal{GQ}_{5})$ (right).

######  Remark 7.11.

Given the embedding $\iota:Q_{2n-2}\hookrightarrow Q_{2n-1}$, we obtain the
induced homomorphism on ordinary cohomology $\iota^{*}:H^{*}(Q_{2n-1})\to
H^{*}(Q_{2n-2})$. Note that $\dim Q_{2n-1}=4n-2>4n-4=\dim Q_{2n-2}$;
therefore, $H^{4n-2}(Q_{2n-1})\simeq\mathbb{Z}$ and $H^{4n-2}(Q_{2n-2})=0$.
Thus, $\iota^{*}:H^{*}(Q_{2n-1})\to H^{*}(Q_{2n-2})$ is not injective.
However, by Lemma 7.2 and Proposition 7.9, the induced homomorphism on
equivariant cohomology $\iota^{*}_{T}:H^{*}_{T}(Q_{2n-1})\to
H^{*}_{T}(Q_{2n-2})$ is injective.

##  8\. GKM description for non-effective $T^{1}$-actions on $\mathbb{CP}^{1}$

In [15], we compute the effective $T^{n}$-ation on $Q_{2n-2}$. According to
the main theorem of [15], the generator $X$ that appears in
$H^{*}(\mathcal{GQ}_{2n-2})$ in Theorem 7.8 is not needed (see also Remark
7.5).

In this final section, we observe a similar phenomenon in the equivariant
cohomology of non-effective torus actions by comparing the equivariant
cohomology of $T^{1}$-actions on $Q_{1}\simeq\mathbb{CP}^{1}\simeq S^{2}$,
i.e., we give details of the research announcement [14, Appendix A]. For
convenience, we denote $T^{1}$ as $T$.

Since the Euler number satisfies $\chi(M)=\chi(M^{T})$ (see [13, Theorem
41.1]), we have $\chi((\mathbb{CP}^{1})^{T})=\chi(S^{2})=2$. Therefore, for
every non-trivial $T$-action on $\mathbb{CP}^{1}$, there are exactly two fixed
points, denote by $(\mathbb{CP}^{1})^{T}=\\{p,q\\}$.

Using the differentiable slice theorem, the $T$-representations around
$T_{p}\mathbb{CP}^{1}$ and $T_{q}\mathbb{CP}^{1}$ are rotations of order $n$,
up to sign, for $n\in\mathbb{Z}$. More precisely, for every $T$-action on
$\mathbb{CP}^{1}$, there exists a non-negative integer $n$ such that the
action is weak equivariantly diffeomorphic (i.e., equivariantly diffeomorphic
up to an automorphism on $T$) to the following action:

$\displaystyle t\cdot[z_{0}:z_{1}]=[z_{0}:t^{n}z_{1}],$

where $t\in T$ and $[z_{0}:z_{1}]\in\mathbb{CP}^{1}$. We denote this action by
$\varphi_{n}$ and the equivariant cohomology $H_{T}^{*}(\mathbb{CP}^{1})$ with
respect to this action by $H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})$.

######  Remark 8.1.

Note that the complex quadric $Q_{1}\simeq SO(3)/SO(2)$, with the maximal
torus $T\leavevmode\nobreak\ (\subset SO(3))$-action, is equivariantly
diffeomorphic to $\mathbb{CP}^{1}$ with the standard effective $T$-action
(i.e., $\varphi_{1}$) because there is no non-trivial center in $SO(3)$. On
the other hand, $\mathbb{CP}^{1}(\simeq Q_{1})$ is also diffeomorphic to
$SU(2)/S(U(1)\times U(1))$, but the maximal torus $T\leavevmode\nobreak\
(\subset SU(2))$-action (i.e., $\varphi_{2}$) has a non-trivial center
$\mathbb{Z}_{2}$.

For $n=0$, $\varphi_{0}$ represents the trivial $T$-action. Therefore, we
have:

$\displaystyle H_{\varphi_{0}}^{*}(\mathbb{CP}^{1})\simeq
H^{*}(\mathbb{CP}^{1})\otimes H^{*}(BT)\simeq\mathbb{Z}[x,\alpha]/\langle
x^{2}\rangle,$

where $x$ is a generator of $H^{*}(\mathbb{CP}^{1})$ and $\alpha$ is a
generator of $H^{*}(BT)$. In [12, Remark 4.5], we also show that:

$\displaystyle
H_{\varphi_{1}}^{*}(\mathbb{CP}^{1})\simeq\mathbb{Z}[\tau_{1},\tau_{2}]/\langle\tau_{1}\tau_{2}\rangle\not\simeq
H_{\varphi_{2}}^{*}(\mathbb{CP}^{1})\simeq\mathbb{Z}[u,v]/\langle
u^{2}-v^{2}\rangle,$

where $\tau_{1},\tau_{2},u,v$ are elements of degree two. We will generalize
this to arbitrary $\varphi_{n}$ for all $n\geq 0$.

The Mayer-Vietoris exact sequence of the equivariant cohomology is given by:

$\displaystyle\cdots\longrightarrow
H_{\varphi_{n}}^{j}(\mathbb{CP}^{1})\longrightarrow
H_{\varphi_{n}}^{j}(U_{0})\oplus H_{\varphi_{n}}^{j}(U_{1})\longrightarrow
H_{\varphi_{n}}^{j}(U_{0}\cap U_{1})\longrightarrow
H_{\varphi_{n}}^{j+1}(\mathbb{CP}^{1})\longrightarrow\cdots$

where $U_{0}\simeq\\{[z_{0}:1]\ |\ z_{0}\in\mathbb{C}\\}$ is an invariant open
neighborhood of the fixed point $[0:1]$, $U_{1}\simeq\\{[1:z_{1}]\ |\
z_{1}\in\mathbb{C}\\}$ is an invariant open neighborhood of the fixed point
$[1:0]$, and $U_{0}\cap U_{1}\simeq\\{[z_{0}:z_{1}]\ |\
z_{0}z_{1}\not=0\\}\simeq\mathbb{C}^{*}$. Here, $U_{i}$ is equivariantly
contractible to a point and $U_{0}\cap U_{1}$ is equivariant deformation
retractable to the great circle $S^{1}$. Moreover, since
$H^{*}(BT)\simeq\mathbb{Z}[\alpha]$ for $\deg\alpha=2$ and
$H^{odd}(\mathbb{CP}^{1})=0$, it is well-known that
$H^{*}_{\varphi_{n}}(\mathbb{CP}^{1})\simeq H^{*}(BT)\otimes
H^{*}(\mathbb{CP}^{1})$ as a module. Hence, this sequence is isomorphic to:

$\displaystyle 0\to H_{T}^{2j-1}(S^{1})\to H_{T}^{2j}(\mathbb{CP}^{1})\to
H^{2j}(BT)\oplus H^{2j}(BT)\to H_{T}^{2j}(S^{1})\to 0.$ (8.48)

Note that $H_{T}^{*}(S^{1})$ is the equivariant cohomology of the $n$-times
rotated action of $T^{1}$ on $S^{1}$. Furthermore, the restricted
$T^{1}$-action from $\varphi_{n}$ on $S^{1}$ has the kernel $\mathbb{Z}_{n}$
for $n\geq 2$, $\\{e\\}$ for $n=1$, and $T^{1}$ for $n=0$.

First, consider for $n\geq 1$. By [11, Example 3.41], the ring structure of
the infinite lens space $B\mathbb{Z}_{n}$ is given by:

$\displaystyle
H^{*}(B\mathbb{Z}_{n};\mathbb{Z})\simeq\mathbb{Z}[\alpha]/\langle
n\alpha\rangle$

for $\deg\alpha=2$. Thus, we have:

$\displaystyle H_{T}^{*}(S^{1})=H^{*}(ET\times_{T}S^{1})\simeq
H^{*}(ET/\mathbb{Z}_{n})\simeq
H^{*}(B\mathbb{Z}_{n})\simeq\left\\{\begin{array}[]{ll}\mathbb{Z}&*=0\\\
\mathbb{Z}_{n}&*=2j,j>0\\\ 0&*=2j-1\end{array}\right.$

(Note: $\mathbb{Z}_{1}$ is interpreted as $0$). Therefore, by the Mayer-
Vietoris sequence (8.48), we get the following short exact sequence for $j>0$
and $n\geq 1$:

$\displaystyle 0\longrightarrow
H_{\varphi_{n}}^{2j}(\mathbb{CP}^{1})\longrightarrow\mathbb{Z}\alpha^{j}\oplus\mathbb{Z}\alpha^{j}\longrightarrow\mathbb{Z}_{n}\longrightarrow
0.$

On the other hand, for $n=0$, we have:

$\displaystyle H_{T}^{*}(S^{1})=H^{*}(BT\times S^{1})=H^{*}(BT)\otimes
H^{*}(S^{1})\simeq\mathbb{Z}\quad\text{for all $*\geq 0$},$

and therefore, by the Mayer-Vietoris sequence (8.48), we obtain:

$\displaystyle 0\longrightarrow\mathbb{Z}\alpha^{j-1}y\longrightarrow
H_{\varphi_{0}}^{2j}(\mathbb{CP}^{1})\longrightarrow\mathbb{Z}\alpha^{j}\oplus\mathbb{Z}\alpha^{j}\longrightarrow\mathbb{Z}\alpha^{j}\longrightarrow
0,$

where $y$ is the generator of $H^{*}(S^{1})$. Hence, by the definition of the
Mayer-Vietoris exact sequence, for all $n\geq 1$, we have:

$\displaystyle H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})$
$\displaystyle\simeq\\{f\oplus g\in\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\
|\ f_{0}=g_{0},f_{j}-g_{j}\equiv 0\mod n\\}$ $\displaystyle\simeq\\{f\oplus
g\in\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\ |\ f-g\equiv 0\mod
n\alpha\\},$

where $f=\sum_{i=1}^{k_{1}}f_{i}\alpha^{i}$ and
$g=\sum_{i=1}^{k_{2}}g_{i}\alpha^{i}$. For $n=0$, we have:

$\displaystyle H_{\varphi_{0}}^{*}(\mathbb{CP}^{1})$
$\displaystyle\simeq\\{h\oplus f\oplus
g\in\mathbb{Z}[\alpha]y\oplus\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\ |\
f=g\\}$ (8.49) $\displaystyle\simeq\mathbb{Z}[x,\alpha]/\langle x^{2}\rangle,$
(8.50)

where $x\in H^{2}_{T}(\mathbb{CP}^{1})$ is the image of the generator $y\in
H^{1}(S^{1})$ under the connecting homomorpshim $H^{1}(S^{1})\simeq
H^{1}_{T}(S^{1})\to H^{2}_{T}(\mathbb{CP}^{1})$.

######  Remark 8.2.

Note that for $n=1$, this description corresponds to the GKM description in
the usual sense. Specifically:

$\displaystyle H_{\varphi_{1}}^{*}(\mathbb{CP}^{1})\simeq\\{f\oplus
g\in\mathbb{Z}[\alpha]\oplus\mathbb{Z}[\alpha]\ |\ f-g\equiv 0\mod\alpha\\}.$

Figure 9 illustrates the GKM graph which corresponds to $\varphi_{n}$ for
$n\geq 1$. Note that the trivial action, $\varphi_{0}$, is not a GKM manifold
by definition.

$n\alpha$$-n\alpha$$p$$q$ Figure 9. The GKM graph of $\varphi_{n}$ for $n\geq
1$ is depicted, with the fixed points $p=[1:0]$ and $q=[0:1]$. Note that the
element $\alpha\in\mathfrak{t}^{*}\simeq\mathbb{R}$ can be regarded as the
generator of the character lattice
$\mathfrak{t}^{*}_{\mathbb{Z}}\simeq\mathbb{Z}\simeq H^{2}(BT^{1})$.

In summary, we have the following GKM description for $\varphi_{n}$.

######  Theorem 8.3 (GKM description for non-effective torus action on
$\mathbb{CP}^{1}$).

For every non-trivial $T^{1}$-action on $\mathbb{CP}^{1}$, there is the
following ring isomorphism:

$\displaystyle
H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})\simeq\\{f:\\{p,q\\}\to\mathbb{Z}[\alpha]\
|\ f(p)-f(q)\equiv 0\mod n\alpha\\},$

where $\\{p,q\\}$ denotes the set of fixed points in $\mathbb{CP}^{1}$ and
$n\geq 1$.

We can also get the ring structure as follows:

######  Theorem 8.4.

For $n\geq 0$, the ring structure of the equivariant cohomology
$H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})$ can be expressed as follows:

$\displaystyle
H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})\simeq\mathbb{Z}[\tau_{p},\tau_{q},\alpha]/\langle\tau_{p}\tau_{q},n\alpha-\tau_{p}+\tau_{q}\rangle,$

where

  * •

$\tau_{p},\tau_{q}$ are the equivariant Thom classes associated with the fixed
points $p=[1:0]$ and $q=[0:1]$, respectively.

  * •

$\alpha$ is the pull-back of the generator of
$H^{*}(BT)\simeq\mathbb{Z}[\alpha]$.

###### Proof.

For $n=0$, the statement follows from (8.50). Assuming $n\geq 1$, we need to
show the isomorphism:

$\displaystyle GKM_{\varphi_{n}}(\mathbb{CP}^{1})$
$\displaystyle:=\\{f:\\{p,q\\}\to\mathbb{Z}[\alpha]\ |\ f(p)-f(q)\equiv 0\mod
n\alpha\\}$
$\displaystyle\simeq\mathbb{Z}[\tau_{p},\tau_{q},\alpha]/\langle\tau_{p}\tau_{q},n\alpha-\tau_{p}+\tau_{q}\rangle.$

In $GKM_{\varphi_{n}}(\mathbb{CP}^{1})$, the equivariant Thom classes are
defined as follows:

  * •

$\tau_{p}(p)=n\alpha$ and $\tau_{p}(q)=0$;

  * •

$\tau_{q}(q)=n\alpha$ and $\tau_{q}(p)=0$;

  * •

$\alpha$ is constant, so $\alpha(p)=\alpha=\alpha(q)$.

Moreover, it is easy to check that the relations $\tau_{p}\tau_{q}=0$ and
$n\alpha=\tau_{p}+\tau_{q}$ hold in $GKM_{\varphi_{n}}(\mathbb{CP}^{1})$.
Thus, this defines the ring homomorphism

$\Phi:\mathbb{Z}[\tau_{p},\tau_{q},\alpha]/\langle\tau_{p}\tau_{q},n\alpha-\tau_{p}+\tau_{q}\rangle\to
GKM_{\varphi_{n}}(\mathbb{CP}^{1})$ (8.51)

by taking the equivariant Thom classes and $\alpha$ defined as above.

To show that the map $\Phi$ is an isomorphism, we first consider the module
structure. Since

$\displaystyle\mathbb{Z}[\tau_{p},\tau_{q},\alpha]/\langle\tau_{p}\tau_{q},n\alpha-\tau_{p}+\tau_{q}\rangle\simeq\mathbb{Z}[\tau_{p},\alpha]/\langle\tau_{p}(\tau_{p}-n\alpha)\rangle,$

we have the following module isomorphism:

$\displaystyle\mathbb{Z}[\tau_{p},\tau_{q},\alpha]/\langle\tau_{p}\tau_{q},n\alpha-\tau_{p}+\tau_{q}\rangle\simeq\bigoplus_{i=0}^{\infty}\mathbb{Z}\alpha^{i}\oplus\bigoplus_{i=0}^{\infty}\mathbb{Z}\tau_{p}\alpha^{i}.$
(8.52)

On the other hand, by the congruence relations, every element in
$GKM_{\varphi_{n}}(\mathbb{CP}^{1})$, say
$f,g:\\{p,q\\}\to\mathbb{Z}[\alpha]$, can be written as

$\displaystyle
f(p)=\sum_{i=0}^{l}r_{i}\alpha_{i}+\sum_{i=1}^{m}nk_{i}\alpha^{i},\
f(q)=\sum_{i=0}^{l}r_{i}\alpha^{i}$

or

$\displaystyle g(p)=\sum_{i=0}^{l}r_{i}\alpha^{i},\
g(q)=\sum_{i=0}^{l}r_{i}+\sum_{i=1}^{m}nk_{i}\alpha^{i},$

where $r_{i},k_{i}\in\mathbb{Z}$. Namely, we can write using
$\tau_{p},\tau_{q}=\tau_{p}-n\alpha$ and $\alpha$ as follows:

$\displaystyle\Phi\left(\sum_{i=0}^{l}r_{i}\alpha^{i}+\tau_{p}\sum_{i=0}^{m-1}k_{i+1}\alpha^{i}\right)=f$

or

$\displaystyle\Phi\left(\sum_{i=0}^{l}r_{i}\alpha^{i}+(\tau_{p}-n\alpha)\sum_{i=0}^{m-1}k_{i+1}\alpha^{i}\right)=g.$

Therefore, $\Phi$ is surjective. If $f=0$ (or $g=0$), then $r_{i}=k_{i}=0$ in
the above expressions. Therefore, $\Phi$ is injective. This establishes the
proof. ∎

$p$$q$$n\alpha$$0$$p$$q$$0$$-n\alpha$$p$$q$$\alpha$$\alpha$ Figure 10.
Visializing the generators $\tau_{p},\tau_{q}$ and $\alpha$ (from left) of
$H_{\varphi_{n}}^{*}(\mathbb{CP}^{1})$ for $n\geq 1$.

######  Remark 8.5.

In [20], Zollar also studies the GKM theory of non-effective torus actions in
a more general setting.

## Acknowledgment

S. Kuroki was supported by JSPS KAKENHI Grant Number 21K03262. B. Paul
acknowledges the Chennai Mathematical Institute and the Infosys Foundation for
their financial support. Additionally, a portion of this work was undertaken
during B. Paul’s tenure at the Indian Institute of Technology Madras, and he
expresses gratitude for their support during that time.

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