GKM manifold - survey
Shintarô Kuroki

Abstract. Continued from the previous article about GKM manifold -definition [28], in thisarticle, we survey some fundamental results in the subject. In particular, we introduce theconnection between combinatorics and equivariant topology of GKM manifolds.

1. Introduction
In the previous article [28], we introduce two deﬁnitions of GKM manifolds and the GoreskyKottwitz-MacPherson (GKM) Theorem. Because the GKM theorem may be regarded as a variantof the localization theorem of equivariant cohomology (see e.g. [5, 7]), the GKM theorem canbe applied to compute equivariant cohomology of GKM manifolds. In this article, we survey
the connection between GKM manifolds and combinatorics, and introduce how the equivariant
cohomology of GKM manifolds can be computed by using combinatorics. As a combinatorial
object, in Section 2, we introduce two classes of abstract labeled graphs, called GKM graphs
introduced by Guillemin-Zara in [21] and torus graphs introduced by Maeda-Masuda-Panov in[35]. Some of these labeled graphs are induced by using the information of torus actions on GKM
manifolds and torus manifolds (see Section 3), and convert the geometric data of torus actions
to the combinatorial data of labeled graphs. As a typical application, in Section 4 and 5, we
introduce some combinatorial formula of equivariant cohomology of torus manifolds with vanishingodd degree cohomology [37, 35], called a face ring (of induced torus graphs). The face ring may be
regarded as the generalization of the Stanley-Reisner ring which gives a combinatorial formula ofequivariant cohomology of complete non-singular toric varieties (the Danilov-Jurkiewicz theorem,see [13, 38]) and that of quasitoric manifolds (the Davis-Januszkiewicz theorem, see [9, 6]).
Note that there are many works and generalizations about the relations between equivariant
topology (or geometry) of GKM manifolds and combinatorics (e.g. [1, 14, 15, 16, 18, 19, 20,
22, 25, 26, 33, 34, 39] etc). In this article, we basically survey a part of the works of GuilleminZara [21] (the original work introducing the notion of GKM graphs), Braden-MacPherson [4](introducing the notion of sheaves of moment graphs which deﬁnes topological or geometricalinvariants systematically from GKM graphs or more general labeled graphs) and Maeda-MasudaPanov [35] (introducing the notion of torus graphs and proved the combinatorial formula of its
equivariant cohomology).
2. Abstract GKM graphWe ﬁrst prepare some notation. Let Γ = (V (Γ), E(Γ)) (or (V, E) whenever the graph Γ isclear from the context) be an (abstract) graph comprising a set V (Γ) of vertices and a set E(Γ)
of oriented edges; denote its initial vertex by i(e) and its terminal vertex by t(e). The symbole ∈ E(Γ) represents the edge e with its orientation reversed, i.e., i(e) = t(e) and t(e) = i(e). In
this article, we assume that there are no loops in E(Γ), i.e., for any e ∈ E(Γ), i(e) ̸= t(e), and Γ is
The author was partially supported by Grant-in-Aid for Young Scientists (B)15K17531, Japan Society for
Promotion of Science.
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connected and ﬁnite, i.e., V (Γ) and E(Γ) are ﬁnite sets. Put the subset of all out-going edges as
Ep (Γ)(= Ep ) = {e ∈ E(Γ) | i(e) = p} ⊂ E(Γ).
An abstract graph Γ is called an m-valent graph if |Ep | = m for all p ∈ V , where the symbol |X|
represents the cardinality of a ﬁnite set X.
Let Γ be an m-valent graph. In order to deﬁne a GKM graph, we need to introduce a label
α : E → H 2 (BT ) on edges of Γ, often denoted by (Γ, α), where BT is a classifying space of ann-dimensional torus T . Recall that the cohomology ring of BT is isomorphic to the polynomialring
H ∗ (BT ) ≃ R[α1 , . . . , αn ],where R is a coeﬃcient ring and αi is a variable with deg αi = 2 for i = 1, . . . , n. So its degree 2part H 2 (BT ) is isomorphic to Rn . Take any function α : E(Γ) → H 2 (BT ) and setα(Ep )(= α(Ep (Γ))) = {α(e) | e ∈ Ep } ⊂ H 2 (BT ).
We call a function α : E → H 2 (BT n ) \ {0} for n ≤ m an axial function (resp. torus axial function)on Γ if it satisﬁes the following three conditions:(1): α(e) = −α(e) (resp. α(e) = ±α(e));(2): for each vertex p ∈ V , the set α(Ep ) spans H 2 (BT n ); moreover, α(Ep ) is pairwise linearly independent, i.e., each pair of elements in α(Ep ) is linearly independent in H 2 (BT );(3): for each edge e ∈ E, there exists a bijective map ∇e : Ei(e) → Et(e) such that(1) ∇e = ∇−1e ,(2) ∇e (e) = e, and(3) for each e′ ∈ Ei(e) , α(∇e (e′ )) − α(e′ ) ≡ 0 mod α(e) ∈ H 2 (BT ); this equation iscalled a congruence relation.
The collection ∇ = {∇e | e ∈ E} is called a connection on the labelled graph (Γ, α); we denotethe labelled graph with connection as (Γ, α, ∇).
Definition 2.1 (GKM graph [21] and torus graph [35]). If an m-valent graph Γ is labeled
by an axial function α : E(Γ) → H 2 (BT n ) for some n ≤ m, then such labeled graph is said
to be an (abstract) GKM graph (see [21]). If an n-valent graph Γ is labeled by a torus axial
function α : E(Γ) → H 2 (BT n ), then such labeled graph is said to be an (abstract) torus graph(see [35]). Both labeled graphs are denoted as (Γ, α, ∇) or (Γ, α) (if the connection ∇ is obviously
determined).
The two ﬁgures in Figure 1 are typical examples of GKM graphs and torus graphs.
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Figure 1. The left graph is a GKM graph (this is also a torus graph) and theright one is a torus graph, where α, β are generators of H 2 (BT 2 ).
By deﬁnition, a 2-valent GKM graph (i.e., the boundary of ℓ-gon) is always a torus graph andits connection is uniquely determined (see Figure 1). Moreover, the following combinatorial factsays that connections of most (but not all) of GKM graphs are uniquely determined (see [21] and[35]).
Proposition 2.2. Let (Γ, α) be a GKM graph. If α(Ep ) is three independent (i.e., each threetuple of elements in α(Ep ) is linearly independent) for all vertices p ∈ V , then the connection ∇on (Γ, α) is unique.
Furthermore, a connection on a torus graph is uniquely determined.
Therefore, we may always denote a torus graph as (Γ, α) without connection.
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3. GKM graphs induced from GKM manifolds
In this section, we recall that some of GKM graphs (resp. torus graphs) are induced from
GKM manifolds (resp. torus manifolds) in [21] (resp. in [35]), also see [28, Remark 3.5] and [30].
3.1. GKM graphs induced from GKM manifolds. We ﬁrst recall how GKM manifolds
in the sense of [21] (i.e., [28, Deﬁnition 2.1]) deﬁne GKM graphs. Let M be a 2m-dimensional
GKM manifold with T n -action. Then, the GKM graph (ΓM , αM , ∇M ) is deﬁned by the following
way. The abstract graph ΓM is the orbit space of one-skeleton of M . Namely, the set of vertices V
is the set of ﬁxed points M T , and an edge connecting p, q ∈ V is an embedding 2-sphere connectingp, q ∈ M T , also see [28]. The axial function αM : E → H 2 (BT n ) is deﬁned by using the tangential
representations around ﬁxed points. More precisely, using an invariant almost complex structure
on M and the diﬀerentiable slice theorem, every tangent space Tp M on p ∈ M T decomposes into
the following irreducible complex T n -representation spaces:(3.1)

Tp M = ⊕mi=1 V (αi,p ),where V (αi,p ) is the complex one-dimensional representation space with αi,p ∈ Hom(T n , S 1 ) ≃H 2 (BT n ). From this fact, we also know that the graph ΓM is an m-valent graph. Now V (αi,p )may be regarded as the complex tangent space of the embedded 2-sphere ei,p (this 2-sphere isequivariantly diﬀeomorphic to CP 1 with the standard T 1 -action, see [28, Example 2.3]). Therefore, for Ep = {e1,p , . . . , em,p }, the axial function is deﬁned as αM (ei,p ) = αi,p ∈ H 2 (BT n ). The
connection ∇M is determined by the splitting of the restricted tangent bundle T M of M to theembedded 2-sphere e ∈ E. More precisely, because e is isomorphic to CP 1 , its restricted tangent
bundle T M |e split into the equivariant complex line bundles:
T M | e = ⊕m
i=1 Li,e ,where Li,e is the complex line bundle over CP 1 for all i = 1, . . . , m. Recall that the complex line
bundle L over CP 1 can be classiﬁed by its 1st Chern class c1 (L) ∈ H 2 (CP 1 ; Z) ≃ Z. Therefore,
the complex line bundle Li,e is isomorphic to
Li,e ≡ S 3 ×S 1 Cρi ,where S 1 acts on S 3 ⊂ C2 by the scaler multiplication and on Cρi by the representation ρi :S 1 → S 1 . Note that this representation is determined by ρi (z) = z ai for the integer ai = c1 (Li,e ).
Then, each line bundle Li,e satisﬁes that Li,e |p = V (αi,p ) and Li,e |q = V (αi,q ) for two ﬁxed point
p, q ∈ V connected by the 2-sphere e ∈ E. Therefore, the bijective map (∇M )e : Ep → Eq is
deﬁned by (∇M )e (ei,p ) = ei,q . Consequently, by deﬁnition of GKM manifold, (ΓM , αM , ∇M ) isa GKM graph. This labeled graph (ΓM , αM , ∇M ) is called an induced GKM graph from a GKM
manifold M .
Example 3.1 (toric manifold). By using the arguments in [28, Example 2.3], we have thatthe induced GKM graph from CP 2 is the left GKM graph in Figure 1. More generally, the GKM
graph of a non-singular complete toric variety is determined by the one-skeleton of its T n -orbit
space and the dual of its isotropy weight vectors (i.e., its characteristic functions, see [31] fordetails).
3.2. Torus graphs induced from torus manifolds. We next deﬁne a torus graph from atorus manifold introduced in [35]. Let M be a torus manifold (see the deﬁnition in [28, Example2.5]). By deﬁnition of torus manifold, with the method similar to that demonstrated in Section
3.1, the induced abstract n-valent graph ΓM can be deﬁned as the orbit space of its one-skeleton.
Then, a torus axial function αM : E(ΓM ) → H 2 (BT n ) is deﬁned as follows. We ﬁrst remark thata torus manifold does not have an invariant almost complex structure in general, unless GKM
manifolds (see Section 3.1). This means that there is no canonical way to regard the tangent
space Tp M on p ∈ M T as the complex T n -representation, i.e., the decomposition (3.1) does
not hold canonically. To regard Tp M as the complex representation space, we need the notionof omniorientation (see [24, 6] or [8, 31, 32] also). Let Mi , i = 1, . . . , m, be a characteristic
submanifold of M , i.e., a codimension 2 torus submanifold in M ; equivalently, the ﬁxed point wise3

connected components of some circle subgroups in T n . We ﬁx orientations on a torus manifold
M and all characteristic submanifolds M1 , . . . , Mm . This ﬁxed orientations on M, M1 , . . . , Mm
is called an omniorientation on M and denote it as O. We call a torus manifold M with ﬁxed
omniorientation O an omnioriented torus manifold and denote it as (M, O). Because there areexactly 2 orientations on each manifolds, there are exactly 2m+1 choices of omniorientations on
M , where m is the number of characteristic submanifolds. Let (M, O) be an omnioriented torus
manifold. By using the orientations on Mi and M , we may regard the normal bundle of Mi in M as
the invariant complex line bundle, say νi , for all i = 1, . . . , m. Take a ﬁxed point p ∈ M T . Denote
the restricted bundle of νi to p as νi |p if p ∈ MiT . Because νi |p is a complex one-dimensional
representation space, we may write νi |p = V (αi,p ) for some non-trivial weight αi,p : T n → S 1 .
Therefore, Tp M is isomorphic to⊕i∈Ip νi |p = ⊕i∈Ip V (αi,p ),where Ip ⊂ {1, . . . , m} is the subset which satisﬁes ∩i∈Ip Mi = {p}; therefore, |Ip | = n. Hence,the decomposition (3.1) is canonically determined by the omniorientation O. Moreover, we may
also regard the normal bundle of each invariant S 2 as the complex bundle by the restrictionsof some νi ’s. Therefore, with the method similar to that demonstrated in Section 3.1, we candeﬁne a torus graph (ΓM , αM , ∇M ) from (M, O). By Proposition 2.2, the connection ∇M is
uniquely determined, i.e., the decomposition of the restricted tangent bundle over an invariant S 2
is automatically determined. So the labeled graph (ΓM , αM ) (without connection) deﬁned froman omnioriented torus manifold (M, O) is called an induced torus graph from (M, O).
Remark 3.2. If there is an invariant almost complex structure on a torus manifold M , e.g.,toric manifolds, then there is the canonical omniorientation OC which is determined by the almost
complex structure. In this case, the induced torus graph is a GKM graph (also see [19]). Moregenerally, there is a torus manifold with an invariant stably complex structure, called a unitary
toric manifold (see [36, 32] and [40] for stably complex structure), e.g., quasitoric manifolds. By
using the invariant stably complex structure, there is the canonical omniorientation on M . Unlessthe case when torus manifolds have almost complex structures, a torus graph induced from aninvariant stably complex structure on a torus manifold is not always a GKM graph (see Figure 2).
The main diﬀerence between torus graphs and GKM graphs is the existence of the torus
axial function described in the right ﬁgure in Figure 2. The left one in Figure 2 is induced fromthe omniorientation induced from the canonical S 1 -invariant complex structure on CP 1 ; more
precisely, this is induced by the S 1 -action on CP 1 by [z0 : z1 ] 7→ [z0 : tz1 ] for [z0 : z1 ] ∈ CP 1 andt ∈ S 1 . On the other hand, the right one in Figure 2 is induced from the diﬀerent omniorientation
which is obtained by changing the orientation on the north pole of CP 1 ; this example can be
regarded as the S 1 -action on S 2 ⊂ C ⊕ R by (z, r) 7→ (tz, r) for (z, r) ∈ S 2 and t ∈ S 1 .
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Figure 2. The GKM graph (left) induced from the standard T 1 -action on CP 1 ,and the torus graph (right) induced from the T 1 -action on S 2 ⊂ C ⊕ R.
Example 3.3. From T n -action on S 2n ⊂ Cn ⊕ R deﬁned in [28, Example 2.5], we can deﬁne
torus graphs which are not GKM graphs. For example, the right graph in Figure 2 is the torus
graph induced from the case when n = 1; the right graph in Figure 1 is that of the case whenn = 2. In general, the torus graph induced from the T n -action on S 2n is the graph Γ = (V, E)
such that V = {p, q} and p, q are connected by n edges E = {e1 , . . . , en }, and the axial function
α : E → H 2 (BT n ) is deﬁned by α(ei ) = α(ei ) = αi for some basis {α1 , . . . , αn } of H 2 (BT n ).
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4. Sheaves on GKM (torus) graphs and equivariant cohomology
Let (Γ, α) be a GKM (or torus) graph. In order to introduce the deﬁnition of (graph) equivariant cohomology H ∗ (Γ, α) of (Γ, α), we introduce sheaves on (Γ, α) and its global sections
introduced in [4] (also see [2, 10, 42]). Note that in [4] the notion of sheaves is deﬁned on thelabeled graphs, called moment graphs. From a moment graph, it is easy to induce a GKM graph;moreover, the notion of sheaves also can be generalized to torus graphs naturally.
4.1. Sheaves on labeled graphs. Let us deﬁne the sheaves on GKM graphs and torus
graphs.
Definition 4.1. A sheaf M on (Γ, α) is the datum M = ({Mp }, {Me }, {ρp,e }}) such that
(1) Mp is an H ∗ (BT )-module deﬁned for each p ∈ V (Γ);(2) Me is an H ∗ (BT )-module with α(e)Me = 0, i.e., this also becomes an H ∗ (BT )/⟨α(e)⟩module, for each edge e ∈ E(Γ), where ⟨α(e)⟩ ⊂ H ∗ (BT ) is the ideal generated byα(e) ∈ H 2 (BT );(3) ρp,e : Mp → Me is a homomorphism of H ∗ (BT )-modules for any p ∈ V (Γ) with i(e) = p;
Example 4.2. The following datum, called A = ({Ap }, {Ae }, {ρp,e }), give the typical example
of a sheaf on (Γ, α): Ap = H ∗ (BT ) for all p ∈ V (Γ); Ae = H ∗ (BT )/⟨α(e)⟩ for all e ∈ E(Γ); andρp,e : H ∗ (BT ) → H ∗ (BT )/⟨α(e)⟩ is the natural projection. This sheaf A is called a structure
sheaf on (Γ, α) in [10] (or a sheaf of rings in [4]). Figure 3 shows the structure sheaves of the torus
graphs in Figure 1.
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Figure 3. The structure sheaves on the GKM graph and the torus graph in
Figure 1. In the ﬁgure, R = H ∗ (BT 2 ) ≃ R[α, β] and each arrow represents thenatural projection.
Note that we may regard a sheaf M on (Γ, α) as a sheaf on a “topological space” Γ in theusual sense (see [4], cf [10]). Now we may regard the abstract graph Γ = (V (Γ), E(Γ)) as theabstract set S(Γ) = V (Γ) ⊔ E(Γ) (this is a ﬁnite set). Deﬁne a topology on this abstract set S(Γ)
as follows: a ﬁnite subset U ⊂ S(Γ) is open if and only if U = ∅, U ⊂ E(Γ) or if p ∈ U ∩ V (Γ)
then Ep (Γ) ⊂ U . We denote U ∩ E(Γ) = E(U ) and U ∩ V (Γ) = V (U ). Under this topology, forthe given sheaf M, we deﬁne sections M(U ) of M over an open set U ⊂ S(Γ) as follows:


 ⊕

⊕
M(U ) :=fp ⊕fe ρp,e (fp ) = fe if p = i(e) ∈ U
(4.1)


p∈V (U )e∈E(U )
⊕
⊕
⊕
⊂
Mp ⊕
Me =
Mx ;
p∈V (U )M(U ) :=⊕x∈Ue∈E(U )

Me if V (U ) = ∅;e∈E(U )

M(∅) := {0}.
Then, M(U ) is an H ∗ (BT )-submodule of the H ∗ (BT )-module ⊕x∈U Mx . For an inclusion U ′ ⊂ Uof open subsets in S(Γ), there exists the natural projection ρeU U ′ : ⊕x∈U Mx → ⊕x∈U ′ Mx , i.e., themap induced from the restriction maps deﬁned by ρeU U ′ |Mx = Id if x ∈ U ′ ⊂ U and ρeU U ′ |Mx = 0
5

if x ∈ U \ U ′ . It is easy to check that this projection induces the restricted homomorphism
ρU U ′ : M(U ) → M(U ′ ) (might not be surjective). Hence, the sheaf M induces the presheaf on
the topological space S(Γ) = V (Γ) ⊔ E(Γ). Moreover, it follows from the deﬁnition of the topologyon S(Γ) that this presheaf on S(Γ) satisﬁes the axiom of a sheaf (see e.g. [29]).
4.2. Global sections of sheaves and equivariant cohomology. Now we call the section
M(S(Γ)) global sections of the sheaf M on (Γ, α), and we denote it by H 0 (Γ, α; M). Moreprecisely,


 ⊕

⊕
H 0 (Γ, α; M) =fp ⊕fe ρp,e (fp ) = fe for p = i(e) .
p∈V (Γ)e∈E(Γ)

The following lemma is straightforward:
Lemma 4.3. Let M = ({Mp }, {Me }, {ρp,e }) be a sheaf on (Γ, α). If there exists an H ∗ (BT )module R such that Mp = R for all p ∈ V (Γ), then the following H ∗ (BT )-module isomorphismholds:H 0 (Γ, α; M) ≃ {f : V (Γ) → R | ρp,e (f (p)) = ρq,e (f (q)) for p = i(e), q = t(e)}.
Example 4.4. Let R = ({Rp }, {Re }, {ρp,e }) be the sheaf on (Γ, α) deﬁned by Rp = R for allp ∈ V (Γ), Re = {0} for all e ∈ E(Γ) and ρp,e is the zero-map. Namely, R = ({R}, {{0}}, {0}).
Then, by Lemma 4.3,⊕
R.
H 0 (Γ, α; R) ≃p∈V (Γ)Let A be the structure sheaf on (Γ, α). By Lemma 4.3, its global sections H 0 (Γ, α; A) is
isomorphic to the following H ∗ (BT )-module:(4.2){f : V (Γ) → H ∗ (BT ) | f (p) − f (q) ≡ 0 mod α(e) for p = i(e), q = t(e)} ⊂⊕H ∗ (BT ).

p∈V (Γ)The relation “f (p) − f (q) ≡ 0 mod α(e)” is often denoted as “f (p) ≡α(e) f (q)”. Because⊕p∈V (Γ) H ∗ (BT ) naturally has the structure of an H ∗ (BT )-algebra, H 0 (Γ, α; A) may be regarded
as an H ∗ (BT )-subalgebra of ⊕p∈V (Γ) H ∗ (BT ). Moreover, by using the graded ring structure ofH ∗ (BT ), H 0 (Γ, α; A) may also be regarded as a graded ring. Namely, H 0 (Γ, α; A) has the structure of a graded H ∗ (BT )-algebra. We denote it as H ∗ (Γ, α).
Example 4.5. Let (ΓCP 2 , αCP 2 ) be the GKM graph induced from the standard T 2 -action on
CP , i.e., the left graph in Figure 1. Then, the global sections of its structure sheaf (see the left
sheaf in Figure 3) H ∗ (ΓCP 2 , αCP 2 ) is as follows:2{f : {p, q, r} → H ∗ (BT 2 ) | f (p) ≡α f (q), f (p) ≡β f (r), f (r) ≡α−β f (q)}.
Let (ΓS 4 , αS 4 ) be the torus graph induced from the standard T 2 -action on S 4 ⊂ C2 ⊕ R, i.e., theright graph in Figure 1. Then, the global sections of its structure sheaf (see the right sheaf inFigure 3) H ∗ (ΓS 4 , αS 4 ) is as follows:{f : {p, q} → H ∗ (BT 2 ) | f (p) ≡α f (q) and f (p) ≡β f (q)}.
By the GKM Theorem (see [28, Theorem 3.6]), we have the following theorem:
Theorem 4.6. Let (M, T ) be an equivariantly formal GKM manifold and (ΓM , αM ) be its
induced GKM graph. Then, HT∗ (M ; Q) ≃ H ∗ (ΓM , αM ) as a graded H ∗ (BT ; Q)-algebra, where thestructure sheaf A of (ΓM , αM ) is defined by Ap = H ∗ (BT ; Q) for all p ∈ V (ΓM ).
Motivating by this fact, the graded ring (or the H ∗ (BT )-algebra) H ∗ (Γ, α) deﬁned by (4.2)
is called a (graph) equivariant cohomology of (Γ, α). Note that for an abstract graph (Γ, α) there
might not be a geometric object M which induces (Γ, α). The description (4.2) is also known asa GKM description of an equivariant cohomology.
Moreover, the following theorem is proved by Maeda-Masuda-Panov [35]:6

Theorem 4.7. Let (M, T ) be a torus manifold with H odd (M ) = 0 and (ΓM , αM ) be its induced
torus graph. Then, HT∗ (M ; Z) ≃ H ∗ (ΓM , αM ) as a graded H ∗ (BT ; Z)-algebra,, where the structure
sheaf A of (ΓM , αM ) is defined by Ap = H ∗ (BT ; Z) for all p ∈ V (ΓM ).
Remark 4.8. The GKM descriptions for the other topological invariants, such as equivariant
K-theory, equivariant complex cobordism or more general equivariant complex oriented cohomology theory, are studied in [8, 22, 23] etc.
5. Combinatorial formula of equivariant cohomology ring
In Section 4, we introduce the graph equivariant cohomology H ∗ (Γ, α) of a GKM (and torus)graph (Γ, α). However, the deﬁnition of H ∗ (Γ, α), i.e., the GKM description, does not say anything
about the generators and relations as a ring or an algebra. So, for given (Γ, α), ﬁnding generatorsand relations of H ∗ (Γ, α) is the natural question. In this section, we introduce the Maeda-MasudaPanov theorem [35] which states combinatorially the generators and relations of H ∗ (Γ, α) of atorus graph (Γ, α). This theorem recovers some well-known results in toric geometry and toric
topology: the equivariant cohomology ring of complete non-singular toric varieties by DanilovJurkiewicz [13]; that of quasitoric manifolds by Davis-Januszkiewicz [9]; that of torus manifolds
with vanishing odd degree cohomologies by Masuda-Panov [37].
5.1. Thom class of a torus subgraph. Let (Γ, α) be a torus graph. To describe the
combinatorial formula of H ∗ (Γ, α), we need the notion of a Thom class of a torus subgraph of(Γ, α).
Let Γ′ be an (n − h)-valent subgraph of Γ for 0 ≤ h ≤ n, and ∇ be the unique connection on(Γ, α). We call Γ′ a torus subgraph, if Γ′ is closed under the connection ∇, i.e., for all e ∈ E(Γ′ )with i(e) = p, t(e) = q ∈ V (Γ′ ), the restricted bijection ∇e |Ep (Γ′ ) : Ep (Γ′ ) → Eq (Γ′ ) is well-deﬁned.
This is equivalent to the following condition: there exists an h-dimensional subtorus T ′′ ⊂ T suchthat the composition function αΓ′ := π ◦ α|E(Γ′ ) : E(Γ′ ) → H 2 (BT ′ ) satisﬁes the axiom of atorus axial function, where α|E(Γ′ ) : E(Γ′ ) → H 2 (BT ) is the restricted torus axial function andπ : H 2 (BT ) → H 2 (BT )/H 2 (BT ′′ ) ≃ H 2 (BT ′ ) is the projection for an (n − h)-dimensional torus
T ′ = T /T ′′ . Namely, the restricted labeled graph (Γ′ , αΓ′ ) is again a torus graph.
For an (n − h)-valent torus subgraph Γ′ , the symbol Np (Γ′ ) represents the set of all normaledges of Γ′ on p ∈ V (Γ′ ), i.e., Np (Γ′ ) = Ep (Γ) \ Ep (Γ′ ). Because Γ′ is an (n − h)-valent graph,|Np (Γ′ )| = h. Then, we deﬁne the function τ ′ : V (Γ) → H 2h (BT ) as follows:{ ∏′e∈Np (Γ′ ) α(e) p ∈ V (Γ )
τ ′ (p) =0
p ̸∈ V (Γ′ )By deﬁnition of torus subgraph, it is easy to check that τ ′ ∈ H 2h (Γ, α). We call this element τ ′ aThom class of Γ′ . Figure 4 shows examples of Thom classes of GKM subgraphs in Figure 1. Notethat we formally deﬁne τΓ = 1 ∈ H 0 (Γ, α), i.e., τΓ (p) = 1 for all p ∈ V (Γ), and τ∅ = 0 ∈ H 0 (Γ, α).
Remark 5.1. From geometric point of view, the equivariant Thom class of a codimension 2h
torus submanifold X of an omnioriented torus manifold (M, O) can be deﬁned by the following way(see [36]). Let ν be the normal bundle of X and T h(ν) be its Thom space, i.e, T h(ν) = M/(D(ν))cis the collapsing space of M on (D(ν))c , where D(ν)c is the complement of the unit disk bundleof ν embedded into M . Because ν has the induced orientation from the omniorientation (in fact,
this becomes a complex h-dimensional vector bundle), there is the following isomorphism, called
the Thom isomorphism:HT∗ (X) → HT∗+2h (T h(ν)).
On the other hand, there is the induced homomorphism H ∗ (T h(ν)) → HT∗ (M ) from the collapsing
map M → T h(ν). Therefore, by taking the composition of these homomorphisms, we have thefollowing homomorphism:
φX : HT0 (X) → HT2h (M ).
7

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Figure 4. Thom classes of some torus subgraphs of torus graphs in Figure 1.
Then, we can deﬁne the equivariant Thom class of the codimension 2h torus submanifold X by
φX (1) = τX ∈ H 2h (M ). Note that ι∗X (τX ) = eT (ν)(= cTh (ν)) ∈ H 2h (X), where ι∗X : HT∗ (M ) →HT∗ (X) is the induced homomorphism from the inclusion ιX : X → M and eT (ν) (resp. cTh (ν)) is
the equivariant Euler class (resp. top Chern class) of ν. Because of the deﬁnition of a torus graph,
the torus subgraph Γ′ is induced from some torus submanifold X; therefore, the Thom class τ ′ of
Γ′ is the combinatorial interpretation of the Thom class τX of X.
5.2. The combinatorial formula of H ∗ (Γ, α) of torus graphs. Let (Γ, α) be a torus
graph. To state the Maeda-Masuda-Panov theorem, we prepare some notation. The symbolZ[τH | H ⊂ Γ] represents the polynomial ring with integer coeﬃcient generated by Thom classesof all torus subgraphs H ⊂ Γ. We set deg τH = 2h if H is an (n − h)-valent graph, and deﬁneτΓ = 1 and τ∅ = 0. For two torus subgraphs G, H, if G ∩ H ̸= ∅, then there exists the uniqueminimal torus subgraph that contains both G and H (see [37, Proposition 5.2]); we denote it as
G ∨ H. Now we may state the Maeda-Masuda-Panov theorem:
Theorem 5.2 (Maeda-Masuda-Panov). Let (Γ, α) be a torus graph. Then, its graph equivariant cohomology H ∗ (Γ, α) is isomorphic to the following ring as the graded ring:(5.1)

Z[Γ, α] := Z[τH | H ⊂ Γ]/I

where the ideal I is generated by the following polynomials:∑
τG τH − τG∨H
τK ,
K∈G∩H

where K ∈ G ∩ H runs through all connected components of G ∩ H.
Note that if G ∩ H = ∅ then τG τH = 0 in Z[Γ, α], because τ∅ = 0. The ring Z[Γ, α] in Theorem
5.2 is called a face ring of (Γ, α).
The face ring Z[Γ, α] also has the structure of a graded H ∗ (BT )-algebra by the following fact
(also see [36, 32]): for all α ∈ H 2 (BT ), there exist integers k1 , . . . , km such that the following
map induces an injective homomorphism π ∗ : H ∗ (BT ) → Z[Γ, α]:π ∗ : α 7→ α(:= ατΓ ) =

m
∑

ki τi

i=1

where m is the number of all (n − 1)-valent torus subgraphs Γ1 , . . . , Γm in (Γ, α) and τi is theThom class of Γi , i.e., deg τi = 2, for all i = 1, . . . , m. We also note the following fact:
Proposition 5.3. The isomorphism H ∗ (Γ, α) ≃ Z[Γ, α] in Theorem 5.2 is also a graded
H (BT )-algebra isomorphism.
∗

8

Using Theorem 4.7 and Theorem 5.2, we have the following fact proved in [37].
Corollary 5.4. Let M be a torus manifold. If H odd (M ) = 0, then its equivariant cohomology
H (BT ; Z)-algebra is isomorphic to the face ring (5.1) which is generated by the Thom classes of
all torus submanifolds.
∗

Example 5.5. Let (ΓS 4 , αS 4 ) be the torus graph induced from the T 2 -action on S 4 (see theright graph in Figure 1). Then, ΓS 4 is constructed from two edges (1-valent torus subgraphs) e1(the left edge), e2 (the right edge) and two vertices (0-valent torus subgraphs) p, q; we put theseThom classes as τ1 , τ2 , τp and τq , respectively. By Theorem 5.2, we have that its graph equivariant
cohomology H ∗ (ΓS 4 , αS 4 ) is isomorphic to the following ring (cf Example 4.5):
Z[τ1 , τ2 , τp , τq ]/⟨τ1 τ2 − (τp + τq ), τp τq ⟩≃Z[τ1 , τ2 , τp ]/⟨τ1 τ2 − (τp + τq ), τp (τ1 τ2 − τp )⟩,where deg τ1 = deg τ2 = 2 and deg τp = deg τq = 4. By Corollary 5.4, this ring is isomorphic toHT∗ (S 4 ; Z).
Moreover, by deﬁnition of Thom classes, π ∗ (α) = τ2 and π ∗ (β) = τ1 (also see Figure 4). Because H odd (S 4 ; Z) = 0, its ordinary cohomology H ∗ (S 4 ; Z) is obtained by HT∗ (S 4 ; Z)/H >0 (BT ; Z).
Therefore, we have thatH ∗ (S 4 ; Z) ≃ HT∗ (S 4 ; Z)/H >0 (BT ) ≃ Z[ΓS 4 , αS 4 ]/Im π >0 ≃ Z[ΓS 4 , αS 4 ]/⟨α, β⟩ ≃ Z[τp ]/⟨τp2 ⟩.
Example 5.6. Let (ΓCP 2 , αCP 2 ) be the torus graph induced from the T 2 -action on CP 2 (seethe left graph in Figure 1). Then, ΓCP 2 is constructed from three edges pq, pr, qr and threevertices p, q, r; we put these Thom classes as τpq , τpr , τqr and τp , τq , τr , respectively. By the
relation of Z[ΓCP 2 , αCP 2 ] in Theorem 5.2, it is easy to get the following equivalence relations:τpq τpr = τp , τpq τqr = τq , τqr τpr = τr .
Therefore, we may reduce the generators of Z[ΓCP 2 , αCP 2 ] into only generators with degree twoand the graph equivariant cohomology H ∗ (ΓCP 2 , αCP 2 ) is isomorphic to the following ring (cfExample 4.5):Z[τpq , τpr , τqr ]/⟨τpq τpr τqr ⟩.
By Corollary 5.4, this ring is isomorphic to HT∗ (CP 2 ; Z).
Moreover, by deﬁnition of Thom classes, π ∗ (α) = τpr − τqr and π ∗ (β) = τpq − τqr . Therefore,
with the method similar to that demonstrated in Example 5.5, we have that3
H ∗ (CP 2 ; Z) ≃ HT∗ (CP 2 ; Z)/⟨α, β⟩ ≃ Z[τpq ]/⟨τpq⟩.

Note that, in Example 5.6, the generators of Z[Γ, α] reduce to the generators with degree two.
The following corollary says that there is a combinatorial reason for this phenomenon.
Corollary 5.7. Let (Γ, α) be a torus graph. Let Γ1 , . . . , Γm be all (n − 1)-valent torus
subgraphs and τ1 , . . . , τm be their Thom classes respectively. Assume that for every pair Γi , Γj ,
Γi ∩ Γj = ∅ or connected. Then, the graph equivariant cohomology H ∗ (Γ, α) is isomorphic toZ[τ1 , . . . , τm ]/I(5.2)

where the ideal I is generated by the following monomials:∏∩
τi for I ⊂ {1, . . . , m} such that
Γi = ∅.
i∈I

i∈I

The ring (5.2) is called a Stanley-Reisner ring. By Corollary 5.7, we have the following
geometric fact:
Corollary 5.8. Let M be a torus manifold. If H ∗ (M ) is generated by the 2nd degree cohomology, then its equivariant cohomology is isomorphic to the Stanley-Reisner ring (5.2) generatedby Thom classes of all characteristic submanifolds.
9In particular, if M is a complete non-singular toric variety or a quasitoric manifold, then M
satisﬁes the assumption of Corollary 5.8. Therefore, we get the well-known Danilov-Jurkiewicz
theorem and Davis-Januszkiewicz theorem from Theorem 5.2.
Remark 5.9. For the other class of GKM graphs (in particular, induced from ﬂag manifolds),
the ring structures of their graph equivariant cohomology are studied in [12, 13, 41, 42, 43].
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Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku,
Tokyo, 153-8914, Japan
E-mail address: kuroki@ms.u-tokyo.ac.jp

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