Classifications of homogeneous complexity one GKM manifolds and
GKM graphs with symmetric group actions
Dedicated to Professors Mikiya Masuda, Masaharu Morimoto and Kouhei Yamaguchi on their 60th birthday.

東京大学大学院数理科学研究科 黒木 慎太郎
Shintarô Kuroki
Graduate School of Mathematical Sciences, The University of Tokyo
Abstract. In this article, we ﬁrst give a classiﬁcation of simply connected, complexity one
GKM manifolds with extended transitive G-actions. This is proved by applying the method
to classify the homogeneous torus manifolds. Motivated by this result (Theorem 1.2), we nextclassify the 3-valent complexity one GKM graphs with certain S3 -actions.

1. Introduction2mn

Let (M , T ) be a pair of 2m-dimensional (compact, connected, simply connected) manifoldwith (almost) eﬀective n-dimensional torus action, where an almost eﬀective means that the T n action on M 2m has a ﬁnite kernel. If there is a ﬁxed point and the one-skelton of its orbit space
has the structure of a graph, then we call (M, T ) a (generalized) GKM manifold (see [GKM,
GuZa, Da, Ku1, MMP]). Note that this deﬁnition is slightly wider than the original deﬁnition
in [GuZa], i.e., we do not assume the existence of an equivariant almost complex structre. By
using the diﬀerentiable slice theorem, it is easy to check that n ≤ m. So the extremal class of
GKM manifolds would be the class when m = n. Such a GKM manifold is known as a torus
manifold (see [Ma99, HaMa]). The torus manifold is deﬁned by Hattori-Mausda in 2003 as the
topological generalization of toric manifolds (i.e., non-singular, complete, toric varieties viewed ascomplex analytic space) in algebraic geometry.
One of the motivations of toric geometry in algebraic geometry is to study the automorphismgroups of toric varieties (see [Co, De, Od]). Due to the results of Demazure and Cox, theroot systems of fans or Cox rings determine the Lie algebras of automorphism groups of toricvarieties. On the other hand, in this two decades, motivated by the study of Davis-Januszkiewicz[DaJa], the notions in toric geometry have been translated into the notions in topology, andnow it is called toric topology (see [BuPa, ToricTop]). In toric topology, more general classof manifolds with topological torus T -actions, such as torus manifolds, is studied. Moreover, theproblems studied in algebraic geometry inspire topologists to study new topological problems,such as cohomological rigidity problem (see [CMS]). In particular, from the topological point ofview, the study of automorphism groups may be regarded as the study of extended G-actions of
T -actions (see [Ku3, Ku4]). Assume that G is a compact Lie group. Motivated by the works
of automorphism groups of toric manifolds, the extended G-actions of a torus manifold (and a
symplectic toric manifold) are completely classiﬁed by several mathematicians in toric topology(and in symplectic geometry), see [KuMa, Ma10, MT, Wi].
In algebraic geometry and symplectic geometry, the manifolds with complexity one torus
actions (not only toric manifolds) are also studied by several mathematicians (see [ADHL, KaTo,
The author was partially supported by Grant-in-Aid for Scientiﬁc Research (S)24224002, Japan Society for
Promotion of Science.
1

2

COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIESMo]). In particular, Arzhantsev-Derenthal-Hausen-Laface study the automorphism groups of suchmanifolds in [ADHL]. The purpose of this article is to study the manifolds with complexity one
torus actions from topological point of view. More precisely, we study the extended actions ofa complexity one GKM manifold, i.e., a GKM manifold (M 2m , T n ) with m = n + 1, and give apartial answer to the following problem:
Problem 1.1. When does a complexity one GKM manifold admit an extended (M 2n+2 , G)action? Here, G is a compact (connected) Lie group with maximal torus T n .
In this article, we solve Problem 1.1 for the case of simply connected complexity one GKM
manifolds with transitive extended actions, called a homogeneous complexity one GKM manifold.
In order to state our main result, we deﬂne some terminology. We call M an irreducible if M has
the following property: if the manifold has the decomposition M = M1 × M2 then M2 = {∗}. If
M is not irreducible, then we call M a reducible. The following theorem is the 1st main result:
Theorem 1.2. Let (M 2m , T m−1 ) be a simply connected homogeneous GKM manifold with acomplexity one torus action. Then, M has the following decomposition:
M 2m = M1 × · · · Mk × M 2nsuch that Mi ’s are homogeneous irreducible simply connected torus manifolds, i.e., a complex
projective space or an even dimensional sphere (see [Ku3]), and M 2n is a simply connected,
irreducible, homogeneous, complexity one GKM manifold. Furthermore, we have n = 3 or 4 andif n = 3, M is one of the following manifolds:
A2 : Fℓ(C3 ) ∼= SU (3)/T 2 ;∼B2 : Q3 = SO(5)/SO(3) × SO(2);B2 : CP 3 ∼= Spin(5)/Sp(1) × T 1 ;3 ∼C2 : CP = Sp(2)/Sp(1) × T 1 ;G2 : S 6 ∼= G2 /SU (3),if n = 4, M is one of the following manifolds:
A3 : G2 (C4 ) ∼= SU (4)/S(U (2) × U (2));
C3 : HP 2 ∼= Sp(3)/Sp(2) × Sp(1);D3 : Q 4 ∼= SO(6)/SO(4) × SO(2),where G2 (C4 ) ∼= Q4 .
As a consequence of Massey in [Ma62] and well-known results, we can easily check which
manifolds in Theorem 1.2 have stably complex structures.
Corollary 1.3. Let M be a homogeneous, irreducible, complexity one GKM manifold. If
there is a T -invariant stably complex structure on M , i.e., a unitary GKM manifold, then M is
one of the followings:
Fℓ(C3 ),Q3 ,CP 3 ,S6

or G2 (C4 ).

Furthermore, every manifold as above also has a T -invariant almost complex structure.
Remark 1.4. In Theorem 1.2, the manifolds Fℓ(C3 ), Q3 , HP 2 and G2 (C4 ) are not torus
manifolds. This can be proved by using their cohomology rings (see [BuPa]). On the other hand,
CP 3 and S 6 are torus manifolds; furthermore, they are unitary toric manifolds (see [Ma99]).
From the unitary GKM manifold, we can deﬁne a labelled graph, so-called genelraized GKM
graph. Moreover, if a GKM manifold (M, T ) has an extension (M, G) such that G preserves theunitary structure, then its Weyl group W (G) = NG (T )/T acts on the GKM graph induced from(M, T ). Motivated by this facts and Theorem 1.2, we may ask the following question:
Problem 1.5. When does a 3 (resp. 4)-valent GKM graph admit a rank 2 (resp. 3) Weylgroup actions?
In this article, we also give a partial answer to Problem 1.5. More precisely, we also classify
the 3-valent GKM graphs (not generalized GKM graph) with certain S3 -actions. The 2nd main
result can be stated as follows:COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

3

Theorem 1.6. Let (Γ, A) be a 3-valent complexity one GKM graph with an S3 -action. Assume
that for every Weyl subgroup W ′ ⊂ S3 and every vertex p ∈ V (Γ), there is a GKM subgraph(Γ′ , A|E(Γ′ ) ) such that W ′ acts on it transitively and p ∈ V (Γ′ ). Then, (Γ, A) is one of the GKM
graphs in Figure 1.
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Figure 1. The list of complexity one GKM graphs with certain S3 -symmetries.

Furthermore, by using the invariants in [Ta] or [Ku5], we also have the following corollary:
Corollary 1.7. In Theorem 1.6, (Γ, A) extends to the torus graph if and only if (2), (3), (4).
The organization of this article is as follows. In Section 2, we prove Theorem 1.2. In Section
3, we quickly recall GKM graphs with Weyl group symmetry. In Section 4, we give a sketch of
the proof of Theorem 1.6.
2. Proof of Theorem 1.2 and observations
In this section, we prove Theorem 1.2 with the method similar to that demonstrated in [Ku3].
Moreover, we also give the method to construct inﬁnitely many complexity one GKM manifolds
with extended G-actions.
2.1. Proof of Theorem 1.2. Let (M 2n , T n−1 ) be a simply connected GKM manifold. Assume (M 2n , T n−1 ) has an extended (M 2n , G), where G is a compact (connected) Lie group withmaximal torus T n−1 . Then, we may put M = G/H such that rank G = rank H = n − 1 anddim G/H = 2n for some closed subgroup H ⊂ G. Because M is simply connected, H is a connected
subgroup.
e H)e be the universal covering of (G, H). Then, by using Borel and De Siebenthal’sLet (G,
result in [BoSi] and the assumption that the T n−1 -action is almost eﬀective, we havee = G1 × · · · × Gm
Ge = H1 × · · · × Hm ,H
where Gi is simply connected simple Lie group and Hi is its closed subgroup for i = 1, . . . , m suchthat rank Gi = rank Hi = ni and dim Gi − dim Hi = 2di . Because the Ti -action on Gi /Hi has
a ﬁxed point where Ti is a maximal torus of Gi and Hi , we have ni ≤ di for all i. Moreover, wehave the following lemma:
Lemma 2.1. We may assume di = ni for all i = 1, . . . , m − 1 and dm = nm + 1.

4

COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIESProof. By using the assumption, we have thatn1 + · · · + nm = d1 + · · · + dm − 1 < d1 + · · · + dm .
If ni = di for all i, then it is easy to show the contradiction to this inequality. Therefore, we may
assume nm < dm . If nm ≤ dm − 2, then it follows from ni ≤ di
d1 + · · · + dm − 1 = n1 + · · · + nm ≤ d1 + · · · + dm−1 + (dm − 2).
This also gives a contradiction. Thus, the equation nm = dm − 1 holds. Hence, by the above
equation and ni ≤ di , we also have ni = di for i ̸= m.

By Lemma 2.1, we have ni = di for i ̸= m. Hence, it follows from the main theorem of [Ku3]that we have the following decomposition:
e=
G

a
∏

SU (ℓi + 1) ×

i=1

e=
H

a
∏

b
∏

Spin(2mj + 1) × Gm

j=1

S(U (1) × U (ℓi )) ×b∏

Spin(2mj ) × Hm ,

n=1

i=1

where mj ≥ 2, a+b = m−1. Here, rank Gm = rank Hm = nm and dim Gm /Hm = 2dm = 2nm +2.
Consequently, the problem is reduced into the classiﬁcation of (Gm , Hm ). To classify (Gm , Hm )we consider the following two cases.
CASE 1: Assume Hm is a maximal same rank subgroup of a compact connected simply
connected Lie group Gm . Then, by using the table [Ku3, Table 1, 2], we have that (Gm , Hm ) is
one of the following:
A3 : (SU (4), S(U (2) × U (2))), i.e., M 8 = G2 (C4 );B2 : (SO(5), SO(3) × SO(2)), i.e., M 6 = Q3 ;C2 : (Sp(2), Sp(1) × T 1 ), i.e., M 6 = CP 3 ;C3 : (Sp(3), Sp(1) × Sp(2)), i.e., M 8 = HP 2 ;D3 : (SO(6), SO(4) × SO(2)), i.e., M 8 = Q4 ;G2 : (G2 , SU (3)), i.e., M 6 = S 6 .
Note that Q4 ∼= G2 (C4 ).
CASE 2: Assume Hm is a non-maximal same rank subgroup of Gm such that rank Gm =rank Hm = n and dim Gm /Hm = 2n + 2. Then, there is a maximal same rank subgroup Km
such that Hm ⊂ Km ⊂ Gm . Because dim Gm /Hm = 2n + 2 > dim Gm /Km (i.e., dim Gm /Km ≥2rank Gm ), we have that dim Gm /Km = 2n. Therefore, Gm /Km must be the homogeneous torus
manifold and Km /Hm ∼= SU (2)/T 1 ∼= S 2 . Hence, by [Ku3], (Gm , Km ) is one of the following:(SU (k + 1), S(U (1) × U (k)));(Spin(2k + 1), Spin(2k)).
Because Km /Hm ∼= S 2 , we also have k = 2. Therefore, we have (Gm , Km , Hm ) is one of thefollowings:(SU (3), S(U (1) × U (2)), T 2 );(Spin(5), Spin(4), Sp(1) × T 1 ),where Spin(4) ∼= Sp(1)×Sp(1) and Spin(5) ≃ Sp(2). Because dim SU (3)/T 2 = 6 and dim Spin(5)/Sp(1)×1
T = 6, we have that (Gm , Km ) is one of the following:
A2 : (SU (3), T 2 ), i.e., M 6 = Fℓ(C3 );B2 : (Spin(5), Sp(1) × T 1 ), i.e., M 6 = CP 3 .
Consequently, we establish Theorem 1.2.

COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

5

2.2. Observations. By using the GKM manifolds appearing in Theorem 1.2, we can construct other GKM manifolds; in particular, we can construct inﬁnitely many complexity one GKM
manifolds with extended (SU (3) × T n )-actions. Here, we show the construction.
Example 2.2. Let X be a 2n-dimensional torus manifold with n ≥ 2 and ρ : T 2 → T n be afaithful representation. Deﬁne the manifold Mρ (X) by the twisted product
SU (3) ×T 2 X
2by the standard (right) T -action on SU (3) and the T 2 -action on X via ρ. Because the elementsof the image ρ(T 2 ) commutes with the elements in T n , the manifold Mρ (X) has the (T 2 × T n )action by the left T 2 -action on the SU (3)-factor and the T n -action on the 2n-dimensional torus
manifold X. Because there is a ﬁbre bundle structure X → Mρ (X) → SU (3)/T 2 , we also havedim Mρ (X) = 2n + 6. Moreover, it is easy to check that its one-skeleton has the structure ofa ﬁbre bundle over the one-skeleton of SU (3)/T 2 whose ﬁbre is that of X. Therefore, this is
a complexity one GKM manifold. Moreover, there is an extended SU (3) × T n -action because
SU (3) acts naturally on the SU (3)-factor in Mρ (X). Because there are inﬁnitely many torus
manifolds X, we can construct inﬁnitely many complexity one GKM manifolds with extended
SU (3) × T n -actions.
Example 2.3. Let Y be a (2n − 4)-dimensional torus manifold for n > 2, and σ : T 1 → T n−2
be a faithful representation. Deﬁne the twisted product Nσ (Y ) by
S 5 ×T 1 Ysuch that T 1 acts on S 5 ⊂ C3 diagonally and on a torus manifold Y via σ. Then, Nσ (Y ) is a
2n-dimensional torus manifold with extended (SU (3) × T n−2 )-actions (see [Ku4] for details of
this construction). Here, SU (3) acts on the S 5 -factor transitively. Now we can add the G2 -factor
as follows:
G2 ×SU (3) Nσ (Y )≃ G2 ×SU (3) (S 5 ×T 1 Y )≃ G2 ×SU (3) ((SU (3)/SU (2)) ×T 1 Y )≃ (G2 /SU (2)) ×T 1 Y.
This is an Nσ (Y )-bundle over S 6 (or a Y -bundle over G2 /S(U (2) × U (1)), also see [Ku2, Example1.4]). Therefore, dim(G2 ×SU (3) Nσ (Y )) = 2n + 6. Moreover, because σ(T 1 ) commutes with
T n−2 , the structure of a Y -bundle over G2 /S(U (2) × U (1)) induces the (T 2 × T n−2 )-action on
G2 ×SU (3) Nσ (Y ). Because both of the ﬁbre and the base space have structures of GKM manifolds,(G2 ×SU (3) Nσ (Y ), T n ) is a (2n + 6)-dimensional GKM manifold, i.e., complexity 3 GKM manifoldwith an extended G2 × T n−2 -action. However, this might not be a complexity one GKM manifold(see [Ta] or [Ku5]).
Similarly, we can also construct other GKM manifolds (might not be a complexity one GKM
manifold) with extended non-abelian Lie group actions, by using the other irreducible homogeneouscomplexity one GKM manifolds.
In particular, we can generalize the construction in Example 2.2 as follows.
Proposition 2.4. Let G be a compact, connected, non-abelian Lie group with rank G = n,
and X be a 2m-dimensional torus manifold such that n ≤ m. Then, for any faithful representation
ρ : T n → T m , the following manifold is a (dim G − n + 2m)-dimensional GKM manifold with the
T n+m -action:
G ×T n X
where T n is a maximal torus in G and acts on X via ρ.
Furthermore, this has the extended (G × T m )-action and a complexity (dim G − 3n)/2.
3. GKM graphs
In this section, we recall GKM graphs introduced in [GuZa], and prepare to prove Theorem
1.6.

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COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

3.1. Notations. Let Γ = (V (Γ), E(Γ)) be an abstract graph, where V (Γ) is the set of vertices
and E(Γ) is the set of oriented edges of Γ. Let e ∈ E(Γ). We denote its initial vertex by i(e),the terminal vertex by t(e), (e.g. i(pq) = p and t(pq) = q) and the reversed oriented edge of e by
ē. It is easy to check that i(e) = t(ē) and t(e) = i(ē). Put the set of all outgoing edges from thevertex p by Ep (Γ), i.e., the set of all edges e such that i(e) = p. We say Γ is an m-valent graph if#Ep (Γ) = m for all p ∈ V (Γ).
A map between two graphs Γ = (V (Γ), E(Γ)) and Γ′ = (V (Γ′ ), E(Γ′ )) is deﬁned as the pairof maps f = (fV , fE ) : Γ → Γ′ such that the following diagram commutes
E(Γ)
↓

fE

−→

E(Γ′ )↓fV

V (Γ) −→ V (Γ′ )where two vertical maps are the maps taking the initial vertex, i.e., e 7→ i(e). In other wards, themap of vertices preserves the edges. An automorphism group of Γ, say Aut(Γ), is deﬁned by theset of all maps f on Γ such that both of fV and fE are bijective.
Let H ∗ (BT n ) be the cohomology ring of BT over Z-coeﬃcient, i.e., H ∗ (BT n ) is isomorphic
to the polynomial ring Z[α1 , . . . , αn ] in the variables αi ∈ H 2 (BT ) (i = 1, . . . , n).
3.2. Abstract GKM graph. Throughout of this paper, Γ is an m-valent (connected) graph,
where n ≤ m. Put
A : E(Γ) → H 2 (BT ).
If A satisﬁes the following three conditions:(1) A(e) = −A(ē);(2) the set {A(Ep (Γ))} is pairwise linearly independent for all p ∈ V (Γ);(3) there is a bijective map ∇e : Ep (Γ) → Eq (Γ) for p = i(e) and q = t(e) such that(a) ∇e = ∇−1e ;(b) ∇e (e) = ē;(c) A(∇e (f )) − A(f ) ≡ 0 mod A(e) for every f ∈ Ep (Γ) (called a congruence relation),then the map A is called an axial function on Γ and the collection ∇ = {∇e | e ∈ E(Γ)} is calleda connection. We call the pair (Γ, A) a GKM graph, where Γ is an m-valent graph and A is anaxial function on Γ.
Assume an axial function A : E(Γ) → H 2 (BT n ) satisﬁes that the image of A spans H 2 (BT n ).
Then, we call the number m − n a complexity of (Γ, A).
Remark 3.1. Let (M, T ) be a 2m-dimensional GKM manifold with invariant almost complex
structure. Then, the one-skeleton of M/T induces the m-valent graph ΓM . Moreover, we can deﬁne
the axial function AM by the complex tangential representation on each ﬁxed point (to deﬁne thiscomplex structure canonically, we need a complex structure around ﬁxed points). Namely, them-valent GKM graph (ΓM , AM ) is deﬁne by an almost complex 2m-dimensional GKM manifold(M, T ).
More generally, we can also deﬁne the generalized GKM graph, i.e., the condition (1) in theaxial function A is changed to the condition A(e) = ±A(ē), from an unitary GKM manifold (M, T )(i.e., there is an invariant stably complex structure on M ) such as an even dimensional sphere.
We omit the precise deﬁnition of the (generalized) GKM graph induced from a GKM manifold inthis article (see [Da], [GuZa], [Ku1] or [MMP] for precise deﬁnition).
Note that a more general labelled graph (i.e., there might not exist any connections) can bedeﬁned by an omnioriented GKM manifold (i.e., there is a ﬁxed orientations on M and all invariant
2-spheres) such as a quaternionic projective space whose dimension is greater than four.
3.3. Automorphism group of GKM graph. Let us deﬁne the automorphism group of a
GKM graph (Γ, A) with A : E(Γ) → H 2 (BT n ) ≃ Zn . Let f ∈ Aut(Γ) and ρ ∈ GL(n; Z). We call

COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

7(f, ρ) is an automorphism on (Γ, A) if the following diagram commutes:
E(Γ)↓
2
H (BT )

fE

−→

E(Γ)
↓
ρ
2
−→ H (BT )where the vertical maps are both of A. We call the set{(f, ρ) ∈ Aut(Γ) × GL(n; Z) | (f, ρ) is an automorphism on (Γ, A)}an automorphism group of (Γ, A) and denote it by Aut(Γ, A).
The following proposition is one of the motivations to considering the 2nd main result in thispaper (Theorem 1.6) (also see [Kaj]).
Proposition 3.2. If an almost complex GKM manifold (M, T ) extends to an almost complex(M, G), then W (G) = NG (T )/T acts on (ΓM , AM ), i.e., W (G) ⊂ Aut(ΓM , AM ).
Proof. Let (M, T ) be a GKM manifold with an almost complex structure J. Assume that
there is an extended G-action (M, G) preserving J. Let g ∈ W (G) and Sα2 be an invariant 2-spherein (M, T ), where α ∈ t∗ is the element corresponding to Sα2 (i.e., the element appearing in thetangential representation on Sα2 ) Then, gSα2 satisﬁes thatT gSα2===(gT g −1 )gSα2(because g ∈ NG (T ))2
gT Sα
gSα2 (because Sα2 is T -invariant).

This implies that gSα2 is an invariant 2-sphere; therefore, W (G) ⊂ Aut(Γ).
By deﬁnition, the axial function AM is induced by the complex structure around ﬁxed points
induced from J. Therefore, it is easy to check that W (G) ⊂ Aut(ΓM , AM ), because G preserves
the almost complex structure J of M .

By the proof described as above, we may thinkgSα2 = Sβ2
where β = g ∗ α ∈ t∗ for g ∗ ∈ W (G). Therefore, the Weyl group W action on an abstract GKM
graph (Γ, A) can be deﬁned as follows. It is well known that W is generated by the reﬂections ont; in particular, this preserves the weight lattice (see [Hu]). Hence, for g ∈ W , the isomorphism g
on tZ induces the isomorphism g ∗ : t∗Z → t∗Z , i.e., the dual of g ∈ W . Therefore, the automorphismg : (Γ, A) → (Γ, A) is deﬁned by(3.1)

g ∗ (A(e)) = A(gE (e)),

where e ∈ E(Γ), gE : E(Γ) → E(Γ) is the bijective map induced from g ∈ W ⊂ Aut(Γ) (we often
abuse two notations gE and g).
Remark 3.3. If the extended action G does not preserve the almost complex structure J(even more generally the stably complex structure) on M , then W (G) do not act on (ΓM , AM ).
However, if M is a unitary GKM manifold and there is an extended G-action (which might notpreserve the stably complex structure), then g ∈ W (G) action changes the relation (3.1) as follows:g ∗ (A(e)) = ±A(gE (e)).
Therefore, g ∈ W (G) gives the equivalence between (Γ, A) and (Γ, A′ ) where A′ satisﬁes A(e) =±A′ (e) for all edges e ∈ E(Γ). In [KuMa], we also study the extended actions for the cases when
G does not preserve the stably complex structures (more generally omniorientations) on torus
manifolds by introducing root systems on torus graphs.

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COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

4. Sketch of a proof of Theorem 1.6
Let (Γ, A) be an abstract m-valent GKM graph. Motivated by Theorem 1.2, we classify thefollowing GKM graphs:(1) Γ is a 3 or 4-valent graph;(2) (Γ, A) is a complexity one, i.e., if A : Γ → H 2 (BT n ) then n = 2 (for m = 3) or 3 (form = 4), called a (3, 2)-type GKM graph or (4, 3)-type GKM graph, respectively;(3) if (Γ, A) is a (3, 2)-type (resp. (4, 3)-type), there is the symmetric group S3 , the signed
symmetric group S2± or the dihedral group D6 (resp. S4 or S3± ) action on it;(4) for every Weyl subgroup W ′ ⊂ W and every vertex p ∈ V (Γ), there is a GKM subgraph(Γ′ , A|E(Γ′ ) ) such that W ′ acts on it eﬀectively and p ∈ V (Γ′ );The assumption (4) as above is induced from the fact that there is a connected K-orbit ofp ∈ M T for all K such that T ⊂ K ⊂ G.
In this paper, we only prove the case when (Γ, A) is a (3, 2)-type GKM graph with thesymmetric group S3 -action. For the other cases, we can prove similarly (it will be shown insomewhere of the future article).
4.1. The property of S3 -orbits. Let W = S3 . Then, it is well-known that this is a Coxeter
group and there is the following representation:W==⟨σ1 , σ2 | σ12 = σ22 = (σ1 σ2 )3 = 1⟩
{1, σ1 , σ2 , σ1 σ2 , σ2 σ1 , σ1 σ2 σ1 }.

In this case, there are four Weyl subgroups:{1}, ⟨σ1 ⟩, ⟨σ2 ⟩, ⟨σ1 σ2 σ1 ⟩(≃ Z2 ),and there is another subgroup
Z3 = ⟨σ1 σ2 ⟩ = ⟨σ2 σ1 ⟩.
Therefore, if there is a W action on Γ = (V (Γ), E(Γ)), then there are the following possible orbit
types for p ∈ V (Γ):(4.1)

W (p) ≃ W/W, W/Z3 , W/Z2 , or W/{e}.

By choosing generators of H 2 (BT ) = Zα ⊕ Zβ, for p ∈ V (Γ), we may assume that
Ap = {A(e) | e ∈ Ep (Γ)} = {α, β, k1 α + k2 β}
for some non-zero integers k1 , k2 . From the next subsection, we consider the cases appeared in(4.1) in case by case.
4.1.1. The case when W (p) ≃ W/W . In this case, W acts on Ep (Γ); therefore it also acts on
Ap . Because H 2 (BT )W = {0}, the symmetric group W acts on Ap transitively; therefore, it also
acts on Ep (Γ) transitively. Consequently, we may assume that k1 = k2 = −1 andσ1 : α 7→ β, −α − β 7→ −α − β;σ2 : α 7→ −α − β, β 7→ β.
This implies that the neighborhood of this case is Figure 2.

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Figure 2. This is the axial function around p ∈ V (Γ)W , where ϵ = ±1.

COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

9

4.1.2. The case when W (p) ≃ W/Z3 . We claim that this case does not occur. Because|W (p)| = 2 and our assumption (4), there is an edge e ∈ E(Γ) such that V (e) = W (p) = {p, q}
and σ1 ∈ W such that σ1 (e) = ē. Because A(e) = −A(ē)(= α), we may assumeσ1 : α 7→ −α.

(4.2)

On the other hand, the subgroup Z3 = {1, σ1 σ2 , σ2 σ1 } acts on Ep (Γ) = {e, e′ , e′′ }. Therefore,
because σ1 (e) = ē ∈ Eq (Γ) and the assumption (4), σ2 (ē) = e. However, this implies that
σ2 : −α 7→ α. This gives a contradiction to (4.2). Consequently, there is no vertex p ∈ V (Γ) suchthat W (p) ≃ W/Z3 .
4.1.3. The case when W (p) ≃ W/Z2 . Because |W (p)| = 3 and the assumption (4), in this casethere is a triangle GKM subgraph (∆, A|∆ = A′ ) ⊂ (Γ, A) such that V (∆) = W (p) = {p, q, r}.
Because the symmetric group W acts on (∆, A′ ) transitively, we may assume that
σ1 : p 7→ p,
σ2 : p 7→ q,

q 7→ r;r 7→ r.

Moreover, considering the axial functions around ∆, we can easily check that this case may
assume the axial functions appeared in Figure 3 (the case when A(pq) = ϵα and A(pr) = ϵβ) or
Figure 4 (otherwise, i.e., the case when A(e) = ϵα and A(e′ ) = ϵβ for e ∈ Ep (Γ) \ Ep (∆) ande′ ∈ Eq (Γ) \ Eq (∆)).

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S

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N Șș

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N Șș
Figure 3. The axial function around ∆ such that V (∆) = W (p), where k is a
non-zero integer and ϵ = ±1. In this case, σ1 : α 7→ β and σ2 : β 7→ −α + β.

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TUȜșȜ̓ ȘșȜ Șș
Figure 4. The axial function around ∆ such that V (∆) = W (p), where ϵ, ϵ′ =±1. In this case, σ1 : β 7→ −α − β and σ2 : α 7→ β.
4.1.4. The case when W (p) ≃ W/{e}. In this case, |W (p)| = 6. By using the assumption (4),
it is easy to check that W acts on Γ transitively and the axial functions are the labelles appearedin Figure 5 or Figure 6.

10

COMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

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Figure 5. The 1st case: σ1 : α 7→ −α, β 7→ α + β; σ2 : β 7→ −β, α 7→ α + β,where k is a non-zero integer.
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șȘȘWFigure 6. The 2nd case: σ1 : α 7→ −α, β 7→ −α + β; σ2 : β 7→ −β, α 7→ α + β,where k is a non-zero integer.
4.2. The proof of Theorem 1.6 for the case when there is an S3 -action. Finally, inthis section, we prove Theorem 1.6 by combining the facts described in Section 4.1. If W = S3
acts on (Γ, A) transitively, then this case is one of the cases of Figure 5 and Figure 6. These are
the cases (5), (6) in Theorem 1.6.
Assume that W acts on (Γ, A) non-transitively. Then, there are three possible orbits, i.e.,
Figure 2, Figure 3 and Figure 4, say type 1, type 2 and type 3 respectively.
If there is a type 1 orbit, then this orbit must be connecting with one of the following orbits:(1) the type 1 orbit with distinct ϵ’s (this is the case (1) in Theorem 1.6);(2) the type 3 orbit with distinct ϵ’s (this is the case (2) in Theorem 1.6).
Assume that there is no type 1 orbit but there is a type 2 orbit. Then, because there is a W -action
on whole Γ, this case is only the connecting two copies of type 2 (with diﬀerent signs of k’s). ThisCOMPLEXITY ONE GKM MANIFOLDS WITH SYMMETRIES

11

is the case (3) in Theorem 1.6. Assume that there is no type 1 orbit but there is a type 3 orbit.
Then, similarly, this case is only the connecting two copies of type 3 (with distinct ϵ’s). This is
the case (4) in Theorem 1.6.
This establishes Theorem 1.6.
Remark 4.1. In the end of this article, we show some geometric models for GKM graphs
appearing in Theorem 1.6:
• The GKM graph (1) is induced from (S 6 , T 2 ), where S 6 ≃ G2 /SU (3) (there is a SU (3)(⊂G2 )-extended action);• The GKM graph (2) is from (CP 3 , T 2 ), where CP 3 ≃ P (V (−α) ⊕ V (−β) ⊕ V (α + β) ⊕ C)
(there is a SU (3)-extended action on the ﬁrst three coordinates);• The GKM graph (3) is from S 5 ×T 1 P (γ ⊗k ⊕ ϵ) (there is a transitive SU (3)-action on
the S 5 -factor);• The GKM graph (4) is from the connected sum of two copies of (CP 3 , T 2 )’s which induce
the GKM graph (2) (there is an SU (3)-extended action because this connected sum is
an SU (3)-equivariant);• The GKM graphs (5), (6). These cases are still not known, that is, which GKM manifolds
induce these GKM graphs? Some of them must be obtained from the projectivizationsof equivariant complex 2-dimensional vector bundles over CP 2 , which are not split intoline bundles (see [Kan]), because the projectivization of Whitney sum of line bundles
is isomorphic to S 5 ×T 1 P (γ k ⊕ ϵ). In particular, if k = 1 for the GKM graph (5),this is obtained from the ﬂag manifold SU (3)/T 2 ≃ SU (3) ×U (2) CP 2 , which can be
regarded as the projectivization of some complex 2-dimensional vector bundle over CP 2
(see [KuSu]).
Acknowledgment
This article is based on author’s talk in RIMS Conference “The Topology and the Algebraic
Structures of Transformation Groups” from May 26th to May 30th 2014. The author is gratefulto the organizer Takao Sato for giving him the chance to give a talk in the conference. He also
would like to thank Prof. Takashi Tsuboi for providing him with excellent working conditions.
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Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku,
Tokyo, 153-8914, Tokyo, Japan
E-mail address: kuroki@ms.u-tokyo.ac.jp