GKM GRAPHS INDUCED BY GKM MANIFOLDS WITH
SU (2)-SYMMETRIESSHINTARÔ KUROKI

1. Introduction
A GKM manifold is a 2m-dimensional manifold M 2m equipped with an eﬀective T n action whose one and zero dimensional orbits have the structure of a graph, where n ≤ m.
Let Γ be the induced m-valent graph of the GKM manifold (M 2m , T n ). We may identify
the ﬁxed points of (M 2m , T n ) with the vertices of Γ. Moreover, we can label each outgoingedge of Γ around a vertex p by its tangential representation, called an axial function
α : E → H 2 (BT n ) where here E is the set of oriented edges1 of Γ. This labeled graph(Γ, α) is called a GKM graph induced by the GKM manifold. On the other hand, a GKM
graph can be deﬁned abstractly by the labeled graph (Γ, α) which satisﬁes some propertiesof GKM graphs induced by GKM manifolds (see Section 2 or [3, 6, 7, 8, 13] for detail).
Let (M 2m , T n ) be a compact GKM manifold, and G a non-abelian, compact Lie groupwhose maximal torus is T n . In this article, we study compact GKM manifolds with extended G-actions (also see [2, 9, 10, 11, 15, 17, 18, 19] for the related topics). For technicalreasons, we assume the followings:(1) a GKM manifold M 2m has an almost complex structure which is compatible with
the T n -action;(2) G preserves almost complex structure J on M , i.e., G ⊂ Diﬀ(M, J );e of G has the SU (2)-factor;(3) the universal covering G(4) there are codimension two characteristic submanifolds in (M, T ); we denote all ofthem by F = {X1 , . . . , Xk }.
The goal of this article is to introduce a property of GKM graphs induced by GKM manifolds with extended G-actions as above (see Theorem 4.1).
The details of this article for more general cases will be appeared in the forthcoming
paper [12].
Date: June 29, 2010.
2010 Mathematics Subject Classification. Principal:57S25, Secondly:05C25; 55N91; 55R91.
Key words and phrases. Characteristic submanifold, Equivariant cohomology, GKM manifold, GKM
graph, Weyl group.
The author was supported in part by Basic Science Research Program through the NRF of Korea fundedby the Ministry of Education, Science and Technology (2010-0001651) and the Fujyukai Foundation.
1In E, we distinguish the two same edges pq and qp by regarding that their orientations are diﬀerent.
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S. KUROKI

2. Definition of GKM graphs
Let Ep be the set of all out going edges from the vertex p. By the assumption (1)in Section 1, the GKM graph (Γ, α) induced by the GKM manifold has the following
properties:• Γ is an m-valent graph, i.e., |Ep | = m for all vertices p;• α(e) = −α(ē), where e and ē are the same edge but their orientations are diﬀerent,e.g., if e = pq then ē = qp;• {α(ei ) | ei ∈ Ep } is pairwise linearly independent, i.e., α(ei ) and α(ej ) are linearly
independent if ei ̸= ej ;• if two vertices p and q are connected by an edge (called f ), there is a bijective¯map ▽f : Ep → Eq such that ▽f¯ = ▽−1
f , ▽f (f ) = f , and α(e) − α(▽f (e)) ≡0 (mod α(f )) for e ∈ Ep , (the collection of maps ▽ = {▽f | f ∈ E} is calledconnection).
On the other hand, if the given labeled graph (Γ, α), where α : E → H 2 (BT n ), satisﬁes
the properties above then we call (Γ, α) a GKM graph in this article.
3. Basic properties of GKM manifolds with SU (2)-symmetries
Assume that the GKM manifold (M 2m , T n ) equipped with an extended G-action satisﬁes
all the assumptions (1)–(4) mentioned in Section 1.
Let W be the Weyl group of SU (2), i.e., W ≃ Z2 . We let r ∈ t∗ ≃ H 2 (BT ; R) denotea simple root of SU (2). As is well-known, the root system corresponds to the elements inW which act on t∗ as the reﬂections (e.g. see [16, Chapter 5]); so, we let σ ∈ W denote thereﬂection corresponding to the simple root r.
Let π : ET ×T M → BT be the projection of the Borel construction of (M, T ), andπ ∗ : H ∗ (BT ) → HT∗ (M ) be the induced homomorphism. The element τi ∈ HT2 (M ),i = 1, . . . , k, represents the equivariant Thom class of codimension two characteristic
submanifold Xi ∈ F. We denote the set of such equivariant Thom classes by F ∗ . Then,
there is the W-action on F ∗ induced by the W-action on F.
In order to state Theorem 4.1, we ﬁrst introduce the following lemma.
Lemma 3.1. Assume σ(Xs ) = Xt , where Xs , Xt ∈ F . Then the following equation holds:π ∗ (r) = τs − τt .
4. Main theorem
Let (Γ, α) be an abstract GKM graph, and HT∗ (Γ, α) its graph equivariant cohomology.
Now π ∗ : H ∗ (BT n ) → HT∗ (Γ, α) is deﬁned by π ∗ (x) = x (x ∈ H ∗ (BT n )), i.e., the constantfunction x : p 7→ x for all vertices p.
Abstractly, we assume that (Γ, α) satisﬁes the property in Lemma 3.1. That is, there is
the element r ∈ H 2 (BT n ) such that the following E.q. (4.1) holds:(4.1)

π ∗ : r 7→ τ1 − τ2 ,

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where τ1 , τ2 ∈ HT2 (Γ, α) are the Thom classes of some (m − 1)-valent GKM subgraphs Γ1 ,Γ2 , respectively.
Now we may state the following theorem.
Theorem 4.1. Suppose that there is r ∈ H 2 (BT n ) such that E.q. (4.1) holds for some
GKM subgraphs Γ1 and Γ2 . Then, one of the following cases occur:
The 1st case: if Γ1 ∩ Γ2 = ∅, there is the GKM fiber bundle ρ : (Γ, α) → (I, αI ),where I is the compact 1-valent graph (i.e., two vertices p, q and one edge e = pq)
and αI satisfies that αI (p) = r and αI (q) = −r;e αThe 2nd case: otherwise, there is the GKM blow-up (Γ,e) → (Γ, α) along Γ1 ∩ Γ2ste
such that (Γ, αe) satisfies the 1 case.
The geometric interpretation of this theorem is as follows (also see [1]). The 1st casecorresponds to that M is T n -equivariantly diﬀeomorphic to the crossed product SU (2) ×S 1
N for some (2m − 2)-dimensional GKM manifold N . The 2nd case is otherwise, i.e., M
does not decompose into the crossed product; however, there is the codimension-4 GKM
f → M along X and M
f is equivariantly
submanifold X such that there is the blow up M
diﬀeomorphic to the crossed product SU (2) ×S 1 N for some N .
See [12] for more general results (in particular, for GKM manifolds with larger symmetries) and further studies.
Acknowledgements. The author would like to thank all speakers and participants in
KAIST Toric Topology Workshop 2010 from February 22nd to 26th 2010. I also would
like to express my gratitude to Professor DongYoup Suh for providing me excellent circumstances to do research.
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S. KUROKI[12] S. Kuroki, GKM manifolds with large symmetries and their GKM graphs, in preparation.
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Department of Mathematical Sciences, KAIST, Daejeon 305-701, R. Korea
E-mail address: kuroki@kaist.ac.kr