Low-dimensional GKM theory Oliver Goertsches Panagiotis Konstantis and Leopold Zoller October 13 2022

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Low-dimensional GKM theory

Oliver Goertsches∗

, Panagiotis Konstantis†

, and Leopold Zoller‡

October 13, 2022

Abstract

GKM theory is a powerful tool in equivariant topology and geometry that can be used

to generalize classical ideas from (quasi)toric manifolds to more general torus actions.

After an introduction to the topic this survey focuses on recent results in low dimensions,

where the interaction between geometry and combinatorics turns out to be particularly

fruitful.

Contents

1 Introduction 1

2 GKM theory and equivariant cohomology 3

2.1 Associating a graph to a torus action . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Examples ....................................... 5

2.3 Equivariantcohomology ............................... 8

2.4 GKMmanifolds.................................... 11

2.5 The abstract notion of a GKM graph and geometric structures . . . . . . . . . . 14

2.6 TheGKMcorrespondence .............................. 19

3 Dimension ≤421

4 Dimension ≥822

5 Dimension 624

5.1 Exotic Hamiltonian actions and non-equivariant rigidity . . . . . . . . . . . . . . 24

5.2 Realization of GKM ﬁbrations with geometric structure . . . . . . . . . . . . . . 24

5.3 The smooth GKM correspondence in dimension 6 . . . . . . . . . . . . . . . . . 26

1 Introduction

The seminal work of Thomas Delzant [13] is a milestone in the ﬁeld of torus actions, showing

the wonderful interplay between torus actions, geometric structures and combinatorics. The

so-called Delzant correspondence comprises two statements: rigidity, stating that a symplectic

manifold M2nof dimension 2non which a torus of rank nis acting in a Hamiltonian fashion – i.e.,

a toric symplectic manifold – is completely determined, up to equivariant symplectomorphism,

∗Philipps-Universit¨at Marburg, email: goertsch@mathematik.uni-marburg.de

†Philipps-Universit¨at Marburg, email: pako@mathematik.uni-marburg.de

‡Ludwig-Maximilians-Universit¨at M¨unchen, email: leopold.zoller@mathematik.uni-muenchen.de

arXiv:2210.06234v1 [math.AT] 12 Oct 2022

by the image of its moment map, which is a so called Delzant polytope. And secondly, realization:

given the combinatorial datum of a Delzant polytope, one can construct a toric symplectic

manifold whose momentum image is the given Delzant polytope. Similar correspondences have

also been observed for other types of torus actions, for example quasitoric [12] or torus manifolds

[47,48].

What these results have in common is that are naturally located in the realm of complexity

zero actions, where the complexity of a torus action is deﬁned to be the diﬀerence 1

2dim M−rk T.

If one wishes to generalize such correspondences between certain types of torus actions and

combinatorial objects, there are several natural ways to proceed. One would be to increase

the complexity gradually, i.e., to consider now actions of complexity one. Recently this has

become an active ﬁeld of research, both from the Hamiltonian and the topological point of

view. Various properties that are known in the (quasi)toric case, such as the orbit space of the

action or its equivariant cohomology, were investigated, see for instance [42,7,8,37], and there

also begin to emerge classiﬁcation results for certain subclasses of actions, such as that of tall

symplectic complexity one actions [41] or certain topological complexity one actions in general

position [6].

Another approach to obtain results in higher complexity is given by GKM theory, the

topic of this survey. This theory and its main players, GKM actions or GKM manifolds (see

Deﬁnition 2.28), are named after Goresky, Kottwitz and MacPherson [31]. The celebrated

Chang-Skjelbred lemma (Theorem 2.22) makes it possible, under the assumption of equiv-

ariant formality (Deﬁnition 2.20), to compute the equivariant cohomology of a torus action

entirely in terms of the 1-skeleton (Remark 2.5) of the action. In GKM theory one relaxes

the assumption on complexity completely, but instead imposes a condition on the set of zero-

and one-dimensional orbits to be as simple as possible. More precisely, the ﬁxed point set is

assumed to be ﬁnite, and the one-skeleton has to be a union of T-invariant 2-spheres, as is

the case in the (quasi)toric setting. The orbit space of the 1-skeleton then naturally leads to a

combinatorial object, speciﬁcally a graph, where the vertices correspond to the ﬁxed points and

the edges to the invariant 2-spheres. Together with a natural labelling of the graph by weights

of the isotropy representations at the ﬁxed points, we arrive at the GKM graph of the action,

which acts as a replacement for the polytopes one associates to actions in the (quasi)toric world.

This brings us back to the Delzant correspondence which we now may try to generalize to the

GKM setting. Note that it is possible to deﬁne the notion of an abstract GKM graph, a purely

combinatorial object (cf. Section 2.5). The most basic version of the GKM correspondence is

the map

{GKM manifolds} −→ {abstract GKM graphs}

sending a GKM manifold to its GKM graph, and one may ask about rigidity and realization

statements, i.e., in how far this map is a bijection – modulo appropriate isomorphisms on

both sides, see Section 2.6. There are several variants of this correspondence, depending on

the chosen coeﬃcient ring in cohomology, as well as on the presence of additional invariant

geometric structures.

The purpose of this survey is twofold: in Section 2we wish to give a comprehensible

introduction to GKM theory focusing both on rational and integral aspects of the theory –

see also [57], or [29, Section 11] for other general introductions to GKM theory. In the later

sections we discuss recent developments on the GKM rigidity and realization questions.

We argue in Section 3that in dimension ≤4, GKM manifolds are automatically torus

manifolds, so that this theory adds nothing new to the picture in these dimensions. In dimension

8 and higher, which are discussed in Section 4, there are no available results on GKM realization

beyond the classical results in complexity zero. GKM rigidity fails (Theorem 4.1), which is

reasonable since in dimension 8 and higher manifolds cannot be distinguished by cohomology

and characteristic classes.

The study of the GKM correspondence turns out to be the most fruitful in low dimensions,

especially in dimension 6, which is the topic of Section 5. Note that the most natural case is to

consider an action of the two-torus T2on a simply-connected 6-manifold, as for the three-torus

we would land in the realm of complexity zero actions and in case of a circle action we cannot

obtain a GKM action. Therefore in dimension 6 GKM theory naturally intersects with the

theory of complexity one actions. It turns out that, at least in the setting without additional

geometric structures, in this dimension we have a fairly complete picture.

Concerning the rigidity question in dimension 6, the GKM graph of a simply-connected inte-

ger GKM manifold with certain assumptions on the stabilizers encodes both the non-equivariant

diﬀeomorphism and the equivariant homeomorphism type. For the ﬁrst statement, which is

Theorem 5.2 below, one uses the result from [58,39,62] that an oriented, simply-connected,

closed 6-manifold is uniquely determined up to diﬀeomorphism by cohomological data (see [51]

for a nice overview on this topic). For the second statement see Theorem 5.5. On the other hand

rigidity fails in presence of nontrivial discrete isotropies even in dimension 6 (Example 5.6). As

an application of these rigidity statements, it follows that Tolman’s [55] and Woodward’s [61]

examples of Hamiltonian non-K¨ahler actions and the Eschenburg ﬂag are non-equivariantly

diﬀeomorphic and equivariantly homeomorphic, see Example 5.1. In particular, it follows that

Tolman’s and Woodward’s examples carry a (non-invariant) K¨ahler structure.

With regard to the realization question in dimension 6 we mention two results. Some 3-

valent GKM graphs are graph-theoretically ﬁbrations over n-gons, which suggests that such a

graph ﬁbration could be geometrically realized by a projectivization of a complex rank 2 vector

bundle over a (quasi)toric 4-manifold or S4(see Section 5.2). This turns out to be true, see

Theorem 5.3. The realization problem in dimension 6 culminated in a recent project, where we

showed that all abstract 3-valent GKM graphs with the necessary conditions can be realized as

6-dimensional GKM manifolds (cf. Theorem 5.4). While in Theorem 5.3 we investigated the re-

alization question for ﬁbrations also with additional invariant almost complex, symplectic, and

K¨ahler structures, the general realization problem remains open in the setting with geometric

structures as of now.

Acknowledgements. This work is part of a project funded by the Deutsche Forschungsge-

meinschaft (DFG, German Research Foundation) - 452427095.

2 GKM theory and equivariant cohomology

2.1 Associating a graph to a torus action

In GKM theory one investigates a certain class of torus actions on smooth manifolds by means

of an associated combinatorial object, its GKM graph. In this section we will explain the

construction of this object.

Let T=Tk=S1×· · ·×S1denote a compact k-dimensional torus. Its Lie algebra tcontains

the lattice Zt= ker exp. We also identify the Lie algebra Lie(S1) of S1with Rsuch that Z⊂R

is the kernel of the exponential map on S1. Then for any λ∈Hom(T, S1) we may consider dλ

as an element of the weight lattice Z∗

t= Hom(Zt,Z)⊂t∗inside the dual of the Lie algebra. In

fact this establishes an isomorphism Hom(T, S1)∼

=Z∗

t∼

=Zkof Abelian groups, where for the

last identiﬁcation we use the dual basis of the standard basis of t= (Lie(S1))k=Rk. Under

these identiﬁcations, which will be used frequently, the tuple (l1, . . . , lk)∈Zkcorresponds to

the homomorphism Tk→S1with (t1, . . . , tk)7→ tl1

1·. . . ·tlk

Remark 2.1. We recall that any representation of Ton a complex vector space Vdecomposes

as V=V0⊕LλVλ, where V0is the subspace of ﬁxed vectors and, for a character λ:T→

S1⊂C,Vλ={v∈V|t·v=λ(t)vfor all t∈T}is the weight space of λ. The elements

06=α:= dλ ∈Z∗

tsuch that Vλ6= 0 are called the weights of the representation, and we

also write Vαinstead of Vλ. In case we are dealing with a representation of Ton a real

vector space V, we can also introduce a notion of weights, as elements in Z∗

t/±1: decomposing

V=V0⊕Was T-modules, we ﬁnd an invariant complex structure on W, turning Winto a

complex representation. The weights of the real representation are by deﬁnition ±α, where α

runs over the weights of the complex representation W. The weight space of ±αis by deﬁnition

V±α=Vα⊕V−α. Note that ±αcorresponds to a character λ:Tn→S1up to sign and that

ker λis independent of the sign choice. The action of ker λon V±αis trivial.

Let us consider a T-action on a compact connected smooth manifold M. We assume that

the action has ﬁnitely many ﬁxed points, i.e., that the ﬁxed point set

MT={p∈M|tp =pfor all t∈T}

is ﬁnite. For each p∈MT, we may consider the isotropy representation of Ton TpM. The

ﬁniteness of the ﬁxed point set implies that there are no ﬁxed vectors in TpMas otherwise the

equivariant exponential map would yield a positive dimensional ﬁxed manifold through p. As

a consequence, TpMdecomposes as a sum of irreducible 2-dimensional real T-representations

with nontrivial weights. Observe that this necessarily implies that Mis of even dimension in

case MTis nonempty. As of now several of these weights might belong to the same weight

space in the above sense.

However we imposes an additional assumption of the weights: to be precise we note that

the notion of pairwise linear independence is meaningful for elements of a vector space that

are well-deﬁned only up to sign, and demand that for each p, the weights of the isotropy T-

representation at pare pairwise linearly independent in Z∗

t/±1⊂t∗/±1. In particular, all

weight spaces of TpMare 2-dimensional.

Remark 2.2. Recall that, given an action of a compact Lie group Kon a smooth manifold M,

the set of K-ﬁxed points forms a submanifold of M(possibly with connected components of

varying dimension). Thus, given a weight α∈Z∗

t/±1 of the isotropy representation at pwith

character λ, we can consider the submanifold Mker λof M, and its component Ncontaining p.

Its tangent space at pis TpN= (TpM)α, i.e., equal to the weight space of α.

Thus under the above assumptions, for any weight αof the isotropy representation at p,

there is a 2-dimensional T-invariant submanifold Ncontaining p, such that TpN=Vα. By the

equality of Euler characteristics χ(N) = χ(NT) ([44], see also [29, Theorem 9.3] for a proof using

equivariant cohomology) it follows that Nis a closed surface with positive Euler characteristic,

and hence diﬀeomorphic to S2or RP2. We can rule out the latter case by assuming that M

is orientable: then, as the T-action induces an orientation on the normal bundle of N, the

submanifold Nis orientable as well. Now we have N=S2and there are exactly two T-ﬁxed

points in Nas χ(N) = 2. We have shown:

Proposition 2.3. Let M2nbe a compact, connected, orientable smooth manifold, equipped with

an action of a torus Twith ﬁnite ﬁxed point set, such that for all p∈MTthe weights of the

irreducible factors of the isotropy representation at pare pairwise linearly independent. Then

for each of the nweights αthere is an invariant two-sphere Sαthrough psuch that Tp(Sα)is

the weight space of α.

Remark 2.4. Any T-invariant two-sphere Sin Marises in the way described above. In fact,

denoting by H⊂Tthe subgroup of elements acting trivially on S, we obtain a character

λ:T→T/H ∼

=S1. As χ(S) = 2, the sphere Scontains two T-ﬁxed points pand q; the

corresponding isotropy representations admit the weight α=±dλ, whose weight spaces are

tangent to S. In particular, under the assumptions of the proposition there exist only ﬁnitely

many T-invariant two-spheres in M.

This allows us to construct a labelled graph. In the situation of the proposition we

(i) draw one vertex for each element in MT,

(ii) draw one edge for each T-invariant two-sphere in M, connecting the vertices corresponding

to the two T-ﬁxed points in S.

(iii) label each edge with the corresponding weight of the isotropy representation, as an element

of Z∗

t/±1.

Remark 2.5. The one-skeleton of a T-action on a smooth manifold Mis the union of at most

one-dimensional orbits

M1:= {p∈M|dim T p ≤1}.

Any T-invariant two-sphere is contained in the one-skeleton M1. The assumptions we made

in Proposition 2.3 have the purpose to ensure that the part of M1which is connected to the

ﬁxed points can be encoded purely combinatorially in the labelled graph constructed above.

Note in particular that the vital assumption of linear independence of the weights is the key

of reducing the wild world of possible candidates for Nas above and opening the door to

a purely combinatorial description. However in order to complete the deﬁnition of a GKM

manifold, as we will in Section 2.4, there is one crucial condition missing which will ensure that

M1does in fact see all (equivariant) cohomological information about M. This condition was

named equivariant formality in [31] and is an equally strong and natural condition from the

world of equivariant cohomology. We will introduce and discuss it in Section 2.3 after a brief

introduction to equivariant cohomology (cf. Deﬁnition 2.20). For now we just note two things:

ﬁrstly, in the setup of Proposition 2.3 the condition of equivariant formality is equivalent to

the vanishing of the odd dimensional (non-equivariant) cohomology of M. Hence equivariant

cohomology is not needed to give the deﬁnition of a GKM manifold. Secondly, the additional

condition of equivariant formality nicely completes our set of assumptions in two ways: it will

not only tell us how the topology of Mis encoded in M1but also assure that M1is precisely the

graph of two spheres constructed above, which in combination tells us that all the cohomological

information is encoded in the labelled graph.

As an addendum to the latter point, observe that as of now the one-skeleton of the action

might be larger than the union of invariant two-spheres. For example, with the current set of

conditions nobody prevents us to consider a free circle action, for which this associated graph

would be empty, and the one-skeleton would be the whole manifold. Slightly more interesting,

we could also consider the equivariant connected sum along a neighborhood of a regular orbit

of an action with isolated ﬁxed points and an action whose one-skeleton consists entirely of

one-dimensional orbits. For example, the equivariant connected sum of the standard T2-action

on S4(see Example 2.6 below) with the product S3×S1, where T2acts in standard fashion on

S3⊂C2and trivially on S1, has a disconnected one-skeleton: one component consists of the

two invariant two-spheres encoded in the graph of the action, and two further components are

invariant two-tori without ﬁxed points.

2.2 Examples

Example 2.6. Consider M=S2n, the 2n-dimensional sphere, as the unit sphere in Cn⊕R.

Then the n-dimensional torus T=Tn=S1× · · · × S1acts on S2nby rotating in the ncomplex

coordinates. The action has exactly two ﬁxed points, (0,...,0,±1), which are connected by

the ninvariant two-spheres S2n∩(0 ⊕ · · · ⊕ 0⊕C⊕0⊕ · · · ⊕ 0⊕R). The corresponding weights

are, up to sign, the elements of the dual basis {ei}of the standard basis of the Lie algebra t.

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Low-dimensionalGKMtheoryOliverGoertsches*,PanagiotisKonstantis,andLeopoldZollerOctober13,2022AbstractGKMtheoryisapowerfultoolinequivarianttopologyandgeometrythatcanbeusedtogeneralizeclassicalideasfrom(quasi)toricmanifoldstomoregeneraltorusactions.Afteranintroductiontothetopicthissurveyfocusesonrec...

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