GKM manifolds are not rigid
aa r X i v : . [ m a t h . A T ] S e p GKM manifolds are not rigid
Oliver Goertsches ∗ , Panagiotis Konstantis † , and Leopold Zoller ‡ September 21, 2020
Abstract
We construct effective GKM T -actions with connected stabilizers on the total spacesof the two S -bundles over S with identical GKM graphs. This shows that the GKMgraph of a simply-connected integer GKM manifold with connected stabilizers does notdetermine its homotopy type. We complement this by a discussion of the minimality ofthis example: the homotopy type of integer GKM manifolds with connected stabilizersis indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lowernumber of fixed points. Regarding geometric structures on the new example, we find analmost complex structure which is invariant under the action of a subtorus. In additionto the minimal example, we provide an analogous example where the torus actions areHamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integerGKM manifolds. There is a rich and successful history of describing manifolds with torus actions through dis-crete objects. Most famously this is illustrated by the bijective correspondence between toricvarieties and their fans or more specifically the correspondence between symplectic toric mani-folds and Delzant polytopes. In [16] Masuda proved that the equivariant isomorphism type of atoric manifold, when considered as a complex variety, is determined by its integral equivariantcohomology ring (or equivalently by its GKM graph [5]). The more ambitious cohomologicalrigidity problem, in its original form posed by Masuda–Suh in [18, Problem 1], asks if a toricmanifold is determined up to (nonequivariant) homeomorphism by its integral cohomology ring.While the problem in this form is unsolved as of today, several variants have been considered inthe literature. For instance, when restricting attention to certain (generalized) Bott manifolds,it was seen to be true, see [2], Sections 2.1 and 2.2 and references therein.Motivated by the success in the toric case it is natural to look for similar results in ageneralized setting. Prominent candidates are the classes of quasitoric manifolds and torusmanifolds which were also considered in the light of the cohomology rigidity problem in [3, 1, 29](note that the latter fails for torus manifolds, see [18, Example 3.4]). The main object ofstudy in the present article is the class of so-called (integer) GKM manifolds, named afterGoresky, Kottwitz and MacPherson [9], which in particular generalizes toric manifolds. Theseare compact manifolds with vanishing odd-degree (integral) cohomology, equipped with anaction of a compact torus, whose fixed point set is finite, and whose one-skeleton is a unionof invariant 2-spheres. To such actions one associates a labelled graph, its GKM graph, which ∗ Philipps-Universit¨at Marburg, email: [email protected] † Universit¨at zu K¨oln, email: [email protected] ‡ Ludwig-Maximilians-Universit¨at M¨unchen, email: [email protected]
Theorem 1.1.
On the total spaces of the two S -bundles over S there exist GKM T -actionswith identical GKM graph. While the two spaces in question have identical integral cohomology ring and characteristicclasses (as needs to be the case for any counterexample) they are not homotopy equivalent byLemma 3.1 below due to differing fifth homotopy groups. In particular this also shows thatthe equivariant cohomological rigidity problem [16, Theorem 1.1] does not generalize to integerGKM manifolds. Regarding geometric structures, we find an almost complex structure whichis invariant under a two dimensional subtorus (see Theorem 3.9):
Theorem 1.2.
The total space of the nontrivial S -bundle over S admits an almost complexstructure invariant under a circle action (even an action of the two-dimensional torus) withexactly four fixed points. This implies an example of a simply connected 8-manifold with an invariant almost complexcircle action with four isolated fixed points which is not diffeormophic to S × S . In [15]Kustarev constructs many non-simply-connected 6-manifolds with almost complex circle actionsfixing exactly two points. Crossing these examples with S one obtains non-simply-connected8-manifolds each endowed with an almost complex circle action with four isolated fixed points.We remark that the significance of our example is due to the vanishing of the odd integercohomology. This forces that each even Betti number must be equal to one (remember that theEuler characteristic must be equal to the number of isolated fixed points). As a further remarkwe note that by [13] our example is unitary cobordant to S × S furnished with its standardalmost complex structure.While the above examples are of course not symplectic, slight modifications give rise to thefollowing Theorem 1.3.
On the total spaces of two C P -bundles over C P which are not homotopyequivalent, there exist Hamiltonian GKM T -actions with identical x-ray. The notion of x-ray encodes in particular the GKM graph and is recalled in Section 5.2. Asa consequence, we show in Proposition 5.6 that symplectic cohomological rigidity, as defined in[22], does not hold for Hamiltonian integer GKM manifolds.We also discuss positive answers to the rigidity question in special cases with respect tothe number of fixed points, dimension, and complexity of the action, where the complexityof a T r -action on M n is defined as n − r . The situation for simply-connected integer GKMmanifolds with connected stabilizers, as known to the authors, is depicted in the table below.In particular this shows that the counterexample from Theorem 1.1 is simultaneously minimalwith respect to all three of the above parameters. Details are discussed in Section 4.Of the results in the table, only rows 4 and 6 are original to this paper. The first row followsfrom [21, Theorem p. 537] (alternatively, the arguments in [29, Section 6] for 6-dimensional torus2anifolds are also applicable). The second row is a consequence of [6, Theorem 3.1]. The thirdrow makes use of the generalized Poincar´e conjecture [23] and is, together with the fourthrow, discussed at the end of Section 4. Note that we do not know if any exotic sphere admitsan action of GKM type. In the remaining fifth row the first two columns are deduced from[30, Theorem 3.4] and the third follows from [29, Theorem 4.1] (for the latter we remark thatequivariant homeomorphisms preserve the GKM graph up to isomorphism).type: homotopy homeomorphism diffeomorphism equivariant homo-topy/homeo/diffeodim ≤ ≤ Acknowledgements.
We are grateful to Yael Karshon and Susan Tolman for valuable re-marks on a previous version of the paper, including a simplified construction of the action onthe bundle E in Section 3. We wish to thank Michael Wiemeler for several helpful comments.This work is part of a project funded by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) - 452427095. In this paper we consider actions of compact tori T on closed manifolds M . Given such anaction, we denote its fixed point set by M T , and its one-skeleton by M = { p ∈ M | dim T p ≤ } . Definition 2.1 ([9]) . We say that a T -action on a closed manifold M is (integer) GKM if1. H odd ( M, Z ) = 02. M T is finite3. M is a finite union of T -invariant 2-spheres.For any GKM T -action on M , the quotient M /T is homeomorphic to a graph, with onevertex for each fixed point, and one edge for each invariant 2-sphere. The tangent spaceof any such sphere in each of its two fixed points is an invariant subspace of the respectiveisotropy representation. We attach as a label to its edge in the graph the corresponding weight,considered as an element in Z ∗ t / ±
1, where t is the Lie algebra of T and Z ∗ t ⊂ t ∗ its weightlattice. This labelled graph will be called the GKM graph of M .If M is equipped with a T -invariant almost complex structure, then the above-mentionedweights are naturally elements of Z ∗ t . In this setting, we attach instead to any oriented edge thecorresponding weight at the initial vertex, and sometimes speak about the signed GKM graph of the action. Example 2.2.
Any toric symplectic manifold is of GKM type. In this case, the GKM graph isgiven by the one-skeleton of the momentum polytope; the labels of the edges are given by theprimitive vectors pointing in direction of its slopes.The (signed) GKM graph of an action is of relevance because under a certain connectednessassumption on the isotropy groups it encodes various topological properties of M and theaction, such as its (equivariant) cohomology algebra [4, Corollary 2.2], and its (equivariant)characteristic classes [6, Proposition 3.5]. 3 The minimal example
We construct two simply-connected GKM T -manifolds in dimension 8 which are not (non-equivariantly) homotopy equivalent but have the same GKM graph. The underlying manifoldswill be S × S as well as the total space of the non-trivial S -bundle over S . We begin by recalling the construction and show that the total spaces of the two S -bundlesover S are indeed not homotopy equivalent. Principal SU(2)-bundles over S are given aspullbacks of the universal bundle SU(2) → S ∞ → H P ∞ along maps S → H P ∞ . The space H P ∞ has a CW-structure with cells only in dimensions which are multiples of 4. In particularits 7-skeleton is H P = S and thus S ֒ → H P ∞ induces an isomorphism on homotopy groupsin dimensions up to 6. We now use the nontrivial generator f : S → S of π ( S ) ∼ = Z to pullback the universal bundle. The resulting commutative diagramSU(2) / / (cid:15) (cid:15) P / / (cid:15) (cid:15) S (cid:15) (cid:15) SU(2) / / S ∞ / / H P ∞ induces a commutative diagram · · · / / π ( S ) / / f ∗ (cid:15) (cid:15) π (SU(2)) / / id (cid:15) (cid:15) π ( P ) / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / π ( H P ∞ ) / / π (SU(2)) / / / / π ( H P ∞ ) / / · · · in which the rows are the long exact homotopy sequences of the bundles in question. By thechoice of f , the map f ∗ : π ( S ) → π ( H P ∞ ) = π ( S ) is surjective. Furthermore, the connect-ing homomorphism π ( H P ∞ ) → π (SU(2)) is an isomorphism since S ∞ is contractible. As aconsequence the connecting homomorphism π ( S ) → π (SU(2)) is surjective. In combinationwith π ( S ) = 0 this implies π ( P ) = 0. Finally, we obtain a C P -bundle X → S from P by factoring out the fiberwise action of the circle S(U(1) × U(1)). Since this action is free weobtain a fiber bundle S → P → X . Again, via the long exact homotopy sequence, this impliesthe following well known Lemma 3.1.
We have π ( X ) = 0 whereas π ( S × S ) = π ( S ) = Z . We construct a GKM action on X . To do this, we show that the map f : S → S in theconstruction above can be chosen to be T -equivariant with respect to certain actions. Thenwe lift the action on S to an action of the restricted universal bundle which descends to theprojectivization. In this way X will be an equivariant pullback and will naturally carry a T -action.Consider the T -action on S ⊂ C given by ( s, t, u ) · ( v, w, z ) = ( su, tw, uz ). The subcircles K = { ( s, s, ∈ T } , K = { ( s, , s ) ∈ T } , and K = { ( s, , ∈ T } generate T . Lemma 3.2.
The orbit space S /K is homeomorphic to S ⊂ C ⊕ R ⊕ C in a way suchthat the induced action of K corresponds to ( s, , s ) · ( v, h, w ) = ( sv, h, sw ) and K acts as ( s, , · ( v, h, w ) = ( sv, h, w ) for ( v, h, w ) ∈ S .Proof. Recall the suspension homeomorphism Σ S n → S n +1 given by[( a , . . . , a n ) , t ] ( ϕ ( t ) a , . . . , ϕ ( t ) a n , t ) , ϕ ( t ) = √ − t . Applying this twice, we see that the K -action corresponds tothe doubly suspended diagonal action on Σ S . In particular its orbit space is Σ S = S as claimed. To prove the statement on the actions we need to recall the explicit form of thehomeomorphism S /S = C P ∼ = S .Let S = { ( v, h ) ∈ S ⊂ C ⊕ R | h ≥ } be the upper hemisphere and denote by A ⊂ S the equator. Then S /A is homeomorphic to C P via ( v, h ) [ v, h ]. On the other hand S /A is homeomorphic to S by the stretching map ( v, h ) ( α ( h ) v, h − α ( h ) ∈ [0 ,
1] is asuitable scaling factor. Finally, note that for z ∈ C with | z | ≤ t = Im( z ), s = ϕ ( t ) − Re(z),we have ϕ ( s ) ϕ ( t ) = ϕ ( | z | ). Thus, under the double suspension map, when explicitly defined asabove, the preimage of a point ( v, w, z ) ∈ S is given by (cid:2) ( ϕ ( | z | ) − v, ϕ ( | z | ) − w ) , s, t (cid:3) with s, t as above. Piecing everything together we obtain S ∼ = Σ S → Σ C P ∼ = Σ S /A ∼ = Σ S ∼ = S ⊂ C ⊕ R ⊕ C given by( v, w, z ) [( ϕ ( | z | ) − v, ϕ ( | z | ) − w ) , s, t ] ∈ Σ S [( ϕ ( | z | ) − β ( w ) v, ϕ ( | z | ) − | w | ) , s, t ] ∈ Σ S /A [ (cid:0) α (cid:0) ϕ ( | z | ) − | w | (cid:1) ϕ ( | z | ) − β ( w ) v, ϕ ( | z | ) − | w | − (cid:1) , s, t ] ∈ Σ S (cid:0) α (cid:0) ϕ ( | z | ) − | w | (cid:1) β ( w ) v, | w | − ϕ ( | z | ) , z (cid:1) ∈ S , where β ( w ) is 1 if w = 0 and w/ | w | otherwise. Since | z | is invariant under multiplicationwith S , we see that the identification S /K ∼ = S above is equivariant with respect to S -multiplication in the first and third coordinate.We now consider the suspended T -action on S . Proposition 3.3.
The orbit space S / ( K K ) is homeomorphic to S in a way that the orbitmap f : S → S ( K K ) ∼ = S is T -equivariant with respect to the T -action on S in which K and K act trivially, while K acts via ( s, , · ( v, w, h ) = ( sv, w, h ) . Furthermore, f definesa generator of π ( S ) .Proof. As we have seen in the previous lemma, the projection S → S /K ∼ = S correspondsto the double suspension Σ g of the Hopf map g : S → S . It also follows from the lemma that S → S /K ∼ = Σ C P ∼ = Σ S ∼ = S can be identified with Σ g and can be chosen K -equivariantwith respect to the action ( s, , · ( v, w ) = ( sv, w ) on S . Thus S → S / ( K K ) ∼ = S isthe composition Σ g ◦ Σ g which is known to give a generator of π ( S ), cf. [11, p. 475]. Thesuspension is a generator of π ( S ) by the Freudenthal suspension theorem and satisfies theequivariance property as claimed in the lemma.Consider the quaternionic Hopf bundle S −→ S −→ S which is given by dividing out the quaternionic diagonal action by right multiplication on S ⊂ H and identifying the orbit space H P with S . The orbit map factors as S → C P → H P ∼ = S where the first map divides by complex diagonal multiplication from the right (viewing C as asubset of H ). The second map is a fiber bundle with fiber C P , and as the quaternionic Hopfbundle is the restriction of the universal SU(2)-bundle S ∞ → H P ∞ to H P , by construction X
5s the pullback of C P → S along f . We will now construct compatible actions on C P → S which pull back to X .On S ⊂ H consider the S -action from the left defined by s · ( p, q ) = ( sp, sq )where p, q ∈ H . This commutes with the actions of the complex and quaternionic diagonalsfrom the right. In particular C P → S is equivariant with respect to the induced circle action. Proposition 3.4. On S ⊂ C ⊕ R this action can be identified with s · ( v, w, h ) = ( s v, w, h ) .On the fibers of C P → S over the fixed points of S , the induced action is equivariantlydiffeomorphic to the rotation s · ( v, h ) = ( s v, h ) on S ⊂ C ⊕ R .Proof. We recall the identification H P ∼ = S . Denote by S the hemisphere { ( q, h ) ∈ S ⊂ H ⊕ R | h ≥ } and by A the equator at height 0. Now the map ( q, h ) [ q : h ] inducesa homeomorphism S /A ∼ = H P . For h ∈ R we have s · [ q : h ] = [ sq : sh ] = [ sqs − : h ].The conjugation action s · q = sqs − leaves the complex plane fixed and rotates the j - k -planeat double speed. Thus with the correct identification S /A ∼ = S this is indeed the actiondescribed in the proposition. The fixed points are the points [ v : w ] ∈ H P with v, w ∈ C . For v, w ∈ C , not both zero, the map q [ vq : wq ] C induces a diffeomorphism from S /S ontothe fiber over [ v : w ], where the quotient S /S is by quaternionic multiplication with complexnumbers from the right, and the notation [ · : · ] C describes points in the quotient of S by thecomplex diagonal action from the right. Using that v, w ∈ C , the fiber inclusion S /S → C P satisfies sqS [ vsq : wsq ] C = s · [ vq : wq ] C , and is thus equivariant with respect to quaternionic multiplication with S from the left on S /S . This can be identified with the action from the proposition.The S -action on the bundle C P → S has kernel generated by −
1. We divide by thecorresponding subgroup to make the action effective. Then we pull back the S -actions alongthe homomorphism ψ : T → S , ( s, t, u ) → st − u − . By Proposition 3.3 the resulting T -actionon S is the one with respect to which the map f : S → S is equivariant. The space X is thusthe pullback of the T -equivariant bundle C P → S along the T -equivariant map f : S → S and thus inherits a T -action.For later purposes we will need Remark 3.5.
Since the total space of C P → S is obtained by dividing out a subcircle actionof SU(2), we have that the C P -bundle C P → S has structure group SU(2) inherited from theprincipal SU(2)-bundle S → S . Therefore the structure group of the pullback C P -bundle X → S is SU(2) as well. Remark 3.6.
The map f may be approximated T -equivariantly by a smooth map. Thisdoes not affect the pullback, so we obtain a smooth structure on X with respect to which the T -action is smooth. Theorem 3.7.
The -skeleton of the T -action on X is T -equivariantly homeomorphic to the -skeleton of the product T -action on S × S which acts on S via ( s, t, u ) · ( v, h ) = ( st − u − v, h ) and on S via ( s, t, u ) · ( v, w, z, h ) = ( sv, tw, uz, h ) for v, w, z ∈ C , h ∈ R . This has the following immediate
Corollary 3.8. On X and S × S there exist GKM T -actions with isomorphic GKM graphs. The corresponding GKM graph is given by61 , , , , , , , , , , , ,
1) (1 , − , − , − , − Proof of Theorem 3.7.
Denote by A ⊂ S the 1-skeleton of the T -action. The 1-skeleton of X is certainly contained in p − ( A ), where p : X → S is the projection. By naturality of thepullback p − ( A ) is the pullback of C P → S along f | A . Recall that f was defined as thesuspension of the orbit map S → S / ( K K ). The space A is the suspension of the 1-skeleton B ⊂ S which consists of the three disjoint circles containing those elements ( v, w, z ) ∈ S wheretwo coordinates are zero. As derived in Lemma 3.2, the map S → S /K ∼ = S ⊂ C ⊕ R ⊕ C can be explicitly described as( v, w, z ) (cid:0) α (cid:0) ϕ ( | z | ) − | w | (cid:1) β ( w ) v, | w | − ϕ ( | z | ) , z (cid:1) . From this it follows that under S → S / ( K K ) ∼ = S , the space B maps to three distinct pointswhich are fixed by K . Thus the suspended map sends A to three joined lines in ( S ) K . Fromthe description of the K -action (see also Proposition 3.3) we deduce that ( S ) K = ( S ) T ∼ = S .As f | A is not surjective onto this S it is homotopic within S to the constant map at somepoint x ∈ S . This homotopy is automatically equivariant because S is fixed under T . Byhomotopy invariance of the pullback, p − ( A ) is T -equivariantly isomorphic to the diagonalaction on A × F x where F x denotes the fiber of C P → S over x . In combination withProposition 3.4 it follows that the 1-skeleton of X has the desired form. The manifold X does not admit a T -invariant almost complex structure as no structure of asigned GKM graph that is compatible with the GKM graph of X can admit a connection in thesense of [10] (to see this, note that transport along a horizontal edge via a connection can notmap the other two horizontal edges onto themselves or one another, so they would both needto map to the single adjacent vertical edge). However we now argue that there is an almostcomplex structure on X that is invariant under a 2-dimensional subtorus. We want to stressthat this restricted action will not be GKM.Recall that X is a T -equivariant C P -bundle over S , where T acts on S in the standardway ( s, t, u ) · ( v, w, z, h ) = ( sv, tw, uz, h ) in S ⊂ C ⊕ R . It is well-known that S admits analmost complex structure that is invariant unter an action of the compact Lie group G . Theaction of a maximal torus T ⊂ G on S can be identified as a subaction of our T -action,namely as that of the subtorus T := { ( st, s − , t ) } ⊂ T , see e.g. [10, Example 1.9.1]. Now, thetangent bundle T X decomposes as
T X = π ∗ T S ⊕ V F , where V F is the subbundle consistingof the tangent spaces of the fibers of X → S . On π ∗ T S we obtain a T -invariant almostcomplex structure by pulling back the T -invariant almost complex structure from S . For V F consider a fiber of X over S which can be identified with C P . We choose an SU(2)-invariant7lmost complex structure on C P , which defines an almost complex structure on the fiber.Since the structure group is SU(2) this definition does not depend on the identification of thefiber with C P , cf. Remark 3.6. This defines an almost complex structure on V F . Moreoverthe T -action leaves this almost complex structure invariant. To see this recall that the T -action is induced by a circle action on S which commutes with the action of SU(2) of theprincipal bundle (see the paragraph before Proposition 3.4). In particular this means that thecircle action preserves the fibers of C P → S and the transformations are given by elementsof the structure group SU(2). Thus the circle action respects the almost complex structure onthe fibers and consequently so does the pullback T -action on V F . Altogether one obtains a T -invariant almost complex structure on X by restricting the T -action on V F to T . Now, wearrive at the following theorem, which gives an example of an almost complex circle action onan 8-manifold with four isolated fixed points such that the odd cohomlogy vanishes and whichis not diffeomorphic to S × S . Theorem 3.9.
The manifold X , which is not diffeomorphic to S × S , admits an almostcomplex structure invariant under a circle action (even an action of the two-dimensional torus)with exactly four fixed points. In this section we observe that our example is optimal with regards to three properties: com-plexity, dimension, and number of fixed points.
Complexity.
For an effective T r -action with non-empty fixed point set on a compact 2 n -dimensional manifold one has r ≤ n and the number n − r is called the complexity of theaction. Thus the previously constructed T -action on X is of complexity 1. Complexity 0manifolds of the above type are also known as torus manifolds. Theorem 3.4 in [30] statesthat the nonequivariant homeomorphism type of a simply-connected torus manifold M with H odd ( M, Z ) = 0 is encoded in the face poset of its orbit space together with a function thatassociates to each face the corresponding isotropy group. If we denote by T the acting torus,then the face poset is – as a partially ordered set – equivalent to the closed orbit type stratifi-cation χ which is the collection of all connected components of M H where H runs through allsubgroups of T , ordered by inclusion. On this poset one considers the function that associatesthe respective principal isotropy group to a connected component of M H . We stress that thehomeomorphism type of M is determined just by the combinatorial data of the poset togetherwith the function which associates the isotropy groups and does not require specific knowledgeof the isotropy submanifolds.For GKM manifolds this information is encoded in the GKM graph: Let M be a GKM T -manifold (not necessarily of complexity 0) with GKM graph Γ. For a subgroup H ⊂ T , setΓ H ⊂ Γ to be the minimal subgraph that contains all edges whose weights – understood aselements in Hom( T → S ) – vanish on H . Define χ Γ to be the set consisting of all connectedcomponents of subgraphs of the form Γ H , partially ordered by inclusion. Lemma 4.1.
There is bijection χ → χ Γ which preserves the partial ordering such that for some N ∈ χ its principal isotropy group is given by the intersection of all kernels of the weights ofall edges emanating from a single vertex in the corresponding element of χ Γ .Proof. The one-skeleton of M H is encoded in the minimal (possibly disconnected) subgraphΓ H ⊂ Γ. As the odd cohomology of (every component of) M H vanishes, see [17, Lemma 2.2],the T -action on every component of M H is again GKM and, since GKM graphs are alwaysconnected, the connected components of Γ H are the GKM graphs of the components of M H .8he principal isotropy type on a component of M H can be reconstructed in the way describedin the lemma, since the weights at a fixed point determine the action on a neighbourhood. Thisalso implies the injectivity of the correspondence: if N ⊂ M H and N ′ ⊂ M H ′ are connectedcomponents corresponding to the same element of χ Γ then the principal isotropy groups of N and N ′ agree and are equal to some U containing both H and H ′ . Thus both are componentsof M U and have nonempty intersection, which implies N = N ′ . Dimension.
The main theorem in [7] implies that for a simply-connected 6-dimensional inte-ger GKM manifold M whose stabilizer groups satisfy a certain assumption which is satisfied ifthey are connected, the GKM graph encodes the nonequivariant diffeomorphism type of M . Indimension 4, even the equivariant diffeomorphism type is known to be encoded by [21, Theo-rem p. 537] (or by the arguments for 6-dimensional torus manifolds in [29, Section 6]). Thus,our 8-dimensional examples of GKM manifolds that are not homotopy equivalent but have thesame GKM graph have the lowest possible dimension with this property. Number of fixed points.
Our example has exactly four fixed points. We would like to arguethat this is the minimal possible number of fixed points in our situation. For a simply-connectedinteger GKM manifold of positive dimension we always have at least two fixed points. In caseof exactly two fixed points, the manifold has the integer homology of a sphere. In this situation,by collapsing the complement of a disc, one always finds a map to the standard sphere thatinduces an isomorphism in integer homology, which by the homology Whitehead theorem is ahomotopy equivalence. From the (topological) generalized Poincar´e conjecture [23] we deducethat the manifold has to be homeomorphic to the standard sphere.For three fixed points we have the following proposition:
Proposition 4.2.
For a simply-connected integer GKM manifold M in arbitrary dimension,with three fixed points, the GKM graph determines the nonequivariant diffeomorphism type of M .Proof. If the action has precisely three fixed points, then the integral homology of M is Z indegrees 0, n , and 2 n , where 2 n is the dimension of M , and 0 in all other degrees. In this case,by Corollary B of [14] the diffeomorphism type of M is determined by the Pontryagin number p n ( M )[ M ] ∈ Z . Note that the cohomology ring of M is given by Z [ x ] / h x i with x ∈ H n ( M ),thus they have a natural orientation given by x = ( − x ) . Hence the Pontryagin number isencoded in the GKM graph of M by [6]. We will construct a Hamiltonian T -action of GKM type on a space Y which has the signedGKM graph of a diagonal action on C P × C P , while Y is not homotopy equivalent to thelatter product. The T -space Y arises as an equivariant pullback of the bundle X → S along a map k : C P → S . Explicitly we define k as the collapsing map C P → C P / C P ∼ = S , where C P isembedded as those points [ z : . . . : z ] ∈ C P with z = 0. The identification of the quotientspace with S can be done such that k is equivariant with respect to the action( s, t, u ) · [ z : z : z : z ] = [ sz : tz : uz : z ]9n C P and the standard T -action on S ⊂ C ⊕ R . Explicitly, let S = { ( v, w, z, h ) ∈ S | h ≥ } be the upper hemisphere and A ⊂ S the equator. Then one has homeomorphisms C P / C P ← S /A → S , where the first map is defined by ( v, w, z, h ) [ v : w : z : h ] and the second map is defined bystretching along the real coordinate as in the proof of Lemma 3.2.We define Y as the pullback of the T -equivariant bundle X → S along k , which is thesame as the pullback of the S -quotient C P → S of the quaternionic Hopf bundle S → S along f ◦ k , where f is defined in Proposition 3.3. Since k maps the one-skeleton of C P to the one-skeleton of S and the restriction of f to the one-skeleton of S was shown to beequivariantly homotopic to a constant map, the same holds for the composition f ◦ k whenrestricted to the one skeleton of C P . Analogously to Theorem 3.7 we obtain Theorem 5.1.
The -skeleton of the T -action on Y is T -equivariantly homeomorphic to the -skeleton of the product T -action on C P × C P which acts on C P via ( s, t, u ) · [ v, w ] =[ st − u − v, w ] and on C P via ( s, t, u ) · [ z : z : z : z ] = [ sz : tz : uz : z ] . It remains to prove that Y is not homotopy equivalent to the product C P × C P . Whilethe strategy is largely analogous to that of Lemma 3.1, the details are slightly more involved,drawing from classical results on homotopy groups of spheres. Lemma 5.2.
Let g be the Hopf map S → S and i : S ∼ = H P → H P ∞ be the inclusion.Then i ◦ Σ g ◦ Σ g ◦ Σ g defines a non-trivial element of π ( H P ) .Proof. As observed in Proposition 3.3, the map Σ g ◦ Σ g is a generator of π ( S ). In particularit induces a non-trivial map π ( S ) → π ( S ). Since g is the projection of an S -bundle, itinduces isomorphisms on higher homotopy groups, so the composition g ◦ Σ g ◦ Σ g induces anon-trivial map on π . Hence it is a generator of π ( S ).The next step is to show that its two-fold suspension Σ g ◦ Σ g ◦ Σ g is still non-trivial.This follows from the EHP-sequence [12] (or [11] for a modern exposition) which involves thesuspension homomorphism or rather its localization at 2 in a long exact sequence. The part ofthe sequence we are interested in reads · · · → π ( S ) (2) E (2) −−→ π ( S ) (2) → π ( S ) (2) → π ( S ) (2) E (2) −−→ π ( S ) (2) → · · · where E denotes the suspension homomorphism and the index (2) denotes localization at 2.Using that π ( S ) = Z (cf. [28]) and π ( S ) = Z by the Freudenthal suspension theorem aswell as π ( S ) = Z (cf. [25]), the left part of the sequence becomes · · · → Z E (2) −−→ Z → Z → E : π ( S ) → π ( S ) is an isomorphism (as argued in section 3, the two groups are Z andgenerated by g ◦ Σ g and Σ g ◦ Σ g respectively). Thus E : π ( S ) → π ( S ) is injective becausethis holds for the localized map and π ( S ) → π ( S ) (2) is an isomorphism. An analogoussequence exists for the suspension E on S . For odd spheres, no localization is necessary andthe relevant part of the sequence reads · · · → π ( S ) E −→ π ( S ) → π ( S ) → · · · which is Z → Z ⊕ Z → Z . Consequently the left hand map is necessarily injective. In totalwe deduce that the map Σ g ◦ Σ g ◦ Σ g , which is the double suspension of the generator of π ( S ), defines a non-trivial element of order 2 in π ( S ).10t remains to prove that it survives the inclusion into H P ∞ . It suffices to consider theinclusion into the 8-skeleton H P . The latter arises from S by attaching a single 8-cell viathe projection in of the Hopf fibration g ′ : S → S . By homotopy excision the kernel of π ( S ) → π ( H P ) is generated by g ′ . From the long homotopy sequence of S → S → S wededuce that [ g ′ ] ∈ π ( S ) is of infinite order. This implies that the class of Σ g ◦ Σ g ◦ Σ g ,which is torsion, is not contained in the kernel of π ( S ) → π ( H P ). Proposition 5.3.
We have π ( Y ) = Z whereas π ( C P × C P ) = Z .Proof. Denote by Q the pullback of the universal SU(2)-bundle along the map i ◦ f ◦ k : C P → H P ∞ , where i, f, k are as above. We have a principal S -bundle S → Q → Y so it suffices tocompute the homotopy groups of Q . There is a pullback diagramSU(2) / / (cid:15) (cid:15) Q / / (cid:15) (cid:15) C P i ◦ f ◦ k (cid:15) (cid:15) SU(2) / / S ∞ / / H P ∞ which induces a commutative diagram · · · / / π ( C P ) / / ( i ◦ f ◦ k ) ∗ (cid:15) (cid:15) π (SU(2)) / / id (cid:15) (cid:15) π ( Q ) / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / π ( H P ∞ ) / / π (SU(2)) / / / / π ( H P ∞ ) / / · · · on long exact homotopy sequences. The group π ( C P ) is generated by the canonical projection p : S → C P . By [27, Lemma 9.2] the composition k ◦ p is homotopic to the quadruplesuspended Hopf map Σ g . We have f = Σ g ◦ Σ g (cf. proof of Proposition 3.3) and thus byLemma 5.2 we conclude that ( i ◦ f ◦ k ) ∗ ([ p ]) = [ i ◦ Σ g ◦ Σ g ◦ Σ g ] is a non-trivial elementof order 2 in π ( H P ∞ ). Using the fact that π ( H P ∞ ) → π (SU(2)) is an isomorphism and π (SU(2)) = Z we see that π ( Y ) = Z . Recall that the space Y was defined as the total space of the pullback of an equivariant C P -fiber bundle C P → S along f ◦ k : C P → S → S . Let π : Y → C P be the projectionmap, then by Thurston’s construction [24], see also [19, Theorem 6.1.4], Y admits a sympleticstructure ω = ω F + Cπ ∗ ( ω B ), where ω B is the Fubini-Study form on C P , ω F is a closed 2-formwhich restricts to the Fubini-Study form on the fibers (after identifying them with C P up toelements of the structure group) and C > C P is invariant under the structure group of π (cf. Remark 3.6) and the inclusion of a fiber is surjective on cohomology due to the factthat the odd-dimensional cohomology of fiber and base vanishes. As argued in Section 3.2, the T -action on Y preserves the fibers of π , with the transformations between fibers correspondingto elements of the structure group. The restriction of ω F to any fiber is invariant under thestructure group. Thus if we average ω F over T we obtain a T -invariant closed 2-form e ω F whoserestriction to any fiber agrees with that of ω F . Clearly ω B is also invariant with respect to the T -actions, hence e ω = e ω F + Cπ ∗ ( ω B ) is a T -invariant symplectic form (potentially replacing C by a larger constant) on Y . In this way, a nonempty open set in the second cohomologyof Y can be realized as cohomology classes of T -invariant symplectic forms on Y . As Y issimply-connected, the T -action on Y is automatically Hamiltonian with respect to any of thesesymplectic forms. 11 emark 5.4. It is possible to write the fiber bundle C P → S as the projectivization of thevector bundle E := S × SU(2) C → S , where SU(2) acts on C via the standard representation.Thus, the space Y is the projectivization of the T -equivariant complex vector bundle ( f ◦ k ) ∗ E → C P . By [20, Chapter I, § C P admits a holomorphicstructure, which may be no longer equivariant, even if the original bundle was equivariant.Consequently, by [26, Proposition 3.18] its projectivization admits a K¨ahler structure. It followsthat the space Y is (nonequivariantly) K¨ahler. We do not know if it admits a T -invariantK¨ahler structure.In this section, we compare the symplectic structures on Y and C P × C P in regard totwo aspects: their x-ray and the symplectic cohomological rigidity problem [22].We recall the notion of the x-ray of a Hamiltonian action of a torus T on a manifold M withmomentum map µ : M → t ∗ : it is given by the datum of the closed orbit type stratification χ (as a partially ordered set) which consists of the connected components of all submanifolds M H ⊂ M , where H ⊂ T is a subgroup, together with the function that associates to N ∈ χ thepolytope µ ( N ) ⊂ t ∗ . Thus in the GKM case it encodes in particular the (signed) GKM graph(associated to an almost complex structure which is compatible with the symplectic form).However, the information on the lengths of the edges is lost, when passing from the x-ray tothe GKM graph. We have the following addendum to Theorem 5.1: Proposition 5.5. On Y and C P × C P there exist T -invariant symplectic forms with mo-mentum maps whose x-rays coincide.Proof. We have seen already that the one-skeletons of the two actions are equivariantly homeo-morphic. It suffices to prove the existence of respective momentum maps which agree on theone skeleton with respect to this homeomorphism: by Lemma 4.1, χ is encoded in the GKMgraph. For some N ∈ χ , the momentum image µ ( N ) is determined by the image under µ of allfixed points that are contained in N . These are however also determined by the correspondingsubgraph.In order to prove that there are momentum maps on Y and C P × C P which coincide on theone-skeleton (with respect to a fixed homeomorphism), we start by investigating a momentummap µ of the symplectic form ω on Y as constructed above. For a fixed ω , µ is unique up totranslation by some element in t ∗ . Let A be the one skeleton of the action on Y and B bethe one skeleton of C P . Every invariant 2-sphere in A gets mapped by µ to an affine linearsegment in t ∗ . The slope of this segment, when moving from a fixed point p in a sphere tothe other fixed point q , is determined up to sign by the weight of the sphere in t ∗ / ± . Thesign is determined by the orientation on the sphere which is induced by ω . The projection π : Y → C P maps some spheres in A homeomorphically onto invariant spheres in C P (wecall those horizontal) and is constant on the remaining spheres which are precisely the fibersover the fixed points of C P (we call those vertical). From the construction of ω we see that π is not necessarily a symplectomorphism on horizontal spheres (where spheres in C P areequipped with the restriction of ω B ). However if the constant C is large enough, at least π isorientation preserving on horizontal spheres.By Theorem 5.1, the subspace of all horizontal spheres in A decomposes into two connectedcomponents A + h and A − h , each of which gets homeomorphically mapped to B in equivariantand sphere-wise orientation preserving fashion. From the specific weights and orientations wededuce that µ | A ± h has to agree with µ B ◦ π up to translation and rescaling, where µ B is amomentum map for ω B . Now the vertical spheres get mapped to parallel line segments of slope ± (1 , − , −
1) in the standard basis of t ∗ . From the previous considerations we deduce that thesigns of the slope must agree when moving on vertical spheres from A + h to A − h . Also this forces µ | A + h and µ | A − h to be scaled in the same way and implies that all the lengths of the segmentsthat are the images of vertical spheres under µ agree.12o sum up the above discussion, µ | A is determined up to translation, global rescaling aswell as the length and sign of the vertical edges. The same considerations apply not only to Y → C P but also to the trivial bundle C P × C P → C P . Thus it remains to show thatthe above parameters can be manipulated in a way that the momentum maps agree on theone skeleton. Translation can be manipulated arbitrarily and global rescaling of an associatedmomentum map is achieved by rescaling the symplectic form ω . The sign of µ on vertical edgescan be changed by replacing ω = ˜ ω F + Cω B with − ˜ ω F + Cω B . Finally the length of the verticaledges is the only thing that can not be manipulated freely. However we can make it arbitrarilyshort when compared to the basic edges by enlarging the constant C . Thus we can change thesymplectic forms to make the momentum maps agree on the one-skeleton.We wish to demonstrate that the pair of Y and C P × C P are a counterexample to thesymplectic cohomological rigidity problem for integer GKM manifolds. The symplectic variantof the cohomological rigidity problem, as posed in [22], asks for families of symplectic manifoldsthat are distinguished by their cohomology ring and the cohomology classes of their symplecticstructures. Proposition 5.6. On Y and C P × C P there exist symplectic forms that are intertwinedvia the isomorphism on cohomology induced by the equivariant isomorphism of the respectiveone-skeleta from Theorem 5.1.Proof. We observed at the beginning of the section that an open subset of the second cohomol-ogy of Y is realized by symplectic structures on Y ; on C P × C P the same is true for an openand dense subset of the second cohomology group. The assertion is immediate.We observe that even more is true: a closed equivariant extension of the symplectic form ω on a symplectic manifold M with Hamiltonian T -action is an equivariant differential formof the form ω + µ , where µ is a momentum map of the T -action, see [8, Example 4.16]. Onecan choose appropriate momentum maps on Y and C P × C P such that the isomorphism H T ( Y ) ∼ = H T ( C P × C P ) induced by the homeomorphism from Theorem 5.1 intertwinesthe corresponding equivariant cohomology classes. In other words, these examples are notequivariantly symplectically cohomologically rigid. References [1]
Choi, S., and Kuroki, S.
Topological classification of torus manifolds which havecodimension one extended actions.
Algebr. Geom. Topol. 11 , 5 (2011), 2655–2679.[2]
Choi, S., Masuda, M., and Suh, D. Y.
Rigidity problems in toric topology: a survey.
Tr. Mat. Inst. Steklova 275 (2011), 188–201.[3]
Choi, S., Park, S., and Suh, D. Y.
Topological classification of quasitoric manifoldswith second Betti number 2.
Pacific J. Math. 256 , 1 (2012), 19–49.[4]
Franz, M., and Puppe, V.
Exact sequences for equivariantly formal spaces.
C. R.Math. Acad. Sci. Soc. R. Can. 33 , 1 (2011), 1–10.[5]
Franz, M., and Yamanaka, H.
Graph equivariant cohomological rigidity for GKMgraphs.
Proc. Japan Acad. Ser. A Math. Sci. 95 , 10 (2019), 107–110.[6]
Goertsches, O., Konstantis, P., and Zoller, L.
GKM theory and Hamiltoniannon-K¨ahler actions in dimension 6.
Adv. Math. 368 (2020), 107141.137]
Goertsches, O., Konstantis, P., and Zoller, L.
Realization of GKM fibrationsand new examples of hamiltonian non-K¨ahler actions. arXiv 2003.11298v1.[8]
Goertsches, O., and Zoller, L.
Equivariant de Rham cohomology: theory andapplications.
S˜ao Paulo J. Math. Sci. 13 , 2 (2019), 539–596.[9]
Goresky, M., Kottwitz, R., and MacPherson, R.
Equivariant cohomology, Koszulduality, and the localization theorem.
Invent. Math. 131 , 1 (1998), 25–83.[10]
Guillemin, V., and Zara, C.
Duke Math. J. 107 , 2 (2001), 283–349.[11]
Hatcher, A.
Algebraic topology . Cambridge University Press, Cambridge, 2002.[12]
James, I. M.
On the suspension sequence.
Ann. of Math. (2) 65 (1957), 74–107.[13]
Jang, D.
Circle actions on 8-dimensional almost complex manifolds with 4 fixed points.arXiv 2001.10699v1.[14]
Kramer, L., and Stolz, S.
A diffeomorphism classification of manifolds which are likeprojective planes.
J. Differential Geom. 77 , 2 (2007), 177–188.[15]
Kustarev, A. A.
Almost complex actions of the circle with few fixed points.
UspekhiMat. Nauk 68 , 3(411) (2013), 191–192.[16]
Masuda, M.
Equivariant cohomology distinguishes toric manifolds.
Adv. Math. 218 , 6(2008), 2005–2012.[17]
Masuda, M., and Panov, T.
On the cohomology of torus manifolds.
Osaka J. Math.43 , 3 (2006), 711–746.[18]
Masuda, M., and Suh, D. Y.
Classification problems of toric manifolds via topology.In
Toric topology , vol. 460 of
Contemp. Math.
Amer. Math. Soc., Providence, RI, 2008,pp. 273–286.[19]
McDuff, D., and Salamon, D.
Introduction to symplectic topology , third ed. OxfordGraduate Texts in Mathematics. Oxford University Press, Oxford, 2017.[20]
Okonek, C., Schneider, M., and Spindler, H.
Vector bundles on complex projectivespaces . Modern Birkh¨auser Classics. Birkh¨auser/Springer Basel AG, Basel, 2011. Correctedreprint of the 1988 edition, With an appendix by S. I. Gelfand.[21]
Orlik, P., and Raymond, F.
Actions of the torus on 4-manifolds. I.
Trans. Amer.Math. Soc. 152 (1970), 531–559.[22]
Pabiniak, M., and Tolman, S.
Symplectic cohomological rigidity via toric degnera-tions. arXiv 2002.12434v1.[23]
Smale, S.
Generalized Poincar´e’s conjecture in dimensions greater than four.
Ann. ofMath. (2) 74 (1961), 391–406.[24]
Thurston, W. P.
Some simple examples of symplectic manifolds.
Proc. Amer. Math.Soc. 55 , 2 (1976), 467–468.[25]
Toda, H.
Composition methods in homotopy groups of spheres . Annals of MathematicsStudies, No. 49. Princeton University Press, Princeton, N.J., 1962.1426]
Voisin, C.
Hodge theory and complex algebraic geometry. I , english ed., vol. 76 of
Cam-bridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2007.Translated from the French by Leila Schneps.[27]
West, R. W.
Some cohomotopy of projective space.
Indiana Univ. Math. J. 20 (1970/71),807–827.[28]
Whitehead, G. W.
The ( n + 2) nd homotopy group of the n -sphere. Ann. of Math. (2)52 (1950), 245–247.[29]
Wiemeler, M.
Exotic torus manifolds and equivariant smooth structures on quasitoricmanifolds.
Math. Z. 273 , 3-4 (2013), 1063–1084.[30]
Wiemeler, M.
Torus manifolds and non-negative curvature.