Cohomology of GKM fiber bundles

Guillemin, Victor; Sabatini, Silvia; Zara, Catalin

doi:10.1007/s10801-011-0292-6

Cited by 18 publications

(32 citation statements)

References 15 publications

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“…The results of this paper are closely related to the combinatorial results of our recent article [5]. More explicitly, in [5] we develop a GKM theory for fiber bundles in which the objects involved: the base, the fiber and the total space of the fiber bundle, are GKM graphs.…”

Section: Examplesmentioning

confidence: 68%

“…This result will enable one to interpolate between two (in principle) very different descriptions of the ring H * T (G/K), as we have already shown in special cases in [5].…”

Section: Theorem 13 a Functionmentioning

confidence: 71%

“…In Section 4, we construct the group W p and prove the isomorphism (1.10). In Section 5, we describe in more detail the results of Guillemin et al [4] alluded to above and show that the fiber bundles G/K 1 → G/K satisfy the conditions of Theorem 4.2, and in Section 6 we describe some connections between the results of this paper and results of Guillemin et al [5], where we work out the implications of this theory in much greater detail for the classical flag varieties of type A n , B n , C n , and D n .…”

Section: Theorem 13 a Functionmentioning

confidence: 90%

“…By repeating an argument similar to the other examples, in order to produce a In [5], we also show how these iterated invariant classes relate to a better known family of classes generating the equivariant cohomology of flag varieties, namely the equivariant Schubert classes.…”

Section: Examplementioning

confidence: 90%

“…More explicitly, in [5] we develop a GKM theory for fiber bundles in which the objects involved: the base, the fiber and the total space of the fiber bundle, are GKM graphs. We then formulate, in this context, a combinatorial notion of "balanced", show that one has an analog of the isomorphism (1.11) and use this fact to define some new combinatorial invariants for the classical flag varieties of type A n , B n , C n , and D n .…”

Section: Examplesmentioning

confidence: 99%

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Balanced Fiber Bundles and GKM Theory

Guillemin

Sabatini

Zara

2012

International Mathematics Research Notices

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Let T be a torus and B a compact T −manifold. Goresky, Kottwitz, and MacPherson show in [GKM] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H * T (B) as a subring of H * T (B T ). In this paper we prove an analogue of this result for T −equivariant fiber bundles: we show that if M is a T −manifold and π : M → B a fiber bundle for which π intertwines the two T −actions, there is a simple combinatorial description of H * T (M ) as a subring of H * T (π −1 (B T )). Using this result we obtain fiber bundle analogues of results of [GHZ] on GKM theory for homogeneous spaces.

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Section: Examplesmentioning

confidence: 68%

“…This result will enable one to interpolate between two (in principle) very different descriptions of the ring H * T (G/K), as we have already shown in special cases in [5].…”

Section: Theorem 13 a Functionmentioning

confidence: 71%

Section: Theorem 13 a Functionmentioning

confidence: 90%

Section: Examplementioning

confidence: 90%

Section: Examplesmentioning

confidence: 99%

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Balanced Fiber Bundles and GKM Theory

Guillemin

Sabatini

Zara

2012

International Mathematics Research Notices

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Equivariant K-theory of GKM bundles

2012

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Given a fiber bundle of GKM spaces, π : M → B, we analyze the structure of the equivariant K-ring of M as a module over the equivariant K-ring of B by translating the fiber bundle, π, into a fiber bundle of GKM graphs and constructing, by combinatorial techniques, a basis of this module consisting of K-classes which are invariant under the natural holonomy action on the K-ring of M of the fundamental group of the GKM graph of B. We also discuss the implications of this result for fiber bundles π : M → B where M and B are generalized partial flag varieties and show how our GKM description of the equivariant K-ring of a homogeneous GKM space is related to the Kostant-Kumar description of this ring. MSC: Primary 55R91; Secondary 19L47; 05C90

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12, 24 and beyond

Godinho

Heymann

Sabatini

2017

Advances in Mathematics

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We generalize the well-known "12" and "24" Theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action.As an application, we provide bounds on the Betti numbers for certain monotone Hamiltonian spaces which depend on the minimal Chern number of the manifold.

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Cohomology of GKM fiber bundles

Cited by 18 publications

References 15 publications

Balanced Fiber Bundles and GKM Theory

Balanced Fiber Bundles and GKM Theory

Equivariant K-theory of GKM bundles

12, 24 and beyond

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