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arXiv · 1203.1027

Equivariant $K$-theory of GKM bundles

Victor Guillemin·Silvia Sabatini·Catalin Zara
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Abstract

Given a fiber bundle of GKM spaces, $π\colon M\to B$, 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 $π\colon M\to B$ 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.

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BibTeXRIS

Victor Guillemin, Silvia Sabatini, Catalin Zara. 2012-03-05. Equivariant $K$-theory of GKM bundles. https://doi.org/10.1007/s10455-012-9331-3

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