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On the Stiefel-Whitney classes of GKM manifolds

Oliver Goertsches, Panagiotis Konstantis, Leopold Zoller

TL;DR

This work establishes a GKM-style framework for the equivariant cohomology with $\mathbb{Z}_p$-coefficients of integer GKM manifolds under a standard isotropy assumption, circumventing non-injectivity issues of fixed-point restriction. It introduces a precise graph-theoretic description of $H_T^*(M; \mathbb{Z}_p)$ as the image of a map from the integral graph cohomology into $H_T^*(\Gamma; \mathbb{Z}_p) \oplus B^*(\Gamma,p)$, tightly tying equivariant cohomology to the GKM graph data. The paper then defines combinatorial total Stiefel-Whitney classes for GKM graphs, proving they encode the manifold’s equivariant SW-classes and yield spin-structure criteria via a graph-theoretic lens; it also provides decompositions of higher SW-classes in terms of Thom classes. As a striking application, it demonstrates non-realizability of certain $8$-dimensional (and higher) GKM graphs, showing that existing obstructions from lower dimensions are no longer sufficient and highlighting the value of combinatorial SW-classes in distinguishing realizable from non-realizable GKM graphs.

Abstract

We show that under standard assumptions on the isotropy groups of an integer GKM manifold, the equivariant Stiefel-Whitney classes of the action are determined by the GKM graph. This is achieved via a GKM-style description of the equivariant cohomology with coefficients in a finite field $\mathbb Z_{p}$ even though in this setting the restriction map to the fixed point set is not necessarily injective. This closes a gap in our argument why the GKM graph of a $6$-dimensional integer GKM manifold determines its nonequivariant diffeomorphism type. We introduce combinatorial Stiefel-Whitney classes of GKM graphs and use them to derive a nontrivial obstruction to realizability of GKM graphs in dimension $8$ and higher.

On the Stiefel-Whitney classes of GKM manifolds

TL;DR

This work establishes a GKM-style framework for the equivariant cohomology with -coefficients of integer GKM manifolds under a standard isotropy assumption, circumventing non-injectivity issues of fixed-point restriction. It introduces a precise graph-theoretic description of as the image of a map from the integral graph cohomology into , tightly tying equivariant cohomology to the GKM graph data. The paper then defines combinatorial total Stiefel-Whitney classes for GKM graphs, proving they encode the manifold’s equivariant SW-classes and yield spin-structure criteria via a graph-theoretic lens; it also provides decompositions of higher SW-classes in terms of Thom classes. As a striking application, it demonstrates non-realizability of certain -dimensional (and higher) GKM graphs, showing that existing obstructions from lower dimensions are no longer sufficient and highlighting the value of combinatorial SW-classes in distinguishing realizable from non-realizable GKM graphs.

Abstract

We show that under standard assumptions on the isotropy groups of an integer GKM manifold, the equivariant Stiefel-Whitney classes of the action are determined by the GKM graph. This is achieved via a GKM-style description of the equivariant cohomology with coefficients in a finite field even though in this setting the restriction map to the fixed point set is not necessarily injective. This closes a gap in our argument why the GKM graph of a -dimensional integer GKM manifold determines its nonequivariant diffeomorphism type. We introduce combinatorial Stiefel-Whitney classes of GKM graphs and use them to derive a nontrivial obstruction to realizability of GKM graphs in dimension and higher.
Paper Structure (5 sections, 15 theorems, 45 equations)

This paper contains 5 sections, 15 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. GKM actions
  3. A GKM description of $H^*_T(M;\mathbb{Z}_p)$
  4. Equivariant Stiefel--Whitney classes
  5. Non-realizability via Stiefel--Whitney classes

Key Result

Lemma 2.4

For a closed smooth $T$-manifold $M$ satisfying eq:standingassumption such that $H^\ast_T(M;\mathbb{Z})$ is a free $H^\ast(BT;\mathbb{Z})$-module, the sequence is exact, where the middle arrow is induced from the inclusion $M^T\to M$, and the right arrow is the boundary morphism in the long exact sequence in equivariant cohomology of the pair $(M_1,M^T)$.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • Definition 2.9
  • ...and 35 more