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Oriented and unitary equivariant bordism of surfaces

Andrés Angel, Eric Samperton, Carlos Segovia, Bernardo Uribe

Abstract

Fix a finite group $G$. We study $Ω^{SO,G}_2$ and $Ω^{U,G}_2$, the unitary and oriented bordism groups of smooth $G$-equivariant compact surfaces, respectively, and we calculate them explicitly. Their ranks are determined by the possible representations around fixed points, while their torsion subgroups are isomorphic to the direct sum of the Bogomolov multipliers of the Weyl groups of representatives of conjugacy classes of all subgroups of $G$. We present an alternative proof of the fact that surfaces with free actions which induce non-trivial elements in the Bogomolov multiplier of the group cannot equivariantly bound. This result permits us to show that the 2-dimensional SK-groups (Schneiden und Kleben, or ``cut and paste") of the classifying spaces of a finite group can be understood in terms of the bordism group of free equivariant surfaces modulo the ones that bound arbitrary actions.

Oriented and unitary equivariant bordism of surfaces

Abstract

Fix a finite group . We study and , the unitary and oriented bordism groups of smooth -equivariant compact surfaces, respectively, and we calculate them explicitly. Their ranks are determined by the possible representations around fixed points, while their torsion subgroups are isomorphic to the direct sum of the Bogomolov multipliers of the Weyl groups of representatives of conjugacy classes of all subgroups of . We present an alternative proof of the fact that surfaces with free actions which induce non-trivial elements in the Bogomolov multiplier of the group cannot equivariantly bound. This result permits us to show that the 2-dimensional SK-groups (Schneiden und Kleben, or ``cut and paste") of the classifying spaces of a finite group can be understood in terms of the bordism group of free equivariant surfaces modulo the ones that bound arbitrary actions.

Paper Structure

This paper contains 19 sections, 16 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant bordism
  4. Equivariant bordism for families
  5. Long exact sequence for families
  6. Conner-Floyd spectral sequence
  7. Equivariant bordism for adjacent families
  8. $G$ fixed points
  9. Rowlett spectral sequence
  10. Localization at fixed points
  11. Kernel of fixed-point homomorphism
  12. Surfaces without isolated fixed points for any subgroup
  13. Bounding equivariant surfaces
  14. Bounding free actions on surfaces
  15. Torsion of the equivariant bordism group of surfaces
  16. ...and 4 more sections

Key Result

Theorem 1

theorem torsion subgroup bordism. Let $G$ be a finite group and $\mathop{\mathrm{Tor}}\nolimits_\mathbb{Z} (\Omega^{G}_2)$ the torsion subgroup of the unitary or oriented $G$-equivariant bordism of surfaces $\Omega^{G}_2$. Then there is a canonical isomorphism where $(K)$ runs over all conjugacy classes of subgroups of $G$, $W_K=N_GK/K$ and $\tilde{B}_0(W_K)$ is the homology version of the Bogomo

Theorems & Definitions (26)

  • Theorem
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Definition 3.5
  • Proposition 3.6
  • proof
  • ...and 16 more