Oriented and unitary equivariant bordism of surfaces

TL;DR

This paper explicitly computes the equivariant bordism groups of surfaces with finite group actions, revealing their structure through fixed point representations and Bogomolov multipliers, and relates these to the classification of free actions.

Contribution

It provides explicit calculations of equivariant bordism groups of surfaces, linking torsion to Bogomolov multipliers and offering a new proof regarding surfaces with free actions.

Findings

Ranks determined by fixed point representations

Torsion subgroups linked to Bogomolov multipliers

Free actions with non-trivial Bogomolov elements cannot bound surfaces

Abstract

Fix a finite group $G$. We study $\Omega^{SO,G}_2$ and $\Omega^{U,G}_2$, 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.