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Oriented and unitary equivariant bordism of surfaces

Andrés Ángel, Eric Samperton, Carlos Segovia and Bernardo Uribe

Algebraic & Geometric Topology 24 (2024) 1623–1654

Fix a finite group G. We study Ω2SO ,G and Ω2U,G, 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 nontrivial 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.

equivariant bordism, equivariant vector bundle, surface
Mathematical Subject Classification
Primary: 55N22, 57R75, 57R77, 57R85
Received: 30 April 2022
Revised: 13 February 2023
Accepted: 26 February 2023
Published: 28 June 2024
Andrés Ángel
Departamento de Matemáticas
Universidad de los Andes
Eric Samperton
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States
Mathematics Department
Purdue University
West Lafayette, IN
United States
Carlos Segovia
Instituto de Matemáticas
UNAM Unidad Oaxaca
Bernardo Uribe
Max Planck Institut für Mathematik
Departamento de Matemáticas y Estadística
Universidad del Norte

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