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Oriented and unitary
equivariant bordism of surfaces
Andrés Ángel, Eric Samperton, Carlos Segovia and Bernardo Uribe |
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Algebraic & Geometric Topology 24 (2024) 1623–1654
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Abstract |
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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 nontrivial elements in the Bogomolov multiplier of the group cannot equivariantly bound. This result permits us to show that the –dimensional –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. |
Keywords
equivariant bordism, equivariant vector bundle, surface
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Mathematical Subject Classification
Primary: 55N22, 57R75, 57R77, 57R85
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References |
Publication
Received: 30 April 2022
Revised: 13 February 2023
Accepted: 26 February 2023
Published: 28 June 2024
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Authors |
| Andrés Ángel | |
| Departamento de Matemáticas Universidad de los Andes Bogota Colombia |
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| 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 Oaxaca Mexico |
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| Bernardo Uribe | |
| Max Planck Institut für
Mathematik Bonn Germany |
Departamento de Matemáticas y
Estadística Universidad del Norte Barranquilla Colombia |
| https://sites.google.com/site/bernardouribejongbloed/ | |
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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