Geodesic
Oriented and unitary equivariant bordism of surfaces
Ángel, Andrés1 ; Samperton, Eric2 ; Segovia, Carlos3 ; Uribe, Bernardo4
1 Departamento de Matemáticas, Universidad de los Andes, Bogota, Colombia
2 Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, United States, Mathematics Department, Purdue University, West Lafayette, IN, United States
3 Instituto de Matemáticas, UNAM Unidad Oaxaca, Oaxaca, Mexico
4 Max Planck Institut für Mathematik, Bonn, Germany, Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, Colombia
Algebraic and Geometric Topology, Tome 24 (2024) no. 3, pp. 1623-1654

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/agt.2024.24.1623
Keywords: equivariant bordism, equivariant vector bundle, surface

Ángel, Andrés 1 ; Samperton, Eric 2 ; Segovia, Carlos 3 ; Uribe, Bernardo 4

1 Departamento de Matemáticas, Universidad de los Andes, Bogota, Colombia
2 Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, United States, Mathematics Department, Purdue University, West Lafayette, IN, United States
3 Instituto de Matemáticas, UNAM Unidad Oaxaca, Oaxaca, Mexico
4 Max Planck Institut für Mathematik, Bonn, Germany, Departamento de Matemáticas y Estadística, Universidad del Norte, Barranquilla, Colombia 
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Ángel, Andrés; Samperton, Eric; Segovia, Carlos; Uribe, Bernardo. Oriented and unitary equivariant bordism of surfaces. Algebraic and Geometric Topology, Tome 24 (2024) no. 3, pp. 1623-1654. doi: 10.2140/agt.2024.24.1623

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