Oriented and unitary equivariant bordism of 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 nontrivial elements in the Bogomolov multiplier of the group cannot equivariantly bound. This result permits us to show that the 2dimensional SKgroups (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.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.02693
 Bibcode:
 2021arXiv211102693A
 Keywords:

 Mathematics  Algebraic Topology;
 57R85;
 55N22;
 57R75;
 57R77
 EPrint:
 We have generalized the second author's result for all finite groups. Namely, an oriented surface with orientation preserving free action of a finite group equivariantly bounds, if and only if, the homology class the surface quotient defines in the second homology of the group is trivial in the homology Bogomolov multiplier (Thm. 4.17). 29 pages