2023/07/12 by Omar A. Cruz, Cruz, Omar A., Gustavo Andrés Ortega +3
Mathematics · #57M60 (Primary) #57R85 (Secondary) #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2307.05863
openalex publication_date 2023/07/12 · openalex created_date 2023/07/14 · openalex updated_date 2026/07/28
We present the unoriented versions of the Schur and Bogomolov multipliers associated with a finite group G. We show that the unoriented Schur multiplier is isomorphic to the second cohomology group H2(G;\ZZ2). We define the unoriented Bogomolov multiplier as the quotient of the unoriented Schur multiplier by the subgroup generated by classes over the disjoint union of tori, Klein bottles, and projective spaces. We prove that the unoriented Bogomolov multiplier is trivial for abelian, dihedral, symmetric, and alternating groups. Since H2(G;\ZZ2) is trivial for any group of odd order, there are numerous examples where the classical Bogomolov multiplier is nontrivial while its unoriented counterpart is trivial. Nevertheless, we exhibit a group of order 64 for which the unoriented Bogomolov multiplier is nontrivial.