10/04/2025, 15:00 — 16:00 —
Sala P3.10, Pavilhão de Matemática
Rafael Gomes, University of Málaga
Topological realization of finite group actions
Algebraic topology provides a natural framework for realizability problems, as it explores the interplay between algebraic structures and topological spaces. These questions have been around since the 1970's, with Steenrod asking when an algebra is the cohomology of a space and Kahn asking which groups are the group of self-homotopy equivalences of a simply-connected space. Addressing such questions deepens our understanding of both spaces and their associated algebraic structures, making them quite interesting.
In this talk, we present two recent realizability results concerning group actions. First, for any action of a finite group on a finitely presented abelian group, there exists a space that realizes this action as the canonical action of the group of self-homotopy equivalences on the first homology group. Second, we establish that any action of a finite group on a permutation module is the action of the group of self-homotopy equivalences of a space on its homology groups. Additionally, we show that any simplicial complex can be perturbed in a way that reduces the automorphism group to any chosen subgroup without changing the homotopy type.
(joint work with Cristina Costoya and Antonio Viruel)
