Rafael Gomes: Homotopical realization of finite group actions (hybrid seminar)

Realizability questions have had a profound impact in algebraic topology. In the 1970’s, Steenrod asked whether every algebra, over a commutative ring R, is the cohomology of a space with coefficients in R; and also if every ZG-module, for G a group, arises as the integral homology of a G-Moore space. Kahn proposed an investigation of the groups that appear as the group of self-homotopy equivalences of a simply-connected space. These classical problems have been deeply studied, and some have only been settled in the last decade. In this talk, we will give an overview of the history of these problems and explain two recent realizability results concerning group actions. On the one hand, for each k≥1, the action of every finite group on a finitely presentable group arises as the action of the group of self homotopy equivalences of a space on its k-th homotopy group. On the other hand, every action of a finite group on a finitely generated (graded) permutation module appears as the action of the group of self-homotopy equivalences of a space on its homology. Additionally, we present an intermediate result about simplicial complexes: every simplicial complex can be rigidified, in the sense that its automorphism group is reduced to a chosen subgroup, without changing its homotopy type.