On the Homotopical Realization of Group Actions

• Autores: Rafael Mauricio Duarte Gomes
• Directores de la Tesis: Antonio Angel Viruel Arbaizar (dir. tes.) , Cristina Costoya (dir. tes.)
• Lectura: En la Universidad de Málaga ( España ) en 2026
• Idioma: inglés
• Tribunal Calificador de la Tesis: Francisco Romero Ruiz del Portal (presid.) , Aniceto Murillo Mas (secret.) , Lucile Vandembroucq (voc.)
• Programa de doctorado: Programa de Doctorado en Matemáticas por la Universidad de Almería; la Universidad de Cádiz; la Universidad de Granada; la Universidad de Jaén y la Universidad de Málaga
• Enlaces
  • Tesis en acceso abierto en: RIUMA
• Resumen

  • 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 $\Z G$-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 pointed self-homotopy equivalences of a simply-connected space. These classical problems have been deeply studied, and some have only been settled in the last two decades.

    This thesis aims at answering variants of these classical realization problems involving group actions. More precisely, this work gives a partial answer to two questions that can be seen as a combination of the Kahn's problem and the $G$-Moore space problem (or its homotopical dual). First, it is shown that every action of a finite group on a (finite) collection of finitely generated permutation modules arises as the canonical action of the group of (pointed) self-homotopy equivalences of a spaces on its integral homology groups. Second, it is proven that every action of a finite group on a finitely presented (abelian) group appears as the canonical action of the group of pointed self-homotopy equivalences of a space on one of its homotopy groups.

    The strategy to obtain these results combines elements of algebraic topology and combinatorics, relying on graphs, simplicial complexes and finite spaces. This interdisciplinary nature of the methods developed in the thesis leads to additional results of independent interest. In particular, it is established that every simplicial complex can be rigidified, in the sense that its automorphism group is reduced to a chosen subgroup, without changing its homotopy type.