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Realization of permutation modules via Alexandroff spaces

Cristina Costoya, Rafael Gomes, Antonio Viruel

Abstract

We raise the question of the realizability of permutation modules in the context of Kahn's realizability problem for abstract groups and the $G$-Moore space problem. Specifically, given a finite group $G$, we consider a collection $\{M_i\}_{i=1}^n$ of finitely generated $\Z G$-modules that admit a submodule decomposition on which $G$ acts by permuting the summands. Then we prove the existence of connected finite spaces $X$ that realize each $M_i$ as its $i$-th homology, $G$ as its group of self-homotopy equivalences $\E(X)$, and the action of $G$ on each $M_i$ as the action of $\E(X)$ on $H_i(X; \Z)$.

Realization of permutation modules via Alexandroff spaces

Abstract

We raise the question of the realizability of permutation modules in the context of Kahn's realizability problem for abstract groups and the -Moore space problem. Specifically, given a finite group , we consider a collection of finitely generated -modules that admit a submodule decomposition on which acts by permuting the summands. Then we prove the existence of connected finite spaces that realize each as its -th homology, as its group of self-homotopy equivalences , and the action of on each as the action of on .
Paper Structure (4 sections, 7 theorems, 8 equations, 3 figures)

This paper contains 4 sections, 7 theorems, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Graphs and permutation representations
  3. Acyclic finite spaces and permutation representations
  4. Realization of permutation modules

Key Result

Theorem A

Let $n$ be a positive integer, and $G$ be a finite group. If $\{M_i\}_{i=1}^n$ is a collection of finitely generated permutation $\mathbb{Z} G$-modules, then there exist infinitely many non homotopically equivalent connected finite spaces $X$ such that

Figures (3)

  • Figure 1: Hasse diagram of $L_1$
  • Figure 2: Hasse diagram of $W_2=L_1 \vee L_1$
  • Figure 3: Hasse diagram of $\widetilde{W}_2=W_1 \vee L_1$

Theorems & Definitions (13)

  • Theorem A
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:graph_realizing_permutation']}
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 3 more