Realization of permutation modules via Alexandroff spaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929241498583040 |
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| author | Costoya, Cristina Gomes, Rafael Viruel, Antonio |
| author_facet | Costoya, Cristina Gomes, Rafael Viruel, Antonio |
| contents | 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 $H_i(X; \Z)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16675 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Realization of permutation modules via Alexandroff spaces Costoya, Cristina Gomes, Rafael Viruel, Antonio Algebraic Topology Combinatorics Group Theory Representation Theory 20B25, 20C05, 06A06, 06A11, 05E18, 55P10 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 $H_i(X; \Z)$. |
| title | Realization of permutation modules via Alexandroff spaces |
| topic | Algebraic Topology Combinatorics Group Theory Representation Theory 20B25, 20C05, 06A06, 06A11, 05E18, 55P10 |
| url | https://arxiv.org/abs/2308.16675 |