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On the homology of partial group representations · arXivDesk

2404.14650Apr 23, 2024This version is a more compact version of the previous one, focusing primarily on partial representations of groups and omitting sections on the homology of partial actions. This change aims to make the structure of the paper more concise and centered on addressing the main problem. The title has been updated accordingly. Minor typographical errors were also corrected

On the homology of partial group representations

Emmanuel Jerez

Algebraic Topology Group Theory K-Theory and Homology

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Abstract

We study how the partial group (co)homology of a group $G$ with coefficient in a partial representation $M$ can be described using the usual group (co)homology. To address this, we introduce the concept of the universal globalization $Λ(M)$ of a partial group representation $M$ of $G$ . Our main result shows that the partial group homology $H^{\text{par}}_{\bullet}(G, M)$

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is naturally isomorphic to the classical group homology

H_{\bullet}(G, Λ(M))

. We extend this result to the cohomological framework, obtaining a spectral sequence involving the classical group cohomology that converges to the partial group cohomology. Notably, when

G

is countable, the spectral sequence collapses, resulting in a natural isomorphism

H^{\bullet}_{\text{par}}(G, M) \cong H^{\bullet}(G, \operatorname{Hom}_{K_{\text{par}} G}(Λ(K_{par}G), M))

, where

K_{par}G

stands for the partial group algebra of

G

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