Vol. 6, No. 3, 2021

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Loop space homology of a small category

Carles Broto, Ran Levi and Bob Oliver

Vol. 6 (2021), No. 3, 425–480
Abstract

In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod p homology of Ω(BGp), when G is a finite group, BGp is the p-completion of its classifying space, and Ω(BGp) is the loop space of BGp. The main purpose of this work is to shed new light on Benson’s result by extending it to a more general setting. As a special case, we show that if 𝒞 is a small category, |𝒞| is the geometric realization of its nerve, R is a commutative ring, and |𝒞|R+ is a “plus construction” for |𝒞| in the sense of Quillen (taken with respect to R-homology), then H(Ω(|𝒞|R+);R) can be described as the homology of a chain complex of projective R𝒞-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson’s theorem is now the case where 𝒞 is the category of a finite group G, R = 𝔽p for some prime p, and |𝒞|R+ = BGp.

Keywords
classifying space, loop space, small category, plus construction, $p$-completion, finite groups, fusion
Mathematical Subject Classification 2010
Primary: 55R35
Secondary: 20D20, 55R40
Milestones
Received: 12 June 2019
Revised: 27 November 2020
Accepted: 15 December 2020
Published: 11 September 2021
Authors
Carles Broto
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Bellaterra
Spain
Centre de Recerca Matemàtica
Campus de Bellaterra
Cerdanyola del Vallès
Spain
Ran Levi
Institute of Mathematics
University of Aberdeen
Aberdeen
United Kingdom
Bob Oliver
LAGA, Institut Galilée
Université Paris 13
Villetaneuse
France