Abstract
In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod p homology of Ώ(BG∧p), whenGis a finite group,BG∧p is the p-completion of its classifying space, and Ώ(BG∧p)is the loop space ofBG∧p. 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=BG∧p..
| Original language | English |
|---|---|
| Pages (from-to) | 425-480 |
| Number of pages | 56 |
| Journal | Annals of K-Theory |
| Volume | 6 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2021 |
Bibliographical note
Funding Information:C. Broto acknowledges financial support from the Spanish Ministry of Economy through the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445) and FEDERMINECO Grant MTM2016-80439-P and from the Generalitat de Catalunya through AGAUR Grant 2017SGR1725. B. Oliver is partially supported by UMR 7539 of the CNRS R. Levi and B. Oliver were partly supported by FEDER-MINECO Grant MTM2016-80439-P during several visits to the Universitat Autònoma de Barcelona.
Keywords
- Classifying space
- Finite groups
- Fusion
- Loop space
- P-completion
- Plus construction
- Small category