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Talk

July 2, 2024
Bob Oliver (Université PARIS 13): The loop space homology of a small category

Abstract

In an article published in 2009, Dave Benson described, for a finite group G, the mod p homology of the the loop space of the p-completion of BG in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we tried to better understand Benson’s result by generalizing it. Among other things, we showed that when C is a small category, |C| is its geometric realization, R is a commutative ring, and |C|^+_R is a “plus construction” of |C| with respect to homology with coefficients in R, then the homology of the loop space of this plus construction is the homology of any chain complex of projective RC-modules that satisfies certain conditions. Benson’s theorem is then the special case where C is the category associated to a finite group G and R = F_p , so that p-completion appears as a special case of the plus construction.

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