The rank filtration via a filtered bar construction

Arone, Gregory; Lesh, Kathryn

doi:10.2140/agt.2022.22.251

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The rank filtration via a filtered bar construction

Gregory Arone and Kathryn Lesh

Algebraic & Geometric Topology 22 (2022) 251–306

DOI: 10.2140/agt.2022.22.251

Abstract

Suppose $ℱ$ is a special $Γ$ –space equipped with a natural transformation $ℱ \to {Sp}^{\infty}$ . Segal’s infinite loop space machine (Topology 13 (1974) 293–312) associates with $ℱ$ a spectrum, denoted by $k ℱ$ , equipped with a map $k ℱ \to H ℤ$ . In our previous work (Fund. Math. 207 (2010) 29–70), we constructed a filtration of $k ℱ$ by a sequence of spectra, which we called the stable rank filtration of $ℱ$ . Here we give a new construction of the stable rank filtration. The new construction is combinatorial in nature and avoids the process of stabilization. In particular, we construct a sequence of special $Γ$ –spaces whose group completion yields the stable rank filtration.

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Keywords

$\Gamma$–space, rank filtration, bar construction

Mathematical Subject Classification 2010

Primary: 55P47

Secondary: 55N15, 55P42

References

Publication

Received: 22 March 2020

Revised: 6 January 2021

Accepted: 12 February 2021

Published: 26 April 2022

Authors

Gregory Arone
Department of Mathematics Stockholm University Stockholm Sweden

Kathryn Lesh
Department of Mathematics Union College Schenectady, NY United States
http://www.math.union.edu/~leshk

Volume 22, issue 1 (2022)