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The rank filtration via a filtered bar construction

Gregory Arone and Kathryn Lesh

Algebraic & Geometric Topology 22 (2022) 251–306

Suppose is a special Γ–space equipped with a natural transformation Sp . Segal’s infinite loop space machine (Topology 13 (1974) 293–312) associates with a spectrum, denoted by k, equipped with a map k H. In our previous work (Fund. Math. 207 (2010) 29–70), we constructed a filtration of k by a sequence of spectra, which we called the stable rank filtration of . 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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$\Gamma$–space, rank filtration, bar construction
Mathematical Subject Classification 2010
Primary: 55P47
Secondary: 55N15, 55P42
Received: 22 March 2020
Revised: 6 January 2021
Accepted: 12 February 2021
Published: 26 April 2022
Gregory Arone
Department of Mathematics
Stockholm University
Kathryn Lesh
Department of Mathematics
Union College
Schenectady, NY
United States