#### Volume 22, issue 1 (2022)

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The rank filtration via a filtered bar construction

### Gregory Arone and Kathryn Lesh

Algebraic & Geometric Topology 22 (2022) 251–306
##### Abstract

Suppose $\mathsc{ℱ}$ is a special $\Gamma$–space equipped with a natural transformation $\mathsc{ℱ}\to {\mathrm{Sp}}^{\infty }$. Segal’s infinite loop space machine (Topology 13 (1974) 293–312) associates with $\mathsc{ℱ}$ a spectrum, denoted by $k\mathsc{ℱ}$, equipped with a map $k\mathsc{ℱ}\to Hℤ$. In our previous work (Fund. Math. 207 (2010) 29–70), we constructed a filtration of $k\mathsc{ℱ}$ by a sequence of spectra, which we called the stable rank filtration of $\mathsc{ℱ}$. 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 $\Gamma$–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