#### Volume 17, issue 6 (2017)

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Cosimplicial groups and spaces of homomorphisms

### Bernardo Villarreal

Algebraic & Geometric Topology 17 (2017) 3519–3545
##### Abstract

Let $G$ be a real linear algebraic group and $L$ a finitely generated cosimplicial group. We prove that the space of homomorphisms $Hom\left({L}_{n},G\right)$ has a homotopy stable decomposition for each $n\ge 1$. When $G$ is a compact Lie group, we show that the decomposition is $G$–equivariant with respect to the induced action of conjugation by elements of $G$. In particular, under these hypotheses on $G$, we obtain stable decompositions for $Hom\left({F}_{n}∕{\Gamma }_{n}^{q},G\right)$ and $Rep\left({F}_{n}∕{\Gamma }_{n}^{q},G\right)$, respectively, where ${F}_{n}∕{\Gamma }_{n}^{q}$ are the finitely generated free nilpotent groups of nilpotency class $q-1$.

The spaces $Hom\left({L}_{n},G\right)$ assemble into a simplicial space $Hom\left(L,G\right)$. When $G=U$ we show that its geometric realization $B\left(L,U\right)$, has a nonunital ${E}_{\infty }$–ring space structure whenever $Hom\left({L}_{0},U\left(m\right)\right)$ is path connected for all $m\ge 1$.

##### Keywords
cosimplicial groups, spaces of representations
##### Mathematical Subject Classification 2010
Primary: 22E15
Secondary: 55U10, 20G05
##### Publication
Revised: 26 March 2017
Accepted: 30 April 2017
Published: 4 October 2017
##### Authors
 Bernardo Villarreal Department of Mathematics University of British Columbia Vancouver, BC Canada http://www.math.ubc.ca/~bernvh