Cosimplicial groups and spaces of homomorphisms

Villarreal, Bernardo

doi:10.2140/agt.2017.17.3519

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Cosimplicial groups and spaces of homomorphisms

Bernardo Villarreal

Algebraic & Geometric Topology 17 (2017) 3519–3545

DOI: 10.2140/agt.2017.17.3519

Abstract

Let $G$ be a real linear algebraic group and $L$ a finitely generated cosimplicial group. We prove that the space of homomorphisms $Hom (L_{n}, G)$ has a homotopy stable decomposition for each $n \geq 1$ . 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 $Hom (F_{n} ∕ Γ_{n}^{q}, G)$ and $Rep (F_{n} ∕ Γ_{n}^{q}, G)$ , respectively, where $F_{n} ∕ Γ_{n}^{q}$ are the finitely generated free nilpotent groups of nilpotency class $q - 1$ .

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

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Keywords

cosimplicial groups, spaces of representations

Mathematical Subject Classification 2010

Primary: 22E15

Secondary: 55U10, 20G05

References

Publication

Received: 11 July 2016

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

Volume 17, issue 6 (2017)