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Cosimplicial groups and spaces of homomorphisms

Bernardo Villarreal

Algebraic & Geometric Topology 17 (2017) 3519–3545

Let G be a real linear algebraic group and L a finitely generated cosimplicial group. We prove that the space of homomorphisms Hom(Ln,G) has a homotopy stable decomposition for each n 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(FnΓnq,G) and Rep(FnΓnq,G), respectively, where FnΓnq are the finitely generated free nilpotent groups of nilpotency class q 1.

The spaces Hom(Ln,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–ring space structure whenever Hom(L0,U(m)) is path connected for all m 1.

cosimplicial groups, spaces of representations
Mathematical Subject Classification 2010
Primary: 22E15
Secondary: 55U10, 20G05
Received: 11 July 2016
Revised: 26 March 2017
Accepted: 30 April 2017
Published: 4 October 2017
Bernardo Villarreal
Department of Mathematics
University of British Columbia
Vancouver, BC