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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(Fn∕Γnq,G) and Rep(Fn∕Γnq,G), respectively, where Fn∕Γnq 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.
Keywords: cosimplicial groups, spaces of representations
Villarreal, Bernardo 1
@article{10_2140_agt_2017_17_3519,
author = {Villarreal, Bernardo},
title = {Cosimplicial groups and spaces of homomorphisms},
journal = {Algebraic and Geometric Topology},
pages = {3519--3545},
publisher = {mathdoc},
volume = {17},
number = {6},
year = {2017},
doi = {10.2140/agt.2017.17.3519},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.3519/}
}
TY - JOUR AU - Villarreal, Bernardo TI - Cosimplicial groups and spaces of homomorphisms JO - Algebraic and Geometric Topology PY - 2017 SP - 3519 EP - 3545 VL - 17 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.3519/ DO - 10.2140/agt.2017.17.3519 ID - 10_2140_agt_2017_17_3519 ER -
Villarreal, Bernardo. Cosimplicial groups and spaces of homomorphisms. Algebraic and Geometric Topology, Tome 17 (2017) no. 6, pp. 3519-3545. doi: 10.2140/agt.2017.17.3519
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