Cosimplicial Groups and Spaces of Homomorphisms

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 $n\ge 1$. 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 $G$. 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 non-unital $E_{\infty }$-ring space structure whenever $Hom(L_0,U(m))$ is path connected for all $m\ge 1$.

23 pages

References in corpus (3)

• A classifying space for commutativity in Lie groups
• Infinite loop spaces and nilpotent K-theory
• On spaces of commuting elements in Lie groups

Cited by in corpus (5)

• Homological stability for spaces of commuting elements in Lie groups
• Classifying spaces for commutativity of low-dimensional Lie groups
• Poincare series of character varieties for nilpotent groups
• Hilbert-Poincare series for spaces of commuting elements in Lie groups
• On the mod-$\ell$ homology of the classifying space for commutativity