Cosimplicial Groups and Spaces of Homomorphisms

TL;DR

This paper investigates the homotopy properties of spaces of homomorphisms from cosimplicial groups to real algebraic groups, revealing stable decompositions and algebraic structures, especially for compact Lie groups and unitary groups.

Contribution

It establishes homotopy stable decompositions of homomorphism spaces and introduces an $E_inite$-ring space structure on their geometric realizations for certain groups.

Findings

Hom($L_n,G$) admits a homotopy stable decomposition.

Decomposition is $G$-equivariant for compact Lie groups.

The geometric realization $B(L,U)$ has a non-unital $E_inite$-ring space structure.

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 $G$. 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 $m\geq1$.