Cosimplicial groups and spaces of homomorphisms (Q2409983)

For \(G\) a real linear algebraic group and \(L\) a finitely generated cosimplicial group, the author proves that the space of homomorphisms \(\mathrm{Hom}(L_n,G)\) has a homotopy stable decomposition for each \(n \geq 1\), \(\Theta(n):\Sigma \mathrm{Hom}(L_n,G) \simeq \vee_{0 \leq k \leq n} \Sigma(S_k(L_n,G)/ S_{k+1}(L_n,G))\). When \(G\) is a compact Lie group the author shows that the \(G\)-decomposition is \(G\)-equivariant and proposes a stable decomposition (Theorem 1.3) for \(\mathrm{Hom}(F_n/\Gamma^{q}_{n},G)\) and \(\mathrm{Rep}(F_n/ \Gamma^{q}_{n},G)\), respectively. The second part of the paper focuses on the study of the geometric realization of \(\mathrm{Hom}(L,G)\) for a finitely generated cosimplicial group denoted \(B(L,G)\). The spaces \(\mathrm{Hom}(L_n,G)\) are assembled into a simplicial space \(\mathrm{Hom}(L,G)\). For \(G=U\), the geometric realization \(B(L,U)\) has a nonunital \(E_{\infty}\)-ring space structure whenever the \(\mathrm{Hom}(L_0,U(m)),m\geq 1\) are path connected (Theorem 1.4).