Let be a real linear algebraic group and a finitely generated cosimplicial group. We prove that the space of homomorphisms has a homotopy stable decomposition for each . When is a compact Lie group, we show that the decomposition is –equivariant with respect to the induced action of conjugation by elements of . In particular, under these hypotheses on , we obtain stable decompositions for and , respectively, where are the finitely generated free nilpotent groups of nilpotency class .
The spaces assemble into a simplicial space . When we show that its geometric realization , has a nonunital –ring space structure whenever is path connected for all .
"Cosimplicial groups and spaces of homomorphisms." Algebr. Geom. Topol. 17 (6) 3519 - 3545, 2017. https://doi.org/10.2140/agt.2017.17.3519