Universal coverings of simplicial groups

Tuesday, July 03rd, 2012 | Author:

Every simplicial set admits a universal covering space. If the simplicial set is a simplicial group, the covering can be given a group structure, such that the covering map is a group homomorphism. This can be done "at once" without using geometric realizations, just by close analysis of the usual construction of the universal covering.

Let G_\bullet be a simplicial group, PG_\bullet its path space, \Omega G_\bullet its loop space and N_\bullet the kernel of the projection \Omega G_\bullet \rightarrow \pi_1(G_\bullet, id), which consists of all contractible loops. The composition of N_\bullet \hookrightarrow \Omega G_\bullet with the inclusion \Omega G_\bullet \hookrightarrow PG_\bullet yields a monomorphism N_\bullet \hookrightarrow PG_\bullet whose cokernel we denote by \tilde{G}_\bullet.

I claim that this simplicial group already does the job.
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Category: English, Mathematics | 3 Comments