On the topology of the nonabelian tensor product of profinite groups

The properties of the nonabelian tensor products are interesting in different contexts of algebraic topology and group theory. We prove two theorems, dealing with the nonabelian tensor products of projective limits of finite groups. The first describes their topology. Then we show a result of embedding in the second homology group of a pro-$p$-group, via the notion of complete exterior centralizer. We end with some open questions, originating from these two results.