Smash product and tensor product of groups

Question

The smash product acts like a 'tensor product' in the category of pointed spaces (i.e. when the spaces are locally compact Hausdorff smashing is associative and satisfies a tensor-hom adjunction). What's the relation between $\pi_1(X \wedge Y)$ and the group tensor product of $\pi_1(X), \pi_1(Y)$? In particular, does the pushforward of the smash product under the fundamental group functor give a natural way to define a tensor product-like operation in the category of groups?

Answer 1

Since $X \wedge Y$ is the quotient of $X \times Y$ by the wedge $X \vee Y$, and the morphism induced on fundamental groups by $X \vee Y \to X \times Y$ is surjective, under reasonable conditions on base points, and connectivity, $X \wedge Y$ is $1$-connected.

There is a nonabelian tensor product of groups which act on each other, and on themselves by conjugation, whose basic idea is that the commutator map on groups $[\, , \,]: G \times G \to G$ is not bimultiplicative but is a biderivation. For more information, see this bibliography.