Fundamental group smash product

Question

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is?

probably there should be a general way like there is for the wedge sum due to van Kampen's theorem.

Answer 1

Assuming $X,Y$ are CW-complexes, this is just the quotient of $\pi_1(X \times Y,p \times q)$ modulo the normal closure of the image of $\pi_1(X,p) * \pi_1(Y,q) \hookrightarrow \pi_1(X \times Y,p \times q)$. You can prove this by applying Van Kampen's theorem, because the Smash product of $(X,p)$ and $(Y,q)$ is homotopy equivalent to the space obtained from $X \times Y$ by coning off the subcomplex $(X \times q) \cup (p \times Y)$. Verifying this kind of homotopy equivalence (between a CW complex modulo a subcomplex and the same CW complex with the subcomplex coned off) is done for example in Hatcher's Algebraic Topology book.