Homotopy group of product is product of homotopy groups

For two based topological spaces
Suppose $$(X,x_0)$$ and $$(Y,y_0)$$ are based topological spaces. Then, the following is true for the homotopy groups of the topological spaces $$(X,x_0)$$, $$(Y,y_0)$$ and the product space $$(X \times Y,(x_0,y_0))$$:

$$\! \pi_n(X \times Y, (x_0,y_0)) \cong \pi_1(X,x_0) \times \pi_1(Y,y_0)$$

More explicitly, if $$p_X$$ and $$p_Y$$ denote the projections from $$(X \times Y,(x_0,y_0))$$ to $$(X,x_0)$$ and $$(Y,y_0)$$ respectively, then the maps:

$$\! \pi_n(p_X): \pi_1(X \times Y,(x_0,y_0)) \to \pi_1(X,x_0)$$

and:

$$\! \pi_n(p_Y): \pi_1(X \times Y,(x_0,y_0)) \to \pi_1(Y,y_0)$$

then under the isomorphism $$\pi_n(X \times Y, (x_0,y_0)) \cong \pi_n(X,x_0) \times \pi_n(Y,y_0)$$ we get the direct factor projections for the group product.

For two topological spaces without basepoint specification
Suppose $$X$$ and $$Y$$ are both path-connected spaces, or more generally, each of them is a space such that all the path components of the space are homeomorphic to each other. Then, the fundamental groups $$\pi_n(X)$$, $$\pi_n(Y)$$, and $$\pi_n(X \times Y)$$ are all well-defined without specification of basepoint. We then have:

$$\! \pi_1(X \times Y) \cong \pi_1(X) \times \pi_1(Y)$$

Particular cases

 * Fundamental group of product is product of fundamental groups