# Homotopy group of product is product of homotopy groups

## Statement

### 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:

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