Homotopy group of product is product of homotopy groups

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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)

Related facts

Particular cases