Hatcher-Algebraic-Topology · Problem 10

The Fundamental Group

Mathematics — Graph Theory · product-space-commuting-loops

Problem

From the isomorphism

\pi_1\bigl(X \times Y, (x_0, y_0)\bigr) \approx \pi_1(X, x_0) \times \pi_1(Y, y_0)

it follows that loops in

X \times \{y_0\}

and

\{x_0\} \times Y

represent commuting elements of

\pi_1\bigl(X \times Y, (x_0, y_0)\bigr)

. Construct an explicit homotopy demonstrating this.