Computing fundamental groups of products

Question

Let $X$ be a connected graph and $S^{1}$ the usual circle and consider the product $X \times S^{1}$. How would one compute the fundamental group $\pi_{1}(X \times S^{1})$ in this case? I know that one should somehow use a cellular decomposition, but I'm not sure how to get it from there.

Answer 1

You don't need a cellular decomposition. Maps from $A$ to a product $X \times Y$ are the same thing as pairs of maps, one $A \to X$, and one to $A \to Y$.

Indeed, we have an isomorphism $f: \pi_1(X) \times \pi_1(Y) \cong \pi_1(X \times Y)$. To write down the map, pick representatives $\gamma, \beta$ of $\pi_1(X)$ and $\pi_1(Y)$; then $f(\gamma,\beta) = (\gamma,\beta)$ (where $(\gamma,\beta)(\theta) = (\gamma(\theta),\beta(\theta))$.)

This does not depend on the choice of representatives: if you had a homotopy $F: S^1 \times I \to X$ between $\gamma$ and $\gamma'$, then this induces a homotopy $$F': S^1 \times I \to X \times Y, F = F \times \beta.$$

It is easy to see that the map $f$ is surjective; you should show (using essentially the same logic as that the choice of representative did not matter) that this map is injective, too.