Math4202 Topology II (Lecture 28)

We need to find group homomorphism: $\phi:\pi_1(X\times Y,(x_0,y_0))\to \pi_1(X,x_0)\times \pi_1(Y,y_0)$.

Let $P_x,P_y$ be the projection from $X\times Y$ to $X$ and $Y$ respectively.

$$(P_x)_*:\pi_1(X\times Y,(x_0,y_0))\to \pi_1(X,x_0)$$ $$(P_y)_*:\pi_1(X\times Y,(x_0,y_0))\to \pi_1(Y,y_0)$$

Given $\alpha\in \pi_1(X\times Y,(x_0,y_0))$, then $\phi(\alpha)=((P_x)_*\alpha,(P_y)_*\alpha)\in \pi_1(X,x_0)\times \pi_1(Y,y_0)$.

Since $(P_x)_*$ and $(P_y)_*$ are group homomorphism, so $\phi$ is a group homomorphism.

Then we need to show that $\phi$ is bijective. Then we have the isomorphism of fundamental groups.

To show $\phi$ is injective, then it is sufficient to show that $\ker(\phi)=\{e\}$.

Given $\alpha\in \ker(\phi)$, then $(P_x)_*\alpha=\{e_x\}$ and $(P_y)_*\alpha=\{e_y\}$, so we can find a path homotopy $P_X(\alpha)\simeq e_x$ and $P_Y(\alpha)\simeq e_y$.

So we can build $(H_x,H_y):X\times Y\times I\to X\times I$ by $(x,y,t)\mapsto (H_x(x,t),H_y(y,t))$ is a homotopy from $\alpha$ and $e_x\times e_y$.

So $[\alpha]=[(e_x\times e_y)]$. $\ker(\phi)=\{[(e_x\times e_y)]\}$.

Next, we show that $\phi$ is surjective.

Given $(\alpha,\beta)\in \pi_1(X,x_0)\times \pi_1(Y,y_0)$, then $(\alpha,\beta)$ is a loop in $X\times Y$ based at $(x_0,y_0)$. and $(P_x)_*([\alpha,\beta])=[\alpha]$ and $(P_y)_*([\alpha,\beta])=[\beta]$.