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Math4202 Topology II (Lecture 28)
Algebraic Topology
Fundamental Groups of Some Surfaces
Recall from last week, we will see the fundamental group of T^2=S^1\times S^1, and \mathbb{R}P^2, Torus with genus 2.
Some of them are abelian, and some are not.
Theorem for fundamental groups of product spaces
Let X,Y be two manifolds. Then the fundamental group of X\times Y is the direct product of their fundamental groups,
i.e.
\pi_1(X\times Y,(x_0,y_0))=\pi_1(X,x_0)\times \pi_1(Y,y_0)
Proof
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].
Corollary for fundamental groups of T^2
The fundamental group of T^2=S^1\times S^1 is \mathbb{Z}\times \mathbb{Z}.
Theorem for fundamental groups of \mathbb{R}P^2
\mathbb{R}P^2 is a compact 2-dimensional manifold with the universal covering space S^2 and a 2-1 covering map q:S^2\to \mathbb{R}P^2.
Corollary for fundamental groups of \mathbb{R}P^2
\pi_1(\mathbb{R}P^2)=\#q^{-1}(\{x_0\})=\{a,b\}=\mathbb{Z}/2\mathbb{Z}
Using the path-lifting correspondence.
Lemma for The fundamental group of figure-8
The fundamental group of figure-8 is not abelian.