Section 54: The Fundamental Group of the Circle

• If $p$ is a covering map, and $p(e)=b$ , then any path in $B$ starting at $b$ has a unique lifting in $E$ starting at $e$ .
• If $p$ is a covering map, and $p(e)=b$ , then for any continuous $F:I\times I\rightarrow B$ such that $F(0,0)=b$ there is a unique lifting $\tilde{F}$ of $F$ in $E$ such that $\tilde{F}(0,0)=e$ . Moreover, if $F$ is a path homotopy, so is $\tilde{F}$ .
  • It follows, that if $p$ is a covering map and $f$ is path homotopic to $g$ in $B$ , then their liftings starting at the same point in $E$ must end at the same point and be path homotopic.

• If $E$ is path connected, then $\phi$ is surjective.
• If $E$ is simply connected, then $\phi$ is bijective.
• If $E$ is path connected and $B$ is simply connected, then $p$ is a homeomorphism.
• A stronger version:
  • $p_{*}:\pi_{1}(E,e)\rightarrow(B,b)$ is a monomorphism;
  • if $H=p_{*}(\pi_{1}(E,e))$ , then $\Phi:\pi_{1}(B,b)/H\rightarrow p^{-1}(b)$ induced by $\phi$ is injective, and bijective if $E$ is simply connected;
  • if $[f]\in\pi_{1}(B,b)$ , then $[f]\in H$ iff $\tilde{f}\in\pi_{1}(E,e)$ .

• The fundamental group of the torus is isomorphic to $\mathbb{Z}\times\mathbb{Z}$ .