If $p : E \to B$ is a covering map and $E$ is simply connected, then each fiber of $p$ has the cardinality of that of $\pi_1(B)$.

Question

Can anyone provide a source or a proof of the following fact:

If $p : E \to B$ is a covering map and $E$ is simply connected, then each fiber of $p$ has the cardinality of that of $\pi_1(B)$?

Answer 1

Check out $\S$1.3 in Hatcher's book for a full exposition, but here's the idea:

Since $E$ is simply-connected, any two paths between points $e,e'$ in a fiber $p^{-1}(b)$ are homotopic rel endpoints (i.e. if $\gamma_0,\gamma_1: [0,1]\to E$ satisfy $\gamma_0(0)=\gamma_1(0)=e$ and $\gamma_0(1)=\gamma_1(1)=e'$, then they are homotopic through a family of paths $\gamma_t:[0,1]\to E$ such that $\gamma_t(0)=e$ and $\gamma_t(1)=e'$ for all $t$.) Thus there are $|p^{-1}(b)|$ paths in $E$ that start at a chosen basepoint $e \in p^{-1}(b)$ and end at another point in $p^{-1}(b)$, up to homotopy rel endpoints. Any path $\gamma:[0,1] \to B$ representing an element of $\pi_1(B,b)$ can be lifted to a path $\tilde \gamma:[0,1] \to E$ with starting point $\tilde \gamma(0)$ equaling any chosen $e \in p^{-1}(b)$. Since $\gamma(0)=\gamma(1)=b$, the condition $p \circ \tilde \gamma = \gamma$ implies that $\tilde \gamma(1)$ must also lie in $p^{-1}(b)$. Using the homotopy lifting property, we see that any other path $\gamma'$ in $B$ representing the same class as $\gamma$ in $\pi_1(B,b)$ lifts to a path $\tilde \gamma'$ in $E$ (starting at $e$) that is homotopic rel endpoints to $\tilde \gamma$. This gives us a one-to-one correspondence between elements of $\pi_1(B,b)$ and homotopy (rel endpoint) classes of paths starting at $e \in E$ and ending at some other point in $p^{-1}(B)$. As mentioned above, this latter set has cardinality $|p^{-1}(B)|$.

Answer 2

I like the view that algebraic topology is about modelling topology by algebra; in these terms, a useful model of a covering map $p: E \to B$, given in Topology and Groupoids, and, following earlier editions of that book, to some extent in Peter May's "Concise ...." book, is the "covering morphism" of fundamental groupoids $\pi_1 p: \pi_1 E \to \pi_1 B$.

A morphism $q: H \to G$ of groupoids is a covering morphism if for each object $x$ of $H$ and element $g: q(x) \to y$ in $G$ there is a unique element $h$ in $H$ starting at $x$ such that $q(h)=g$. It is helpful to draw pictures of this.

The traditional arguments show that a covering map determines a covering morphism, and a version of traditional arguments show that if the space $B$ has appropriate local conditions then a covering morphism $H \to \pi_1 B$ determines a topology on $E= Ob(H)$ so that $Ob(q) :E \to B$ is a covering map with $\pi_1 E \cong H$.

The standard argument can now be given in the algebraic model, as explained in the books and by squirrel, to show the result required.