# Quail Entry: Hatcher Chap 1

I’m putting this on here now, but I actually first did this around 12/16/09.

### Covering Spaces

hmtpy lifting property (hlp)

$f_t \colon Y \to X$ and have lift $\tilde{f}_0 \colon Y \to \tilde X$ then can lift entire hmtpy. similar to how Van Kampfen is proved, this result is proved by partitioning the domian of a map $f \colon Y \times I \to X$ near a point to construct a unique lift (i.e. locally construct lift), then show they agree on overlap (basically because they agree on something like $\{y\} x I$

(random) Van Kampfen Counter samples (cover of S^1 by two open sets, and suspension of three points).

1. $X'$ covering space of $X$ then $\pi_1 X'$ injects into $\pi_1X$, and image is loops at basepoint that lift to loops upstairs.

if $X$ is path connected then the number sheats in above in a covering space $X'$ is constant

2. and in this case the index [pi-1(X): p_* pi-1(X’)] is equal to the number of sheets.

Lifting Criterion: $Y \to X$ lifts to $X'$ if $\pi_1Y \to \pi_1X$ lifts; in proof the cond on $\pi_1$ says a reasonably natural way of lifting map to X is well defined (make a path and lift it)

$Y$ needs to be path conn. (to define lift)

$Y$ needs to be loc. path. conn. ( to make map cont.)

Uniqueness of lifts, in the above, if $Y$ is path conn. (don’t need loc. path conn) and have to lifts that agree at a point then they are the same lift.

pf: show set where lifts agree is both open and closed; rather set where they agree and and set where they disagree are both open (i.e show any point has nbd where lift is goes to a covering space nbd U maping homeo to im.).

Universal covering space
Need path connected, locally path connected, semilocally simply connected

the last property means every point $latex x$ has a nbd $U$ such that the inclusion $U \to X$ gives trivial map on $\pi_1$.  Latter two properties allow you to define a basis

$X' = \{ [f] | f \mbox{ a path in } X \}$.
Take a $U$ as in def. of semilocally simply conn. $f$ a path from base point to a point in $U$, then $U_[f] = \{ g \circ f | g \mbox{ a path in }U\}$
Now want to get correspondence between path. conn. cov. spaces and subgroups.  To get a cov. to correspond. to a sub gp. H do

$[f] ~ [f']$ if $f(1) = f'(1)$ and $f \circ f'(1-t)$ is a loop in $H$.  Under identification get $X_H$ will be cov. sp.