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.

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