# 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)

and have lift then can lift entire hmtpy. similar to how Van Kampfen is proved, this result is proved by partitioning the domian of a map 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

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

1. covering space of then injects into , and image is loops at basepoint that lift to loops upstairs.

if is path connected then the number sheats in above in a covering space 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: lifts to if lifts; in proof the cond on says a reasonably natural way of lifting map to X is well defined (make a path and lift it)

needs to be path conn. (to define lift)

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

Uniqueness of lifts, in the above, if 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 such that the inclusion gives trivial map on . Latter two properties allow you to define a basis

.

Take a as in def. of semilocally simply conn. a path from base point to a point in , then

Now want to get correspondence between path. conn. cov. spaces and subgroups. To get a cov. to correspond. to a sub gp. H do

if and is a loop in . Under identification get will be cov. sp.

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- Published:
- January 1, 2010 / 7:45 pm

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- Quail, wall scribble

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