From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: local diffeo & RP^n
Date: 8 Aug 2000 17:49:46 -0400
Newsgroups: sci.math
Summary: [missing]

lena <lena@math.nwu.edu> writes:
>
>Ed Hook wrote:
>
>>   I don't know how helpful the following
>>   is, but:
>>
>>   If p: RP^n --> RP^n is a _covering_map_
>>   (which is probably a strictly stronger
>>   condition than "local diffeomorphism"),
>>   then p_# : \pi_1(RP^n) --> \pi_1(RP^n)
>>   is a monomorphism, Since \pi_1(RP^n) = Z/2Z
>>   is finite, p_# is (in fact) an isomorphism.
>>   Since p is a covering map, it follows that
>>   p is a bijection. Since RP^n is a compact
>>   Hausdorff space, p is a (smooth) homeomorphism.
>>   It follows that p is a diffeomorphism (as it
>>   is such locally).
>>
>
>Actually, since RP^n is compact and hausdorff, the local diffeo is indeed a
>covering map.
>However, I'm not fimiliar with the result that says if p# is an iso on the
>Fundamental Groups, then p is a bijection since it is a covering map.

Let p: (X,p^{-1}(*)) --> (Y,*) be any covering map.  Then,
essentially directly from the *definitions* of covering map
and fundamental group, the fiber F = p^{-1}(*) above the basepoint  
* of the base space Y is a set on which \pi_1(Y) has a natural
action; further, as a set-with-\pi_1(Y)-action, F is naturally 
isomorphic to the set of cosets \pi_1(Y)/p_#(\pi_1(X)) with
its natural \pi_1(Y)-action.  In particular, if p_# is iso,
this latter set contains just one element, so F is a one-point
set, so p is a homeomorphism.  The phrase "deck transformation"
is swimming up into my consciousness from murky depths.

No doubt Ed will correct me if I'm totally crazy here.  

>Could you please explain a bit more or provide a few references.

Hu, _Homotopy Theory_.

Lee Rudolph