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 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