From: hook@nas.nasa.gov (Ed Hook) Subject: Re: local diffeo & RP^n Date: 8 Aug 2000 22:22:25 GMT Newsgroups: sci.math Summary: [missing] In article <39907334.26DC9E96@math.nwu.edu>, 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. How do you prove that ?? It's clear that the map is finite-to-one everywhere, but I can (I think) visualize examples where such a map is _not_ a covering map. So I'd be interested in your proof (so that I can figure out where my intuition is going astray.) |> 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: (E,e0) --> (B,b0) be a covering map. Then any loop in B based at b0 can be lifted to a unique path in E starting at e0. And homotopic loops lift to paths that end at the same point in E. One application of these results is that the induced homomorphism p_# : \pi_1(E,e0) --> \pi_1(B,b0) is a monomorphism. (Proof: Suppose that f: ([0,1],{0,1}) --> (E,e0) is a map such that p o f is homotopic rel the boundaries to the constant map to b0. Choose such a homotopy and lift it to a homotopy "upstairs" -- looking at what happens on the boundaries, uniqueness of the lifting forces the lifted map to be a homotopy rel boundary from f to the constant map at e0. That proves the claim.) Another application of this idea: lifting loops based at b0 to paths starting at e0 and looking at the _other_ endpoint defines a right action of \pi_1(B,b0) on the fiber p^{-1}(b0). If E is path-connected, then this action is _transitive_ on the fiber -- and, in that case, you can show easily that two elements of \pi_1(B,b0) send e0 to the same point iff they belong to the same right coset of p_#(\pi_1(E,e0)) in \pi_1(B,b0). In particular, if the fiber is a *finite* set, then its cardinality is equal to the index of p_#(\pi_1(E,e0)) in \pi_1(B,b0). So, after all of that, the last ingredient that you need is that \pi_1(RP^n,whatever) is a _finite_ group. Hence, the fact that p_# is a self-monomorphism implies that p_# is an _auto_morphism. So the index of its image is 1 -- hence, p is one-to-one. (Hmmm -- I just realized that I glossed over p's being onto. Msut it be ?? I guess that's _forced_ by the assumption that p is a covering map, since the definition of such a map says that every point in the basespace has evenly-covered neighborhoods ... and that implies that a covering map is a surjection. Not particularly *satisfying*, though ... ) |> |> Could you please explain a bit more or provide a few references. Hope that the above qualifies as "explaining a bit more". The results in question _should_ be found in any elementary algebraic topology text that deals with covering spaces. However, it's been *many* years since I did this stuff, so any references I gave would be rather old -- certainly, you could find this is Spanier's "Algebraic Topology" or Greenberg's "Lectures on Algebraic Topology" (which, I think, has since appeared in a newer edition authored by Greenberg and Harper(?) -- anyone ?) |> Thanks. |> -- Ed Hook | Copula eam, se non posit Computer Sciences Corporation | acceptera jocularum. NAS, NASA Ames Research Center | All opinions herein expressed are Internet: hook@nas.nasa.gov | mine alone ============================================================================== From: lena Subject: Re: local diffeo & RP^n Date: Wed, 09 Aug 2000 15:14:39 -0500 Newsgroups: sci.math Summary: [missing] Ed Hook wrote: > |> Actually, since RP^n is compact and hausdorff, the local diffeo is indeed a > |> covering map. > > How do you prove that ?? It's clear that > the map is finite-to-one everywhere, but > I can (I think) visualize examples where > such a map is _not_ a covering map. So I'd > be interested in your proof (so that I can > figure out where my intuition is going > astray.) Let X and Y be smooth, compact, hausdorff manifolds of the same dimension, and let f:X --> Y be the local diffeo. Pick y in Y; this is a regular value because of the local diffeo. This implies that f inverse (f^-1(y)) of y is a zero dimensional compact manifold--hence it is a finite, discrete set, say x1, x2,...,xk. Now, pick non-intersecting neighborhoods around each of these points such that f restricted to these neighborhoods is a a homeo; this is easily accomplished because of the local diffeo and the "hausdorfness" at hand. You might also need the fact that a finite intersection of open sets is still open. This shows that f is a covering map. [reformatted --djr] ============================================================================== From: hook@nas.nasa.gov (Ed Hook) Subject: Re: local diffeo & RP^n Date: 9 Aug 2000 21:36:17 GMT Newsgroups: sci.math In article <3991BBAE.1D63B7DF@math.nwu.edu>, lena writes: [quote of previous message deleted --djr] [quote continues through: ] |> You might also need the fact that |> a finite intersection of open sets is still open. You _do_ need to use that fact to get "evenly-covered" neighborhoods of all points in the base space. (Just take f(U_1) \cap f(U_2) \cap ... \cap f(U_k), where the U_i are the above-mentioned neighborhoods of the x_i -- that's a neighborhood of y that's evenly-covered by f ... ) |> This shows that f is a covering map. Yep. It does, indeed :-) Thanks. That was (more or less) the way that I convinced myself that the map had to be finite-to-one. But I stumbled over some sort of mental block before I could get all the way to the end ... -- Ed Hook | Copula eam, se non posit Computer Sciences Corporation | acceptera jocularum. NAS, NASA Ames Research Center | All opinions herein expressed are Internet: hook@nas.nasa.gov | mine alone