It's been a good long while since we've had a maths post here, and I think we're long overdue. You've probably been tossing and turning for hours upon hours in the dead of night, fisted by an overpowering urge to feast upon fatty scraps of technical detail rent from the flesh of my budding mathematical career. And I, dear reader, shall indulge you.
I've been working through Allen Hatcher's Algebraic Topology with some other grads here and one of our faculty.
(Incidentally, my roommate may possibly have just related the complex K-theory of a space to its first cohomology with coefficients in the infinite unitary group U. Which would be crazy. I'm excited.)
Flip to Chapter 2, on homology. Now read it. Come back when you're finished.
All done? Swell.
By now you surely understand why it's important that we can compute degrees of maps between spheres: we need it to do cellular homology. And by now you surely understand what the degree of a map f from the n-sphere to itself is: the nth homology group of the n-sphere is just the group Z of integers, and f induces a homomorphism from Z to itself, which must be multiplication by some integer, which we dub the degree of f. Unfortunately, computing the degree of a map directly is usually horribly difficult, at least for me. Hatcher offers a sort of short-cut, defining local degrees and proving that we can obtain the degree of a map, under suitable hypotheses, as the sum of some local degrees. Doubly unfortunately, local degrees as he defines them aren't much easier to compute than full-fledged degrees. Which is sad. But, mirabile dictu, there is another way one could define local degree, which is quite tractable!
Suppose f is a smooth map of the n-sphere to itself. Suppose y is a regular value of f: at every point in the (finite) inverse image, the pushforward (or Jacobian, or derivative, or [insert notation here]) of f has rank n, making it a linear isomorphism. Then at every point x in the inverse image, f is locally a diffeomorphism: we can choose some neighbourhood V of y, with f-1V a disjoint collection of neighbourhoods of the x's. Given such a neighbourhood U of an x, f induces a map from Hn(U,U-x) to Hn(V,V-y), both groups being Z; so this map is multiplication by some integer, the local degree of f at x. If f is a diffeomorphism on U, then this map is invertible, so the local degree must be +1 or -1. We can also consider the pushforward of f at x; it's a linear isomorphism of the tangent space at x to the tangent space at y, so we can easily check whether it preserves or reverses the orientations on these tangent spaces defined by the standard orientation of the sphere. I claim the local degree at x will be +1 precisely when the pushforward preserves orientations, and -1 when it reverses orientations. Note that this should be given simply by the sign of the determinant of the Jacobian in appropriate coordinate charts, which is immensely amenable to computation.
Why should this be true?
Let's sweep a lot of crap under the rug right now. Let's choose V and our charts so that the image of y is 0, and the image of V is a teeny tiny disk about 0. And let's also choose charts so that the image of x in its chart is likewise 0. Just for shits. So let's look at everything in these coordinates; if F is the coordinate representation of f, then F(0)=0, and for any z in the image of U, we can write F(z)=F(0)+DF(0)z+o(z)=DF(0)z+o(z) where DF(0) is the Jacobian of F at 0, and the 'little o' function dies off very rapidly near the origin.
Here's the kicker: the map DF(0)z+to(z) should be a homotopy between F and DF(0), at least on a small enough neighbourhood. So DF(0) and F must have the same local degrees at 0. Since DF(0) is just an invertible linear map, and GL(n,R) has exactly two path components, DF(0) should be homotopic either to the identity or to the diagonal matrix with all diagonal entries 1 except for the first, which will be -1. Depending on the sign of its determinant. The local degree of DF(0) is then the sign of its determinant, and so we obtain the local degree of F; assuming we've chosen our charts so as not to fuck anything up, we now have the local degree of f at x, too.
Wave your hands in the air! Wave them like you don't care!
So there. That is enough math-posting for today. Fill in the details at your leisure; correct any egregious mistakes in the above. Perhaps I will try to do this rigourously at some point in the near future. It depends on how distracted I get.
We now return you to your regularly-scheduled programming. (You were looking at porn, weren't you?)
Posted by aloysius at February 19, 2004 07:36 PM | TrackBack |