Where do the Stiefel-Whitney numbers (and the general characteristic numbers) come from?

Question

I'm having some troubles here. They are defined as the evaluation of the characteristic classes on the unique fundamental class (that is basically an evaluation). Why? Where do they come from? What is the "physical" meaning of them? Any more insight or resources would be very helpful, since I'm trying to understand them better!

Answer 1

If you want to understand them, few places will get you started better than Milnor and Stasheff's book "Characteristic Classes".

For surfaces in 3-space, Banchoff (ca. 1975, maybe?) has a paper that shows that the poincare dual to $w_1$ is carried by the "fold set" of any (generic) projection along a vector $v$ to a plane, so $w_1 \cup w_1$ is carried by the intersections of the fold sets of projections in two different (generic) directions, $v$ and $w$. The intersection is also the set of critical points of the height, measured in direction $v \times w$, and hence $w_1 \cup w_1$ is the same, mod 2, as the Euler characteristic.

If you follow the relationship of the SW classes to obstructions (chapter 10 of M&S, I believe), a similar sort of argument can be made for other top-dimensional cup-products, I believe, but I've never worked out the details.

There's another possible way to think about the duals of the SW classes:

First triangulate a surface. Then apply one level of barycentric subdivision. Each vertex of the resulting triangulation is at the center of a simplex of the original triangulation: there are the face-centers, the edge-centers (i.e., midpoints), and the "vertex-centers" (which are just the original vertices). Imagine coloring these red, blue, and green, respectively.

There's now a very nice map from the surface to a single triangle: you color the three vertices of the triangle red, green, and blue, and map vertices of your surface to the correspondingly-colored vertices of the triangle, and then extend across faces by linearity, for each face in the barycentric subdivision has one red vertex, one blue, and one green.

The poincare dual to $w_1$ is then (homologous to) the set of singular points of the projection, which amounts to the 1-skeleton of the barycentric subdivision; the p-dual to $w_2$ is (homologous to) the set of vertices of the subdivision, and the p-dual to $w_0$ is (homologous to) the set of faces.

What does $w_0 \cup w_2$ then correspond to? The "intersection product" of these two chains.

For surfaces, this isn't very enlightening, but you can do something exactly analogous for higher dimensions, and there's probably an intuitive explanation of these things in terms of stuff like "if you have two general fields of 3-frames on the 4-skeleton, and try to extend each over the 5-skeleton, the obstructions will give you two 4-chains, and the intersection of these 4-chains will be related to ...."

Definitely look at chapter 10 of M&S for some "physical" meaning.