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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!

Nythra
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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.

John Hughes
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  • Just for readers: I believe you meant chapter 12, called "Obstructions".

    It's enlighening me, slowly! Though I still can't see what is the role of the orientability. I get that it's used for defining a unique (single generated) fundamental class, but is there a way to see it more clearly?

     – Nythra Apr 02 '17 at 15:55
  • Completely right...I was doing it from memory, and grad school was a long, long time ago. :) – John Hughes Apr 02 '17 at 15:56
  • Correct me if I'm wrong. The orientability is used because in a oriented manifold the sum of all n-simplices is a generator for the homology class $H_n$, that is also the p-dual to the total characteristic class! – Nythra Apr 02 '17 at 16:16
  • I think you're wrong here -- because the SW classes are in mod-2 cohomology, and everything is orientable (i.e., has a fundamental class) mod 2. Orientability comes up in the M-and-S discussion of obstructions, as I recall, but that's because knowing that the set of zeroes of a vector field is zero, mod 2, might not be enough to show that there's an everywhere nonzero field (which is what the Euler class, sort of the archetype of obstruction classes) tells you. – John Hughes Apr 02 '17 at 16:31
  • I though it was a general thing: For SW classes, it's a generator because we pick coefficients in $\mathbb{Z}_2$. For anything else (Euler), it's a generator because it's orientable. – Nythra Apr 02 '17 at 16:40
  • Right... I missed the last sentence of your original comment, and was answering a more general question. It helps to have an intuitive notion (slighlty wrong) of poincare duality: take a triangulation. There's a dual "reticulation". (For a surface, join centers of triangles in the triangulation with new edges that cross the midpoint of triangulation edges. For 2-cells, use neighborhoods of triangulation vertices, neighborhoods bounded by chains of new edges.) The PD of a simplex is a cochain that evaluates to "1" on the reticulation piece that "crosses" that simplex. But for the surface case.. – John Hughes Apr 02 '17 at 16:46
  • ..you have to decide "crosses in which direction???" And for that, you need orientability. – John Hughes Apr 02 '17 at 16:47
  • What I still struggle with: we take the fundamental class, that is the nonzero element in $H_n$ (using coefficients in $\mathbb{Z_2}$). We have a duality between homology and cohomology, and use that to evaluate the SW class on the fundamental class. The fundamental class comes from the sum of all simplices, right? what's its PD? And even more, I still can't see a phisical meaning of all this. I fear I'm looking at it in the wrong direction, and chapter 12 in M&S is not helping much... – Nythra Apr 15 '17 at 19:01