We have a quite some characteristic classes:

Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc.

I wonder whether some math experts can use simple words and basic equations to give explanation on their relations and correspondence between them? (it is a more or less on the illuminating the concept.)

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    $\begingroup$ I think this question would be better if you removed the bit about dumb people. Lots of smart people have trouble with these concepts, too. People who have difficulty with graduate level math aren't usually "dumb." $\endgroup$ – Potato May 25 '14 at 0:39
  • $\begingroup$ p.s. see a related one, which is certainly not for the dumb (@ Potato, thanks!). $\endgroup$ – wonderich May 25 '14 at 0:40
  • $\begingroup$ the reason to "dumb people", is that a hope to keep the answer down to earth simple rather than too much equation-involved technical. (which one can read from textbooks or papers.) $\endgroup$ – wonderich May 25 '14 at 0:43
  • $\begingroup$ @ Grigory M, Well, I know them from Nakahara's book "geometry topology and physics" and AG class. $\endgroup$ – wonderich May 25 '14 at 1:53
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    $\begingroup$ My point is that you can ask the same question without importing the connotations and implications "dumb" has. $\endgroup$ – Potato May 25 '14 at 1:59

Here goes.

The classes that are assigned to real bundles are Stiefel-Whitney Classes, and Wu classes. The Wu class is a refinement of the Stiefel-Whitney class, it is sort of like a (Steenrod) square root. Both take on values in $\mathbb{Z_2}$ cohomology. They can be seen as obstructions to nonvanishing sections defined on subsets of the base.

If the bundle $p:E\rightarrow B$ is in addition oriented with fiber a vector space of dimension $r$, then the classes defining the orientation in $H^r(F,F-\{0\})$ can be woven together to form the Euler class. If $X\subset B$ is a cycle of dimenison $r$ it assigns to the cycle the intersection of number of a generic section over the cycle with the $0$-section.

The pontryagin classes are integral cohomology classes assigned to real vector bundles by turning the real vector bundle into a complex vector bundle so that a real basis for each fiber becomes a complex basis for the fibers of the new bundle.

Finally, the chern classes are assigned to complex vector bundles. There are many geometric interpretations of the chern classes, but there was one that was more illuminating to me than the others when I finally learned it, it's due to Grothendieck. You can build a complex vector bundle by covering the base $B$ with open sets $U_{\alpha}$ along with a continuous functions, $U_{\alpha}\cap U_{\beta}\rightarrow GL_r\mathbb{C}$ that act like they should be coordinate changes for trivializations. To build the bundle you treat them as such.

So... a vector bundle is a lot like a matrix, and a matrix has a characteristic polynomial. The chern classes of a bundle are the coefficients of a "characteristic polynomial" of the vector bundle. Specifically, if $p:E\rightarrow B$ is a complex $r$-plane bundle, let $P(E)\rightarrow B$ be the bundle whose fibers are the projective $r-1$-spaces modeled on the fibers. There is a class $\gamma \in H^2(P(E))$ whose restriction to each fiber is the chern class of the canonical bundle over that fiber. The cohomology $H^*(P(E))$ is a free module over $H^*(B)$ under the obvious action with basis $, 1,\gamma,\gamma^2,\ldots, \gamma^{r-1}$. Hence you can write $\gamma^r+c_{1}\gamma^{r-1}+\ldots +c_{r-1}\gamma+c_r=0$ where the $c_i\in H^{2i}(B)$. These coefficients are the chern classes of the bundle.

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  • $\begingroup$ @ Charlie Frohman, thank you so much. I am so glad there are so many math experts here. The answer is brilliant. :-) $\endgroup$ – wonderich May 25 '14 at 1:49
  • $\begingroup$ I also wish to see some other relation between those classes, other than the previously discussed one: w2=c1 mod 2, between Stiefel-Whitney class and Chern class. $\endgroup$ – wonderich May 25 '14 at 2:05
  • $\begingroup$ @ Charlie Frohman, you mean the "Characteristic Classes" one by "JOHN W. MILNOR AND JAMES D. STASHEFF"? Yes, I am going to check it. thanks! $\endgroup$ – wonderich May 25 '14 at 2:24

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