Characteristic Classes  

In General
* Idea: They are global invariants of a bundle, which measure its deviation from being a product bundle; Given a fiber bundle π: EB, its characteristic classes are suitable cohomology classes of the base space B that contain information about E.
* History: Invented by Whitney, Stiefel, ...; Chern-Weil theory is a geometric theory of characteristic classes.
$ Def: For a bundle map f : E → (universal bundle with fiber F) and its induced map of base spaces fb*: BM, the characteristic cohomology class of E determined by c ∈ Hn(M; Λ) is fb*c ∈ Hn(B; Λ).
* Procedure: Have a set of functions c: (universal bundles over B) → Hp(B; \(\mathbb R\)), such that (1) P equivalent to P' implies c(P) = c(P'); (2) f * c(ξ) = c(f *ξ), i.e., c commutes with the pullback of the bundle; Then, we calculate Hp(O(n)/G × O(nk)), evaluate these functions on the universal bundles, and pull back the results from ξ to the principal fiber bundle of interest P by the map f : B → O(n)/(G × O(nk)).
* Remark: One can get the three characteristic classes with real coefficients below as polynomials in F, the invariant polynomials of the Lie algebra (> see Weil Homomorphism).

And Bundle Operations
* Remark: Given the characteristic classes of two bundles, it is easy to express in terms of these the ones corresponding to the Whitney sum of the bundles, but the same is not true for tensor product bundles; In the latter case, it is better to use characteristic polynomials.

Specific Classes
* Types: Several kinds of characteristic classes have been defined, such as
> chern classes,
> euler classes,
> pontrjagin classes,
> stiefel-whitney classes.
The first three are cohomology classes with real coefficients, the fourth ones have coefficients in \(\mathbb Z\)/2.
* Universal characteristic classes: They are the characteristic classes of some universal bundle ξ.
* Total classes: For each of the kinds of characteristic classes listed above, one can define the total classes as the sum of all classes of one kind.

Topological Invariants
* Idea: Numbers constructed integrating characteristic classes.
@ Using torsion: Chandía & Zanelli PRD(97)ht, AIP(98)ht/97.

@ General: Milnor & Stasheff 74; in Spivak 75, v5; in Nash & Sen 83; Zhang 01 [Chern-Weil theory].
@ Secondary characteristic classes: Vaisman 87.
@ Generalizations: Lyakhovich JGP(10)-a0906 [of Q-manifolds].

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