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*: *E* → *B*,
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 *f*_{b}***:
*B* → *M*, the characteristic cohomology class of *E*
determined by *c* ∈ H^{n}(*M*;
Λ) is *f*_{b}**c* ∈
H^{n}(*B*; Λ).

* __Procedure__: Have a set of functions *c*:
(universal bundles over *B*) → H^{p}(*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 H^{p}(O(*n*)/*G*
× O(*n*−*k*)), 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(*n*−*k*)).

* __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.

**References**

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 14 jan 2016