Characteristic
Classes| 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 
:
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 fb*: B → M,
the characteristic cohomology
class of E determined by c 
Hn(M; 
)
is fb*c Hn(B; ).
* Procedure: Have a set
of functions c:
(universal bundles over B) → Hp(B; R),
such that (1) P equivalent to P' implies c(P)
= c(P');
(2) f * c(
)
= c(f *),
i.e., c commutes
with the pullback of the bundle; Then, we calculate Hp(O(n)/G ×
O(n–k)),
evaluate these functions on the universal bundles, and pull back the results
from 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 Z/2.
* Universal characteristic
classes: They are the characteristic classes
of some universal bundle .
* 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, ht/97-in.
References
@ General: Milnor & Stasheff 74; in Spivak 75, v5; in Nash & Sen 83;
Zhang 01 [Chern-Weil theory].
@ Secondary characteristic classes: Vaisman 87.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
22 jun 2008