Chern Classes and Numbers |

**Chern Classes**

* __Idea__: They are
characterisic classes, used for *G* = U(*k*).

* __Notation__: They belong
to *f** H^{i}(Gr(*n*,
*k*, \(\mathbb C\)); \(\mathbb R\)) = H^{i}(*B*;
\(\mathbb R\)), and they are non-zero only if *i* is even, so they are often written
*c*_{i}(*P*) ∈
H^{2i}(*B*; \(\mathbb R\)).

**And Bundle Operations**

* __For the Whitney sum__: *c*(*E* ⊕
*F*) = *c*(*E*) *c*(*F*) (in terms of forms, this means exterior product).

**Examples**

* __For SU(2)__:

*c*_{1}(*P*)
= 0 , *c*_{2}(*P*)
= –\(1\over16\pi^2\)*F*^{a} ∧ *F*^{a} =
\(1\over8\pi\)tr(*F* ∧ *F*) .

* __For a tangent bundle T M__:
The

*

*c*_{0}(*P*)
= 1, *c*_{1}(*P*)
= \({\rm i}\over2\pi\)tr(*F*), *c*_{2}(*P*)
= \(\big({{\rm i}\over2\pi}\big)^2\)\(1\over2\)(tr
*F* ∧ tr *F *–
tr *F* ∧ *F*), *c*_{n}(*P*)
= \(\big({{\rm i}\over2\pi}\big)^n\)det *F* .

**References** > s.a. non-commutative geometry.

@ __In terms of curvature, etc__: Briggs gq/99.

@ __And physics__: Yang PT(12)jan [and quantum numbers].

> __Online resources__:
see Wikipedia page.

**Chern Numbers**

* __Idea__: Roughly speaking,
the number of times a closed surface is wrapped around another closed surface.

$ __Def__: Integrals of Chern
classes, *C*_{j}(*P*):=
∫_{c}* c*_{j}(*P*),
where *c* is a *j*-chain.

* __Chern character__: The
Chern character of a complex vector bundle is most conveniently defined as
the exponential of a curvature of a connection; Its cohomology class does not
depend on the particular connection chosen.

@ __References__: Leonforte et al a1806 [Uhlmann number extension, and fermion systems].

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