Pontrjagin
Classes |

**In General** > s.a. euler classes.

* __Idea__: They are characteristic
classes, used for principal fiber bundles with base space *B* and *G* =
O(*k*) or SO(*k*).

* __Properties__: They
belong to *f* * H^{i}(Gr(*n*,* k*,
\(\mathbb R\)); \(\mathbb R\)) = H^{i}(*B*;
\(\mathbb R\)), and are non-zero only for *i* a multiple of 4, so they
*p*_{i}(*P*) ∈ H^{4i}(*B*; \(\mathbb R\)).

* __Remark__: *p*_{i}(*P*)
= *P*_{i}(*F*), a 2*i*-form, *F* being
o(*n*)-valued.

> __Online resources__:
see *n*Lab page; Wikipedia page.

**And Bundle Operations**

* __Whitney sum__: Given
two bundles *E* and *F*, *p*(*E* ⊕ *F*)
= *p*(*E*) *p*(*F*) (in terms of forms, this means exterior product).

**Examples**

* __Non diffeomorphic, homeomorphic
manifolds__: For an example of different *p*_{i}s,
@ see Milnor Top(64).

**Tangent Bundles**

* __In general__: For a tangent
bundle T*M*, the *p*_{i}(T*M*)s
are invariants of the differentiable structure of *M*.

* __2D manifold__: The first
two are *p*_{0}(T*M*)
= 1, *p*_{1}(T*M*) = 0, and all others are also zero.

* __4D manifold__: The first
two are *p*_{0}(T*M*)
= 1, *p*_{1}(T*M*)
= –\(\big({1\over2\pi}\big)^2{1\over2}\)tr(*F* ∧ *F*).

**Pontrjagin Numbers** > s.a. Cobordism.

$ __Def__: The integrals
of Pontrjagin classes, given by *P*_{j}(*P*):= \(\int\)_{c}
*p*_{j}(*F*), where *c* is a *j*-chain.

**Physics Applications** > see chern-simons theories; parity [violation]; spacetime topology; theories of gravity.

* __Pontryagin density__: Given a curvature 2-form *F*,
the Pontryagin density for a 4-dimensional manifold is the integrand ε^{abcd} tr(*F*_{ab} *F*_{cd}) of the first Pontryagin number, used as a term in the Lagrangian density for certain gauge theories or modified theories of gravity.

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send feedback and suggestions to bombelli at olemiss.edu – modified 22
nov 2016