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 are often
denoted by *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.

* __Pontrjagin density__: Given a curvature 2-form *F*,
the Pontrjagin density for a 4-dimensional manifold is the integrand ε^{abcd}
tr(*F*_{ab} *F*_{cd})
of the first Pontrjagin 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 12 jun 2018