 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 * Hi(Gr(n, k, $$\mathbb R$$); $$\mathbb R$$) = Hi(B; $$\mathbb R$$), and are non-zero only for i a multiple of 4, so they are often denoted by pi(P) ∈ H4i(B; $$\mathbb R$$).
* Remark: pi(P) = Pi(F), a 2i-form, F being o(n)-valued.
> Online resources: see nLab page; Wikipedia page.

And Bundle Operations
* Whitney sum: Given two bundles E and F, p(EF) = p(E) p(F) (in terms of forms, this means exterior product).

Examples
* Non diffeomorphic, homeomorphic manifolds: For an example of different pis, @ see Milnor Top(64).

Tangent Bundles
* In general: For a tangent bundle TM, the pi(TM)s are invariants of the differentiable structure of M.
* 2D manifold: The first two are p0(TM) = 1, p1(TM) = 0, and all others are also zero.
* 4D manifold: The first two are p0(TM) = 1, p1(TM) = −$$\big({1\over2\pi}\big)^2{1\over2}$$tr(FF).

Pontrjagin Numbers > s.a. Cobordism.
\$ Def: The integrals of Pontrjagin classes, given by Pj(P):= $$\int$$c pj(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 εabcd tr(Fab Fcd) of the first Pontrjagin number, used as a term in the Lagrangian density for certain gauge theories or modified theories of gravity.