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 p_i(P) ∈ H⁴ⁱ(B; \(\mathbb R\)).
* Remark: p_i(P) = P_i(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(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_is, @ see Milnor Top(64).

Tangent Bundles
* In general: For a tangent bundle TM, the p_i(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(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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