Pontryagin class - Knowlpedia

Let $E\to X$ be a rank- $r$ topological real vector bundle over a paracompact Hausdorff space. Its complexification $E^{\mathbb C}$ has fibers $E_x\otimes_{\mathbb R}\mathbb C$ , with local trivializations obtained by regarding the real transition matrices as complex matrices. For $k\ge0$ , the $k$ th Pontryagin class is

p_k(E)=(-1)^k c_{2k}(E^{\mathbb C})\in H^{4k}(X;\mathbb Z),

where $c_{2k}$ is the integral Chern class. Thus $p_0(E)=1$ and $p_k(E)=0$ for $2k>r$ . The total Pontryagin class is $p(E)=1+p_1(E)+p_2(E)+\cdots$ .

Chern–Weil representatives

If $X=M$ is a smooth manifold and $E$ is smooth, choose a fiber metric and a compatible connection $\nabla$ with curvature $F_\nabla$ . Its complexification induces a complex connection $\nabla^{\mathbb C}$ . Define the Pontryagin forms by

p_k(\nabla)\;:=\;(-1)^k\,c_{2k}(\nabla^{\mathbb C})\in \Omega^{4k}(M),

where $c_{2k}(\nabla^{\mathbb C})$ is the corresponding Chern–Weil Chern form.

Then:

Each $p_k(\nabla)$ is closed: $d\,p_k(\nabla)=0$ , where $d$ is the exterior derivative.
The de Rham class $[p_k(\nabla)]\in H^{4k}_{\mathrm{dR}}(M)$ is independent of the choice of compatible connection.
The de Rham class $[p_k(\nabla)]$ is the real image of the integral class $p_k(E)$ .

The de Rham representative detects only the real image of $p_k(E)$ , not any torsion in the integral class. Equivalently, this real image is the Chern–Weil class associated to the structure group $O(r)$ (or $SO(r)$ in the oriented case) by applying an $Ad$ -invariant polynomial on $\mathfrak{so}(r)$ corresponding to the $k$ th elementary symmetric polynomial in the squares of the formal curvature eigenvalues.

Naturality

For any continuous map $f:Y\to X$ between paracompact Hausdorff spaces, the topological classes satisfy

p_k(f^*E)=f^*p_k(E).

Examples

Trivial bundle / flat connection. If $E\cong M\times\mathbb R^r$ is trivial, then $p_k(E)=0$ for all $k\ge 1$ . More generally, a flat connection has $F_\nabla=0$ , so its Pontryagin forms vanish and the real (hence rational) images of the positive-degree Pontryagin classes are zero.

Underlying real bundle of a complex line bundle. Let $L\to M$ be a complex line bundle with $c_1(L)=x\in H^2(M;\mathbb Z)$ . For the underlying real rank-2 bundle $L_{\mathbb R}$ one has $p_1(L_{\mathbb R})=x^2\in H^4(M;\mathbb Z),$ because $p_1=(-1)c_2\big((L_{\mathbb R})^{\mathbb C}\big)$ and $(L_{\mathbb R})^{\mathbb C}\cong L\oplus \overline{L}$ .

Dimensional vanishing. If $\dim M < 4k$ , then every $4k$ -form vanishes and hence $p_k(E)=0$ in de Rham cohomology (and therefore in rational cohomology) for degree reasons.