Chern–Weil homomorphism - Imported Knowlpedia Content

Definition

Let $P\to M$ be a principal $G$ -bundle with Lie algebra $\mathfrak g$ , and write

I(G)=\operatorname{Sym}(\mathfrak g^*)^G

for its algebra of invariant polynomials. The Chern–Weil homomorphism of $P$ is the graded-algebra map

\operatorname{CW}_P:I(G)\longrightarrow H_{\mathrm{dR}}^{\mathrm{even}}(M), \qquad p\longmapsto [p(F_A)],

where $A$ is any principal connection and $F_A$ is its curvature. A polynomial of degree $k$ maps to degree $2k$ . The Chern–Weil theorem makes the class independent of $A$ .

Algebra and naturality

With the standard symmetric-product convention for invariant polynomials,

\operatorname{CW}_P(pq) = \operatorname{CW}_P(p)\smile\operatorname{CW}_P(q), \qquad \operatorname{CW}_P(1)=1.

The product on the right is the cohomology-ring product, identified with wedge product in de Rham cohomology. If $f:N\to M$ , then

\operatorname{CW}_{f^*P}(p)=f^*\operatorname{CW}_P(p).

These properties are developed in Bott–Tu, chapter 11.

Standard examples

For a unitary frame bundle, the coefficients of

\det\!\left(I+\frac{i}{2\pi}F_A\right)

give the real images of Chern classes. Trace polynomials on orthogonal bundles produce Pontryagin classes, and the Pfaffian on an oriented even-rank orthogonal bundle produces the Euler class.

Conventions and scope

The notation $\operatorname{CW}_P$ records dependence on the bundle $P$ , even though it does not depend on the auxiliary connection.

References

Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: chapter 11, invariant polynomials and the Chern–Weil homomorphism.
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, vol. II, Wiley Classics, 1996. Publisher record. Relevant: chapter XII, characteristic forms.