Why is the constant relating the chern class and curvature form always $2\pi i$?

Question

I'm reading Milnor's book on Characteristic Classes. In Appendix C, Milnor shows the invariant polynomial of the curvature form and the Chern class differ by powers of $2\pi i$. He first shows that the first Chern class and the trace of the curvature form are multiples of each other by a constant $a$. He calculates this constant for a bundle on a sphere, and finds it is $2\pi i$. Milnor says it's enough to calculate $a$ for one specific case, and that this constant is the same for all vector bundles.

My question is: why is it enough to find $a$ for one specific case? Is there any reason why the constant can't be different for different bundles?

The proof is done on page 306.

Answer 1

The point of the proof is to show that what Milnor calls $\Omega_{12}(M)$ is a characteristic class for complex line bundles $\zeta\to M^2$. It's not an arbitrary element of $H^*(M, \mathbb{C})$; it's a map from bundles to cohomology that's natural and defined in terms of a certain classifying space. On p. 298, Milnor proves that certain forms induce characteristic classes; he then proves on p 306 that $\Omega_{12}(M)$ is a form of that type and thus defines a characteristic class. The only characteristic classes for complex line bundles are of the form $\zeta \to \alpha c(\zeta)\in H^*(X, \mathbb{C}) = H^*(X)\otimes \mathbb{C}$ for some fixed $\alpha\in\mathbb{C}$ (see, for example, the axiomatic definition of the Chern class--- it's determined by its value on the tautological bundle of $\mathbb{CP}^\infty$), and going through the computation for the case of $M = S^2, \zeta = T^*S^2$ determines $\alpha$.

Answer 2

This is meant as a supplement to anomaly’s elegant answer above. In what follows, all manifolds are assumed to be closed and oriented.

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For each smooth manifold $ M $, let

• $ \mathscr{L}(M;\mathbb{C}) $ denote the set of all isomorphism classes of smooth (complex) line bundles over $ M $;
• $ {H^{*}}(M;\mathbb{C}) $ denote the cohomology ring of $ M $.

Hence, we may view $ \mathscr{L}(\bullet;\mathbb{C}) $ and $ {H^{*}}(\bullet;\mathbb{C}) $ as contravariant functors from the category of smooth manifolds to the category of sets.

Milnor essentially establishes that $ \Omega_{12} $ is a natural transformation from $ \mathcal{L}(\bullet;\mathbb{C}) $ to $ {H^{*}}(\bullet;\mathbb{C}) $, i.e., for each morphism $ f: M \to N $ of smooth manifolds, we have the following commutative diagram: $$ \require{AMScd} \begin{CD} \mathscr{L}(N;\mathbb{C}) @>{f^{*}}>> \mathscr{L}(M;\mathbb{C}) \\ @V{{\Omega_{12}}(N)}VV @VV{{\Omega_{12}}(M)}V \\ {H^{*}}(N;\mathbb{C}) @>>{f^{*}}> {H^{*}}(M;\mathbb{C}) \end{CD} $$

It is a well-known result (see Theorem 14.5 of Milnor’s and Stasheff’s book) that any element of the cohomology ring $ {H^{*}}(\mathbb{C P}^{\infty};\mathbb{C}) $ is a polynomial in the first Chern class of the tautological line bundle $ \gamma^{1} $ over $ \mathbb{C P}^{\infty} $. Having said so, let $ P \in \mathbb{C}[X] $ be a polynomial such that $$ [{\Omega_{12}}(\mathbb{C P}^{\infty})](\gamma^{1}) = P({c_{1}}(\gamma^{1})). $$ Let $ M $ be any smooth $ 2 $-manifold and $ E $ any smooth line bundle over $ M $. As $ \gamma^{1} $ is a universal line bundle, there exists a smooth mapping $ f: M \to \mathbb{C P}^{\infty} $ such that $ E = {f^{*}}(\gamma^{1}) $. The naturality of the first Chern class $ c_{1} $ then yields \begin{align} [{\Omega_{12}}(M)](E) & = [{\Omega_{12}}(M)]({f^{*}}(\gamma^{1})) \\ & = {f^{*}}([{\Omega_{12}}(\mathbb{C P}^{\infty})](\gamma^{1})) \\ & = {f^{*}}(P({c_{1}}(\gamma^{1}))) \\ & = P({f^{*}}({c_{1}}(\gamma^{1}))) \\ & = P({c_{1}}({f^{*}}(\gamma^{1}))) \\ & = P({c_{1}}(E)). \end{align} Now, $ [{\Omega_{12}}(M)](E) \in {H^{2}}(M;\mathbb{C}) $, so $ P $ is a linear polynomial with no constant term, i.e., $ P = \alpha X $ for some $ \alpha \in \mathbb{C} $. Therefore, for all smooth $ 2 $-manifolds $ M $ and all smooth line bundles $ E $ over $ M $, $$ [{\Omega_{12}}(M)](E) = \alpha \cdot {c_{1}}(E). $$

Conclusion: We can determine the value of $ \alpha $ by considering the cotangent bundle $ T^{*} \mathbb{S}^{2} $ over $ \mathbb{S}^{2} $.