Abstract
Characteristic classes We view characteristic classes as being defined by generators of the cohomology ring of the classifying space. Accordingly, we study the chohomology rings of the classifying spaces for the classical compact Lie groups and use their generators to define the Chern, Pontryagin and Stiefel-Whitney classes. Moreover, we discuss theWeil homomorphism, which provides a geometric description of characteristic classes in terms of de Rham cohomology, and genera of vector bundles. Finally, we explain a method to construct an approximation of the classifying space in terms of Eilenberg-MacLane spaces, known as the Postnikov tower.
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Notes
- 1.
For \(\alpha \in H^{k}_{R}(X, A)\) and \(\beta \in H^{l}_{R}(Y, B)\), \(\alpha \times \beta = ({{\mathrm{pr}}}_X^*\alpha ) \,{\scriptstyle \cup }\,({{\mathrm{pr}}}_Y^*\beta ) \in H^{k+l}_{R}(X \times Y, A \times Y \cup X\times ~B)\).
- 2.
Recall the notion of orientation of a \(\mathbb {K}\)-vector bundle from Example 1.6.6/1.
- 3.
Note that E itself need not be oriented here.
- 4.
This holds also for \(n = 1\), provided we define \(\mathrm U(0)\) as the trivial group consisting of one element.
- 5.
The assumption made there that \(H_{k}{(X)}\) be finitely generated for every k is met by all topological spaces of CW-homotopy type.
- 6.
That is, \(\beta \) intertwines the homomorphism induced by a continuous mapping in \({\mathbb {Z}}_2\)-cohomology with that induced in integral cohomology.
- 7.
Composition of mod 2 reduction with the integral Bockstein homomorphism yields the Steenrod square.
- 8.
In fact, this is the Stiefel bundle \(\mathrm S_\mathbb {C}(1, n) \rightarrow \mathrm G_\mathbb {C}(1, n)\).
- 9.
In view of Corollary 4.2.17/2, the vanishing of \({{\mathsf {w}}}_1\) follows also from the fact that a real vector bundle admitting a complex or quaternionic structure is necessarily orientable.
- 10.
Every quaternionic Hermitean fibre metric on \(L_n\) provides an isomorphism.
- 11.
The factor \(\frac{1}{k!l!}\) in this definition is dictated by our choice of the wedge product of differential forms, see formula (2.4.17) in Part I. In many textbooks, the coefficient in (4.6.2) is \(\frac{1}{(k+l)!}\) which corresponds to the other common choice of the wedge product. These different conventions lead to different combinatorial factors on the way, but the final formulae for the Chern classes will be the same. We will comment on this at the end of this section in Remark 4.6.10/2.
- 12.
See Remark 4.1.10/1 in Part I for the definitions of the integral and the derivative of a 1-parameter family of differential forms and for the corresponding calculus.
- 13.
A standard reference is [105]. The arguments for the classical compact Lie groups are elementary though, see the discussion below.
- 14.
We will see below that the normalization factor \(4 \pi \) will make the cohomology classes obtained via the Weil homomorphism match the Chern classes. In many textbooks, the factor is \(2\pi \). This will be explained in Remark 4.6.10.
- 15.
Since A is real and the eigenvalues are purely imaginary, they come in conjugate pairs.
- 16.
By definition, \({\mathrm {pf}}(A) = \frac{1}{2^l l!} \sum _{\sigma \in \mathrm S_{2l}} \prod _{i=1}^l A_{\sigma (2i-1),\sigma (2i)}\).
- 17.
An element of \(H^{k}_{A}(K(A, k))\) is called characteristic if under the bijection \(H^{k}_{A}(K(A, k)) \cong {{\mathrm{Hom}}}\big (H_{k}{(K(A, k))}, A\big )\) it corresponds to an isomorphism \(H_{k}{(K(A, k))} \rightarrow A\).
- 18.
Every continuous mapping between CW-complexes is homotopic to a cellular mapping [287, Theorem 4.8].
- 19.
Explained prior to Proposition 3.2.9.
- 20.
The mapping cylinder of \(f : X \rightarrow Y\) is the quotient space of \((X \times I) \sqcup Y\) obtained by identifying each pair \((x, 1) \in X \times I\) with the point \(f(x) \in Y\).
- 21.
See for example [287, Theorem 4.37].
- 22.
If the diagram is commutative, if the rows are exact and if the vertical arrows except for that in the middle are isomorphisms, then the arrow in the middle is an isomorphism, too.
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Rudolph, G., Schmidt, M. (2017). Cohomology Theory of Fibre Bundles. Characteristic Classes. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-0959-8_4
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DOI: https://doi.org/10.1007/978-94-024-0959-8_4
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