11 CHARACTERISTIC CLASSES

Given a fibre $F$ , a structure group $G$ and a base space $M$ , we may construct many fibre bundles over $M$ , depending on the choice of the transition functions. Natural questions we may ask ourselves are how many bundles there are over $M$ with given $F$ and $G$ , and how much they differ from a trivial bundle $M\times F$ . For example, we observed in section 10.5 that an $\mathrm{SU}(2)$ bundle over $S^4$ is classified by the homotopy group ${\pi }_3(\mathrm{SU}(2))\cong \mathbb{Z}$ . The number $n\in \mathbb{Z}$ tells us how the transition functions twist the local pieces of the bundle when glued together. We have also observed that this homotopy group is evaluated by integrating $\mathrm{tr}{\mathcal{F}}^2\in H^4(S^4)$ over $S^4$ , see theorem 10.7.

Characteristic classes are subsets of the cohomology classes of the base space and measure the non-triviality or twisting of a bundle. In this sense, they are obstructions which prevent a bundle from being a trivial bundle. Most of the characteristic classes are given by the de Rham cohomology classes. Besides their importance in classifications of fibre bundles, characteristic classes play central roles in index theorems.

Here we follow Alvarez-Gaumé and Ginsparg (1984), Eguchi et al (1980), Gilkey (1995) and Wells (1980). See Bott and Tu (1982), Milnor and Stasheff (1974) for more mathematical expositions.

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