In this lecture we prove the Leray-Hirsch Theorem, which is a powerful tool for computing the additive structure of the cohomology of a total space of a fibre bundle.

The main ingredient needed for the Leray-Hirsch Theorem is the existence of a cohomology extension of the fibre. Next lecture we will show that such an extension always exists for orientable sphere bundles—this will give the Thom Isomorphism Theorem. On Problem Sheet O there is another famous application, the Gysin Sequence.

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I am in the process of adding a new commenting system to my website. The main advantage of this it that it will allow typing $\mathrm{\LaTeX}$  in comments, which many of you have asked about over the year. However to begin with I’m looking for some friendly “guinea pigs” to test it for me—if you interested in helping (this will take at most five mins of your time) please email me. 🤓 Otherwise the new system will go live next week. Update (17.04): Now live!