We specialise the Leray-Hirsch Theorem from the last lecture to the special case of vector bundles and disk bundles. In this case the existence of a cohomology extension of the fibre is equivalent to the existence of a single cohomology class, called a Thom Class. Thus as an immediate corollary of the Leray-Hirsch Theorem we deduce the Thom Isomorphism Theorem.

We then investigate when a Thom class exists. For $R= \mathbb{Z}_2$, the answer is: always. For $ R = \mathbb{Z}$, this requires us to define what it means for a vector/sphere bundle to be orientable. We show that a Thom class exists in the orientable case.


As mentioned last time, I will shortly be introducing a better commenting system. Thank you to those that have already agreed to help test it—I am still looking for more volunteers so please send me an email if you are interested. Update (17.04): Now live!

Comments and questions?