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.

Remark

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