Some properties vector bundles and Thom spaces

Question

Basic properties of Thom spaces for vector bundles are rather scattered throughout various sources, mostly without proof: Is it right that

• $T(\xi\oplus\zeta)=T\xi\wedge T\zeta$ for two bundles $\xi$ and $\zeta$ over the same base space? Or was that for the external product of vector bundles?

• $T(\xi\oplus\varepsilon)=\Sigma T\xi$ for the trivial line bundle $\varepsilon$?

• If $E$ is the total space of a bundle $\xi$, is then $E\times\mathbb{R}$ the total space of $\xi\oplus\varepsilon$?

If yes, where is this shown?

Answer 1

See e.g. chapter 12 in

Switzer, Robert M. Algebraic topology-homotopy and homology. Springer, 2017.

You have $T(\xi \times \zeta) = T\xi \wedge T\zeta$. The other bullet points are correct.