4 Thom spaces
Let be a vector bundle. Then we define the Thom space to be
Let be a closed embedding in with normal bundle . Then we have an equivalence (in ).
[Proof idea] The first geometric input is the construction of a bundle of closed embeddings over
whose fiber over
elsewhere. This has a very explicit description:
Indeed, the fiber over is , which is canonically isomorphic to . (This construction is known as “deformation to the normal cone”)
The second step shows that in , we have a homotopy pushout squares
To prove this, one uses Nisnevich descent to reduce to the affine case.