# 4 Thom spaces

4.1

Let $E \to X$ be a vector bundle. Then we define the *Thom space* to be

4.2

$\mathrm{Th}(E) \cong \mathbb {P}(E \oplus 1)/\mathbb {P}(E).$

4.3
(Purity theorem)

Let $Z \hookrightarrow X$ be a closed embedding in $\mathrm{Sm}_S$ with normal bundle $\nu _Z$. Then we have an equivalence (in $\mathrm{Spc}_S^{\mathbb {A}^1}$).

$\frac{X}{X \setminus Z} \to \mathrm{Th}(\nu _Z).$
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Indeed, the fiber over $\{ 0\}$ is $\mathbb {P}(\nu _Z \oplus \mathcal{O}_Z) \setminus \mathbb {P}(\nu _Z)$, which is canonically isomorphic to $\nu _Z$. (This construction is known as “deformation to the normal cone”)

The second step shows that in $\mathcal{H}(S)$, we have a homotopy pushout squares

To prove this, one uses Nisnevich descent to reduce to the affine case.

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