4 Thom spaces

Definition 4.1

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

\mathrm{Th}(E) = E / E^\times .

Proposition 4.2

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

Theorem 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).

Proof

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[Proof idea] The first geometric input is the construction of a bundle of closed embeddings over

\mathbb {A}^1

whose fiber over

\{ 0\}

(\nu _Z, Z)

and

(X, Z)

elsewhere. This has a very explicit description:

D_ZX = \mathrm{Bl}_{Z \times _S \{ 0\} } (X \times _S \mathbb {A}^1) \setminus \mathrm{Bl}_{Z \times _S \{ 0\} } (X \times _S \{ 0\} ).

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

$\begin{useimager} \[ \begin{tikzcd} \dfrac{\nu_Z}{\nu_Z \setminus Z} \ar[d] & Z \ar[r] \ar[l] \ar[d] & \dfrac{X}{X \setminus Z} \ar[d]\\ \dfrac{D_ZX}{D_ZX \setminus Z \times \A^1} & Z \times \A^1 \ar[l] \ar[r] & \dfrac{D_ZX}{D_Z X\setminus Z \times \A^1} \end{tikzcd} \] \end{useimager}$

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

Proof

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