Motivic Homotopy Theory — Thom spaces

4 Thom spaces

Definition 4.1

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

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

Proposition 4.2
Th(E)P(E1)/P(E). \mathrm{Th}(E) \cong \mathbb {P}(E \oplus 1)/\mathbb {P}(E).

Theorem 4.3 (Purity theorem)

Let ZXZ \hookrightarrow X be a closed embedding in SmS\mathrm{Sm}_ S with normal bundle νZ\nu _ Z. Then we have an equivalence (in SpcSA1\mathrm{Spc}_ S^{\mathbb {A}^1}).

XXZTh(νZ). \frac{X}{X \setminus Z} \to \mathrm{Th}(\nu _ Z).

Proof idea

The first geometric input is the construction of a bundle of closed embeddings over A1\mathbb {A}^1 whose fiber over {0}\{ 0\} is (νZ,Z)(\nu _ Z, Z) and (X,Z)(X, Z) elsewhere. This has a very explicit description:

DZX=BlZ×S{0}(X×SA1)BlZ×S{0}(X×S{0}). 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}\{ 0\} is P(νZOZ)P(νZ)\mathbb {P}(\nu _ Z \oplus \mathcal{O}_ Z) \setminus \mathbb {P}(\nu _ Z), which is canonically isomorphic to νZ\nu _ Z. (This construction is known as “deformation to the normal cone”)

The second step shows that in H(S)\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.