Motivic Homotopy Theory

Motivic homotopy theory, developed by Vladimir Voevodsky, revolutionizes our understanding of algebraic varieties by applying the powerful machinery of homotopy theory. This Fields Medal-winning work created a new foundation for algebraic geometry and resolved long-standing conjectures.

🏅 Fields Medal 2002

Vladimir Voevodsky (1966-2017)

Awarded for developing motivic homotopy theory and proving the Milnor conjecture, fundamentally transforming algebraic topology and K-theory.

The $\mathbb{A}^1$-Homotopy Category

Fundamental Principle

In motivic homotopy theory, the affine line $\mathbb{A}^1$ plays the role of the interval $[0,1]$ in classical topology:

  • $X \times \mathbb{A}^1$ is "contractible" to $X$
  • Nisnevich topology replaces the usual topology
  • Pointed schemes $(X, x_0)$ form the basic objects

The Milnor Conjecture (Proved by Voevodsky)

For any field $F$ of characteristic not 2:

$$K_*^M(F)/2 \cong H^*_{ét}(F, \mathbb{Z}/2)$$

where $K_*^M(F)$ is Milnor K-theory and $H^*_{ét}$ is étale cohomology.

$\mathbb{A}^1$-Homotopy Equivalence

Motivic Spheres and Stable Homotopy

Classical Spheres

$S^n = \mathbb{A}^n \setminus \{0\} / \mathbb{A}^{n-1} \setminus \{0\}$

Motivic Spheres

$S^{p,q} = S^p \wedge \mathbb{G}_m^{\wedge q}$

Motivic Cohomology

Motivic cohomology groups $H^{p,q}(X, \mathbb{Z})$ satisfy:

  • $H^{p,0}(X, \mathbb{Z}) = CH^p(X)$ (Chow groups)
  • $H^{p,p}(X, \mathbb{Z}) \cong H^p_{Zar}(X, \mathbb{Z})$ for smooth $X$
  • Connects to K-theory via spectral sequences

The Stable Motivic Homotopy Category

Slice Filtration

The slice filtration provides a motivic analog of the Postnikov tower:

Connection to Algebraic K-Theory

The motivic spectrum representing algebraic K-theory:

$$KGL \simeq \bigvee_{n \in \mathbb{Z}} \Sigma^{2n,n} MGL$$

Six Functors Formalism

f*
f*
f!
f!
⊗
Hom

These functors satisfy analogous properties to their topological counterparts, enabling powerful computational techniques.

Key Computations

Motivic Cohomology of $\mathbb{P}^1$

$$H^{p,q}(\mathbb{P}^1, \mathbb{Z}) = \begin{cases} \mathbb{Z} & \text{if } (p,q) \in \{(0,0), (2,1)\} \\ 0 & \text{otherwise} \end{cases}$$

Motivic Homotopy Groups of Spheres

$$\pi_{p,q}(S^{0,0}) = \begin{cases} K_q^M(k) & \text{if } p = q \\ 0 & \text{if } p < q \end{cases}$$

"The idea of motivic homotopy theory is to apply the machinery of homotopy theory to algebraic varieties, where the affine line plays the role of the interval."

— Vladimir Voevodsky

Foundational Papers

$\mathbb{A}^1$-Homotopy Theory of Schemes
Fabien Morel, Vladimir Voevodsky (1999)
Publications IHES

The foundational paper establishing motivic homotopy theory.

The Milnor Conjecture
Vladimir Voevodsky (2003)
Annals of Mathematics

Complete proof of the Milnor conjecture using motivic cohomology.

Motivic Cohomology with Z/2 Coefficients
Vladimir Voevodsky (2003)
Publications IHES

Detailed development of mod 2 motivic cohomology.

Algebraic Cobordism
Marc Levine, Fabien Morel (2007)
Springer

Comprehensive treatment of the motivic analog of complex cobordism.

Revolutionary Applications

Resolution of Classical Conjectures

  • Milnor conjecture (Voevodsky, 2002)
  • Bloch-Kato conjecture (Voevodsky, Rost, et al., 2011)
  • Friedlander-Milnor conjecture

Motivic Galois Groups

Deep connections to Grothendieck's theory of motives and the conjectural motivic Galois group.

Univalent Foundations

Voevodsky's later work on homotopy type theory, providing new foundations for mathematics.

Leading Research Centers

  • Institute for Advanced Study - Voevodsky's home institution
  • University of Munich - Marc Levine's group
  • MIT - Motivic homotopy and K-theory
  • University of Oslo - Paul Arne Østvær's group
  • IHES Paris - French school of motives