Motivic homotopy theory and stable homotopy groups

TL;DR

Motivic homotopy theory extends topological methods to algebraic geometry by using the $ ext{A}^1$-interval and Nisnevich descent to build unstable and stable homotopy theories for smooth schemes over a field $k$. The unstable theory develops motivic spaces, sheaves, and homotopy sheaves, with Milnor–Witt K-theory and Gersten resolutions providing a quadratic refinement of invariants. Stabilization via $ ext{P}^1$ yields the motivic stable homotopy category $ ext{SH}(k)$, where Morel’s plus–minus decomposition and the slice filtration organize stable stems and connect to motivic cohomology and K-theory (via $ ext{KGL}$, $ ext{GW}$, and motivic obstruction theory). The work of Isaksen–Wang–Xu and collaborators then leverages the motivic Adams spectral sequence and the deformation by the class $ au$ to compute stable stems, linking motivic computations to classical stable homotopy and even paving the way toward synthetic homotopy theory. Overall, the notes synthesize foundational principles, unstable and stable frameworks, and modern computational strategies that unify algebraic geometry with stable homotopy theory and illuminate deep connections between motivic and classical invariants.

Abstract

These notes, written version of a Bourbaki talk, survey Morel-Voevodsky's motivic homotopy theory over a field, with a focus on computations of motivic homotopy sheaves, both stable and unstable. We also describe Isaksen-Wang-Xu's applications to the determination of stable stems through the motivic Adams spectral sequence, which also paved the way toward synthetic homotopy theory.

Theorems & Definitions (1)

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