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In This Chapter
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Motivic stable homotopy groups
- Introduction
- The motivic Adams spectral sequence
- Milnor K-theory and the cohomology of a point
- The motivic Steenrod algebra
- The cohomology of the Steenrod algebra
- The ρ-Bockstein spectral sequence
- The motivic Adams spectral sequence
- The motivic effective slice spectral sequence
- C -motivic stable homotopy groups
- C -motivic and classical stable homotopy groups
- η-periodic-phenomena
- Adams differentials
- Motivic nilpotence and periodicity
- R -motivic stable homotopy groups
- Motivic stable homotopy groups over general fields
- Motivic graded commutativity
- Milnor-Witt K-theory
- The first Milnor-Witt stem
- Finite fields
- η-periodic phenomena
- Other motivic spectra
- Motivic Eilenberg-Mac Lane spectra
- Algebraic K-theory
- Motivic Brown-Peterson spectra and truncations
- Hermitian K-theory
- Higher Witt theory
- Future directions
- Bibliography
Abstract
Motivic homotopy theory was developed by Morel and Voevodsky [64] [69] in the 1990s. The original motivation for the theory was to import homotopical techniques into algebraic geometry. For example, it allowed for the powerful theory of Steenrod operations in algebro-geometric cohomology theories. Motivic homotopy theory was essential to the solution of several long-standing problems in algebraic K-theory, such as the the Milnor conjecture [101] and the Bloch-Kato conjecture [103].
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