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Unstable motivic homotopy theory. (English) Zbl 1473.14041

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 931-972 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: Morel-Voevodsky’s \(\mathbb A^1\)-homotopy theory transports tools from algebraic topology into arithmetic and algebraic geometry, allowing us to draw arithmetic conclusions from topological arguments. Comparison results between classical and \(\mathbb A^1\)-homotopy theories can also be used in the reverse direction, allowing us to infer topological results from algebraic calculations. This chapter discusses realization functors to topological spaces, which allow us to see how \(\mathbb A^1\)-homotopy theory combines phenomena associated to the real and complex points of a variety. The standard choice of Grothendieck topology for \(\mathbb A^1\)-homotopy theory is the Nisnevich topology, although the étale topology is also used, producing a different homotopy theory of spaces. The étale realization functor can transport theorems in \(\mathbb A^1\)-homotopy theory to the older and very successful theory of étale homotopy or cohomology.
For the entire collection see [Zbl 1468.55001].

MSC:

14F42 Motivic cohomology; motivic homotopy theory
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry

Citations:

Zbl 1468.55001
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