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Motivic cell structures. (English) Zbl 1086.55013
A fundamental problem in motivic homotopy theory is understanding how spaces can be “built” from each other. In particular, how much can be built by spheres only? The cellular spaces in the paper under review are similar to schemes with algebraic cell decompositions or linear varieties, but are better suited for homotopical arguments. More precisely, a motivic space is stably cellular if its associated motivic suspension spectrum is cellular, i.e. up to stable equivalence can be built from motivic spheres by homotopy colimit constructions.
Cellular spectra are very well behaved; for instance, a map of cellular spectra is a stable equivalence if it induces an isomorphism of homotopy groups.
The authors establish that the general linear, Grassmannian and Stiefel varieties are stably cellular, and likewise the algebraic K-theory and cobordism spectra are cellular. They remark that claims by Hopkins and Morel will then imply that the motivic cohomology spectrum itself is cellular.
The paper also contains further examples (showing e.g. that all toric varieties and some quadrics are stably cellular) and Künneth formulas for cellular spectra, as well as a final chapter on the compact generators of the stable motivic homotopy category.

MSC:
55U35 Abstract and axiomatic homotopy theory in algebraic topology
14F42 Motivic cohomology; motivic homotopy theory
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