Motivic cell structures
This is joint with D. Isaksen.
Abstract: We explore varieties which can be written as homotopy colimits of spheres S^{p,q} in the Morel-Voevodsky homotopy theory of schemes. Basic techniques and examples are given, in particular showing that toric varieties, Stiefel manifolds, and Grassmannians are cellular (the latter is harder than one might think.) We prove some Kunneth-type theorems for the homology/cohomology of cellular objects. A few basic properties of the motivic stable homotopy category are developed along the way.