Fundamental classes in motivic homotopy theory

Galois Equivariance and Stable Motivic Homotopy Theory

For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full a...

Motivic Homotopy Theory of Group Scheme Actions

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic K-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.

Topology Seminar Non-nilpotent elements in motivic homotopy theory

Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite p-local spectra induce nonzero homomorphisms on BP -homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and whilst η : S1,1 → S0,0 induces zero on BP -homology, it is nonnilpotent. Recent work with Haynes Miller has led to a calculation of ηπ∗...

Open Problems in the Motivic Stable Homotopy Theory, I

New families in the homotopy of the motivic sphere spectrum

In [1] Adams constructed a non-nilpotent map v 1 : Σ S/2 −→ S/2. Using iterates of this map one constructs infinite families of elements in the stable homotopy groups of spheres, the v1-periodic elements of order 2. In this paper we work motivically over C and construct a nonnilpotent self map w 1 : Σ S/η −→ S/η. We then construct some infinite families of elements in the homotopy of the motivi...