Abstract:

The failure of homotopy invariance for many cohomology theories in algebraic geometry (K-theory,
crystalline cohomology) has inspired several attempts at developing a non-A^1-invariant version
of Morel--Voevodsky's motivic homotopy category. Here I will explain a new model, based on joint
work with Marc Hoyois and Ryomei Iwasa, of a non-A^1-invariant enhancement of the stable motivic
homotopy category. We show that many basic results in A^1-homotopy theory can be proved in this
category, using the invertibility of P^1 and blowup excision. We also construct a non-A^1-invariant
refinement of the algebraic cobordism spectrum, and prove that algebraic K-theory can be recovered
from it, in an analogue of the classical Conner--Floyd theorem.