Computing framed motives

Abstract

We develop methods for computing framed motives associated with motivic Thom spectra. Our main tool is a motivic Atiyah--Hirzebruch spectral sequence relating framed motives to framed motivic cohomology. As a consequence, after inverting a finite set of primes, the bigraded homotopy sheaves of motivic Thom spectra are computed in terms of framed motivic cohomology. We further analyze the symmetric-group actions inherent in framed correspondences and introduce a theory of torsion framed motivic cohomology that yields new computational descriptions of framed motivic cohomology groups. These constructions lead to a category of permutation-free framed correspondences from which we reconstruct rational stable motivic homotopy theory.