Ongoing work joint with Paul Arne Østvær.
Motivic calculations sometimes facilitate calculations in topology and sometimes they shed light on relations that initially appear mysterious. The calculations of motivic Hochschild homology (MHH) of low chromatic height have mostly been of the latter type, where a lot of zeros in topology have appeared as torsion. The torsion is due to the tension between the topological circle and G_m, which I may comment on if time permits, but the main message in this talk is that the torsion actually holds valuable information. Indeed, climbing up the chromatic tower to cobordism we find that the torsion classes are placeholders for non-torsion in MHH(MGL). Equivariantly, however, torsion reappears.
In this talk I plan to outline yet another approach to the K-theory K(MU), or rather TC(MU), by working motivically over the complex numbers, which potentially can compete or work in tandem with existing approaches (some of which use the word “motivic” in a different sense).
