Abstract:

The talk is based on a joint work with Ahina Nandy and Oliver Röndigs. The algebraic cobordism spectrum MGL was introduced at the dawn of motivic homotopy theory. A formula for the slices of this spectrum was conjectured by Vladimir Voevodsky and proved by Markus Spitzweck assuming the Hopkins-Morel-Hoyois equivalence. The goal of this talk is to present oriented analogues of this computation. More precisely, there are two motivic spectra that might be called oriented algebraic cobordism. One is Panin-Walter MSL based on the special linear groups. The second one is MSL^c and it is build from the structure groups of oriented vector bundles in the sense of Fabien Morel. I will discuss the precise relation between them and compute their slices. If time permits, I will mention a few applications of these results.