Matthias Wendt | A1-connected components of geometric classifying spaces

Seminar on A1-topology, motives and K-theory, November 11, 2021
Matthias Wendt (Universität Wuppertal)
A1-connected components of geometric classifying spaces

In classical algebraic topology, the classifying space of a Lie group G classifies G-principal bundles: isomorphism classes of bundles are in natural bijection with homotopy classes of maps into BG. Algebraic-geometric analogues of this representability theorem are known for rationally trivial torsors under isotropic groups. However, at this point it is unclear what the "geometric" classifying spaces as defined by Morel and Voevodsky actually classify. In the talk I want to explain a first step towards answering that question: the presheaves of isomorphism classes of etale torsors and of A1-connected components become isomorphic after Nisnevich sheafification (under some rather weak restrictions on the group). This implies that the sheaves of A1-connected components are homotopy invariant as conjectured by Morel. For groups satisfying local purity for torsors we can also concretely identify the sheaves of A1-connected components of the classifying space in terms of unramified torsors.
This is joint work with Elden Elmanto and Girish Kulkarni.