A pro-algebraic fundamental group for topological spaces and an arithmetic application.

Time: 9:30 -- 11:00, Feb 28, 2024.

Venue: Room 612, A6, Institute of Mathematics, VAST

Abstract: Using the Tannakian formalism we define and study a pro-algebraic fundamental group for connected topological spaces. Using ideas of Nori and a result of Deligne on fibre functors of Tannakian categories we also define a pseudo-torsor under this fundamental group which can serve as a replacement for the universal covering space in this generality. The group of connected components of the pro-algebraic fundamental group is the pro-étale fundamental group used by Kucharczyk and Scholze to exhibit the absolute Galois group of a field K containing all roots of unity as the fundamental group of an ordinary topological space Y_K. We calculate the pro-algebraic fundamental group of Y_K and show that Y_K also carries some information about the motivic Galois group of K. We also mention a categorical criterion derived from the work of Coulembier for the Tannakian dual of a neutral Tannaka category over a perfect field to be reduced. Part of this work is joint with Michael Wibmer.