John Milnor's attempt at the higher algebraic K-groups of a field
uses the field units as generators and the Steinberg relation as
the only relation. The resulting Milnor K-groups map to the higher
algebraic K-groups Daniel Quillen defined slightly later as homotopy
groups of a certain topological space, and the map is an isomorphism
in degrees 0, 1, and 2. A decade later Andrei Suslin constructed a
Hurewicz-type homomorphism from Quillen's algebraic K-groups to
the Milnor K-groups of a field and proved that the resulting endomorphism
on the n-th Milnor K-group is multiplication by (n-1)! if n>0.
He conjectured that the image of the Hurewicz-type homomorphism
is the same as the image of this endomorphism (hence as small as possible)
and proved the degree 3 case. Recently Aravind Asok, Jean Fasel, and Ben
Williams ingeniously used $\mathbb{A}^1$-homotopy groups of algebraic spheres to settle
the degree 5 case. Also using $\mathbb{A}^1$-homotopy groups, but now for the projective
plane, I will explain how to treat the fourth degree.
Oliver Röndigs
Homotopy groups of spheres, special linear groups, and the Suslin-Hurewicz homomorphism
Monday, 11 September, 2023 - 15:30
Résumé :
Institution de l'orateur :
Universität Osnabrück
Thème de recherche :
Algèbre et géométries
Salle :
4