A1-homotopy of the general linear group and a conjecture of Suslin

Following work of Röndigs-Spitzweck-Østvær and others on the stable A1-homotopy groups of the sphere spectrum, it has become possible to carry out calculations of the n-th A1-homotopy group of BGLn for small values of n. This group is notable, because it lies just outside the range where the homotopy groups of BGLn recover algebraic K-theory of fields. This group captures some information about rank-n vector bundles on schemes that is lost upon passage to algebraic K-theory. Furthermore, this group relates to a conjecture of Suslin from 1984 about the image of a map from algebraic K-theory to Milnor K-theory in degree n. This conjecture says that the image of the map consists of multiples of (n-1)!. The conjecture was previously known for the cases n=1, n=2 (Matsumoto's theorem) and n=3, where it follows from Milnor's conjecture on quadratic forms. I will establish the conjecture in the case n=4 (up to a problem with 2-torsion) and n=5 (in full). This is joint work with Aravind Asok and Jean Fasel.