Motivic spheres and the image of the Suslin–Hurewicz map

Abstract

We show that an old conjecture of A.A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree n to Milnor K-theory in degree n admits an interpretation in terms of unstable A 1 -homotopy sheaves of the general linear group. Using this identification, we establish Suslin's conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3. We do this by linking the relevant unstable A 1 -homotopy sheaf of the general linear group to the stable A 1 -homotopy of motivic spheres.

Vector bundles on algebraic varieties

“…Furthermore, the map on the right fails to be surjective. The unstable description above is not present in the literature, but it is equivalent to the results stated in [AFW20]. In [RSØ19], the stable motivic 1-stem was computed in the terms above: the above sequence is exact on the left stably.…”

“…Indeed, in joint work with T.B. Williams [AFW20] we showed that S n can described using a "Hurewicz map" analyzed by Andrei Suslin [Sus84]. Suslin's conjecture on the image of this map is equivalent to the following conjecture.…”

“…For n = 3, Suslin observed the conjecture was equivalent to the Milnor conjecture on quadratic forms, which was resolved later independently by Merkurjev-Suslin and Rost. The conjecture was established for n = 5 in "most" cases in [AFW20] (see the latter for a precise statement); this work relies heavily on the computation by Østvaer-Röndigs-Spitzweck of the motivic stable 1-stem [RSØ19].…”

Endomorphisms of the projective plane and the image of the Suslin–Hurewicz map

“…This closes the gap between three and five in the set of degrees for which Suslin's conjecture was previously known. See the introduction of [3] for more details on its history, as well as Suslin's original paper [25].…”

“…Along the way, the homotopy modules π 1 P 2 and π 2 P 2 will be determined, based on computations in [23] and [22]. These computations provide an ingredient to complete the program Aravind Asok, Jean Fasel and Ben Williams developed in [3] to prove Suslin's conjecture on the Suslin-Hurewicz homomorphism from Quillen to Milnor K -theory in degree four.…”