Abstract:
We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism φ:MSp→BO of commutative ring T-spectra which sends the Thom class thMSp to the Thom class thBO. Using φ we construct an isomorphism of bigraded ring cohomology theories on the category SmOp/S, ¯¯¯¯φ:MSp∗,∗(X,U)⊗MSp4∗,2∗(pt)BO4∗,2∗(pt)≅BO∗,∗(X,U). The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSpp,q=MSp[q]2q−p, we have an isomorphism ¯¯¯¯φ:MSp[∗]∗(X,U)⊗MSp[2∗]0(pt)KO[2∗]0(pt)≅KO[∗]∗(X,U), where the KO[n]i(X,U) are Schlichting's hermitian K-theory groups.
The first author was supported by Laboratoire J.-A. Dieudonné, UMR 7351 du CNRS, Universite de Nice – Sophia-Antipolis, by the Research Council of Norway (Frontier research group project no. 250399 “Motivic Hopf equations” at the University of Oslo), and by the Russian Foundation for Basic Research (project no. 19-01-00513).
Citation:
I. A. Panin, C. Walter, “On the Relation of Symplectic Algebraic Cobordism to Hermitian K-Theory”, Algebra, number theory, and algebraic geometry, Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 307, Steklov Mathematical Institute of RAS, Moscow, 2019, 180–192; Proc. Steklov Inst. Math., 307 (2019), 162–173