On the relation of symplectic algebraic cobordism to hermitian K-theory

Ivan Panin (1) , Charles Walter (2)

1 PDMI RAS - St. Petersburg Department of V.A. Steklov Mathematical Institute
2 JAD - Laboratoire Jean Alexandre Dieudonné

We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S) there is a unique morphism g : MSp -> BO of commutative ring T- spectra which sends the Thom class th^{MSp} to the Thom class th^{BO}. We show that the induced morphism of bigraded cohomology theories MSp^{*,*} -> BO^{*,*} is isomorphic to the morphism of bigraded cohomology theories obtained by applying to MSp^{*,*} the "change of (simply graded) coefficients rings" MSp^{4*,2*} -> BO^{4*,2*}. This is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory via symplectic cobordism.

Mots clés: K-théorie hermitienne; cobordisme symplectique algébrique; classes de Pontryagin; classes de Thom; changement de coefficients

Domaines: Géométrie algébrique [math.AG]; K-théorie et homologie [math.KT]

Ivan Panin, Charles Walter. On the relation of symplectic algebraic cobordism to hermitian K-theory. 2010. ⟨hal-00531734⟩