
Mathematics 2010
On the motivic commutative ring spectrum BOAbstract: We construct an algebraic commutative ring T spectrum BO which is stably fibrant and (8,4) periodic and such that on SmOp/S the cohomology theory (X,U) > BO^{p,q}(X_{+}/U_{+}) and Schlichting's hermitian Ktheory functor (X,U) > KO^{[q]}_{2qp}(X,U) are canonically isomorphic. We use the motivic weak equivalence Z x HGr > KSp relating the infinite quaternionic Grassmannian to symplectic $K$theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is Spec Z[1/2], this monoid structure and the induced ring structure on the cohomology theory BO^{*,*} are the unique structures compatible with the products KO^{[2m]}_{0}(X) x KO^{[2n]}_{0}(Y) > KO^{[2m+2n]}_{0}(X x Y). on GrothendieckWitt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO^{*,*}(T^{2}) in the same way as multiplication by the GrothendieckWitt class of the symmetric bilinear space <1>.
