On the algebraic cobordism spectra 𝐌𝐒𝐋 and 𝐌𝐒𝐩

Abstract

The algebraic cobordism spectraMSL\mathbf {MSL}andMSp\mathbf {MSp}are constructed. They are commutative monoids in the category of symmetricT∧2T^{\wedge 2}-spectra. The spectrumMSp\mathbf {MSp}comes with a natural symplectic orientation given either by a tautological Thom classthMSp∈MSp4,2(MSp2)th^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(\mathbf {MSp}_2), or a tautological Borel classb1MSp∈MSp4,2(HP∞)b_{1}^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(HP^{\infty }), or any of six other equivalent structures. For a commutative monoidEEin the categorySH(S){SH}(S), it is proved that the assignmentφ↦φ(thMSp)\varphi \mapsto \varphi (th^{\mathbf {MSp}})identifies the set of homomorphisms of monoidsφ:MSp→E\varphi \colon \mathbf {MSp}\to Ein the motivic stable homotopy categorySH(S)SH(S)with the set of tautological Thom elements of symplectic orientations ofEE. A weaker universality result is obtained forMSL\mathbf {MSL}and special linear orientations. The universality ofMSp\mathbf {MSp}has been used by the authors to prove a Conner–Floyed type theorem. The weak universality ofMSL\mathbf {MSL}has been used by A. Ananyevskiy to prove another version of the Conner–Floyed type theorem.