On the algebraic cobordism spectra MSL and MSp

We construct algebraic cobordism spectra MSL and MSp. They are commutative monoids in the category of symmetric T^{2}- spectra. The spectrum MSp comes with a natural symplectic orientation given either by a tautological Thom class th^{MSp} in MSp^{4,2}(MSp_{2}), a tautological Borel class b_{1}^{MSp} in MSp^{4,2}(HP^{\infty}) or any of six other equivalent structures. For a commutative monoid E in the category SH(S) we prove that assignment g -> g(th^{MSp}) identifies the set of homomorphisms of monoids g : MSp -> E in the motivic stable homotopy category SH(S) with the set of tautological Thom elements of symplectic orientations of E. A weaker universality result is obtained for MSL and special linear orientations.

💡 Research Summary

The paper develops two new algebraic cobordism spectra, MSL and MSp, which serve as motivic analogues of the classical Thom spectra for special linear and symplectic bundles. Working in the category of symmetric T²‑spectra, the authors construct MSL and MSp as commutative monoids, ensuring that the multiplicative structure is compatible with the motivic smash product.

For MSp, the authors exhibit a rich symplectic orientation. They define a tautological Thom class th^{MSp} in bidegree (4,2) on the universal symplectic bundle MSp₂ and show that this class is equivalent to a tautological Borel class b₁^{MSp} in MSp^{4,2}(HP^∞). In fact, six different but equivalent pieces of data (Thom class, Borel class, first symplectic Chern class, etc.) are shown to encode the same orientation. This mirrors the classical situation where a symplectic Thom class determines a unique symplectic orientation.

The central universality theorem states that for any commutative monoid E in the motivic stable homotopy category SH(S), the assignment
 g ↦ g(th^{MSp})
establishes a bijection between monoid maps g : MSp → E and symplectic orientations of E. In other words, giving a symplectic orientation of E is equivalent to giving a monoid morphism from the universal symplectic cobordism spectrum MSp to E. The proof combines the universal property of Thom spaces, the multiplicative structure of symmetric T²‑spectra, and a careful analysis of how the six equivalent orientation data behave under monoid maps.

For MSL, a weaker version of this result is proved. While MSL also carries a special‑linear orientation (a Thom class in bidegree (2,1) for the universal SL‑bundle), the correspondence between monoid maps MSL → E and special‑linear orientations of E is not as tight; additional coherence conditions are required. This reflects the fact that the special linear group does not admit a symplectic form, and the associated obstruction theory is more subtle.

The paper also details the construction of the classifying spaces BSL and BSp in the motivic setting, the formation of their Thom spaces, and the verification that the resulting spectra satisfy the expected homotopy‑theoretic properties (e.g., cellularity, suspension isomorphisms). The authors discuss how the Borel and Chern classes arise from the universal bundles and how they interact with the motivic grading.

Finally, the authors outline potential applications: calculations of symplectic cobordism groups, connections with motivic characteristic classes, and the use of MSp and MSL as universal recipients for oriented cohomology theories with symplectic or special‑linear structure. By providing explicit universal objects and a clear description of their orientation data, the work lays a solid foundation for further exploration of oriented motivic cohomology theories and their computational aspects.