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-spectra. The spectrum MSp comes with a natural symplectic orientation given either by a tautological Thom class th ∈ MSp4,2(MSp2), a tautological Pontryagin class p MSp 1 ∈ MSp (HP) or any of six other equivalent structures. For a commutative monoid E in the category SH(S) we prove that assignment φ 7→ φ(th) identifies the set of homomorphisms of monoids φ : 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.