On the algebraic cobordism spectra 𝐌𝐒𝐋 and 𝐌𝐒𝐩

Abstract

The algebraic cobordism spectra M S L \mathbf {MSL} and M S p \mathbf {MSp} are constructed. They are commutative monoids in the category of symmetric T ∧ 2 T^{\wedge 2} -spectra. The spectrum M S p \mathbf {MSp} comes with a natural symplectic orientation given either by a tautological Thom class t h M S p ∈ M S p 4 , 2 ( M S p 2 ) th^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(\mathbf {MSp}_2) , or a tautological Borel class b 1 M S p ∈ M S p 4 , 2 ( H P ∞ ) b_{1}^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(HP^{\infty }) , or any of six other equivalent structures. For a commutative monoid E E in the category S H ( S ) {SH}(S) , it is proved that the assignment φ ↦ φ ( t h M S p ) \varphi \mapsto \varphi (th^{\mathbf {MSp}}) identifies the set of homomorphisms of monoids φ : M S p → E \varphi \colon \mathbf {MSp}\to E in the motivic stable homotopy category S H ( S ) SH(S) with the set of tautological Thom elements of symplectic orientations of E E . A weaker universality result is obtained for M S L \mathbf {MSL} and special linear orientations. The universality of