On the algebraic cobordism spectra MSL and MSp

TL;DR

This paper constructs algebraic cobordism spectra MSL and MSp within the motivic stable homotopy category, establishing their properties and universal features related to symplectic and special linear orientations.

Contribution

It introduces the spectra MSL and MSp as commutative monoids in symmetric T^{2}-spectra and proves their universal properties for orientations in motivic homotopy theory.

Findings

MSp has a natural symplectic orientation via tautological Thom and Borel classes.

Homomorphisms from MSp to other spectra correspond to tautological Thom elements.

A weaker universality result applies to MSL and special linear orientations.

Abstract

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.