On the algebraic cobordism spectra 𝐌𝐒𝐋 and 𝐌𝐒𝐩

Panin, I.; Walter, C.

doi:10.1090/spmj/1748

On the algebraic cobordism spectra $\mathbf {MSL}$ and $\mathbf {MSp}$
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by I. Panin and C. Walter

St. Petersburg Math. J. 34 (2023), 109-141

DOI: https://doi.org/10.1090/spmj/1748

Published electronically: December 16, 2022

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Abstract:

The algebraic cobordism spectra $\mathbf {MSL}$ and $\mathbf {MSp}$ are constructed. They are commutative monoids in the category of symmetric $T^{\wedge 2}$-spectra. The spectrum $\mathbf {MSp}$ comes with a natural symplectic orientation given either by a tautological Thom class $th^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(\mathbf {MSp}_2)$, or a tautological Borel class $b_{1}^{\mathbf {MSp}} \in \mathbf {MSp}^{4,2}(HP^{\infty })$, or any of six other equivalent structures. For a commutative monoid $E$ in the category ${SH}(S)$, it is proved that the assignment $\varphi \mapsto \varphi (th^{\mathbf {MSp}})$ identifies the set of homomorphisms of monoids $\varphi \colon \mathbf {MSp}\to E$ 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 $\mathbf {MSL}$ and special linear orientations. The universality of $\mathbf {MSp}$ has been used by the authors to prove a Conner–Floyed type theorem. The weak universality of $\mathbf {MSL}$ has been used by A. Ananyevskiy to prove another version of the Conner–Floyed type theorem.

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