The geometric diagonal of the special linear algebraic cobordism

TL;DR

This paper constructs a motivic cobordism spectrum, computes the homotopy groups of special linear algebraic cobordism over certain domains, and explores related characteristic numbers and classes.

Contribution

It introduces a motivic version of the $c_1$-spherical cobordism spectrum and computes its homotopy groups, connecting to hermitian K-theory and Calabi-Yau classes.

Findings

Computed the $P^1$-diagonal of the homotopy groups of $ ext{MSL}$

Described the action of the motivic Hopf element $oldsymbol{ au}$

Established a motivic version of the Anderson-Brown-Peterson theorem

Abstract

The motivic version of the $c_1$-spherical cobordism spectrum is constructed. A connection of this spectrum with other motivic Thom spectra is established. Using this connection, we compute the $\mathbb{P}^1$-diagonal of the homotopy groups of the special linear algebraic cobordism $\pi_{2*,*}(\mathrm{MSL})$ over a local Dedekind domain $k$ with $1/2\in k$ after inverting the exponential characteristic of the residue field of $k$. We discuss the action of the motivic Hopf element $\eta$ on this ring, obtain a description of the localization away from $2$ and compute the $2$-primary torsion subgroup. The complete answer is given in terms of the special unitary cobordism ring. An important component of the computation is the construction of Pontryagin characteristic numbers with values in the hermitian K-theory. We also construct Chern numbers in this setting, prove the motivic version of the Anderson-Brown-Peterson theorem and briefly discuss classes of Calabi-Yau varieties in the $\mathrm{SL}$-cobordism ring.