The geometric diagonal of the special linear algebraic cobordism

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 ${\pi }_{2\ast ,\ast }(\mathrm{MSL})$ 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.

arXiv comment41 pages, final version, fixed a minor sign mistake

References in corpus (16)

• From algebraic cobordism to motivic cohomology
• Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem
• Slices of hermitian K-theory and Milnor's conjecture on quadratic forms
• Affine representability results in A^1-homotopy theory II: principal bundles and homogeneous spaces
• Modules over algebraic cobordism
• The Generalized Slices of Hermitian K-Theory
• Motivic Gauß-Bonnet formulas
• Motivic slices and colored operads
• The special linear version of the projective bundle theorem
• Witt sheaves and the $\eta$-inverted sphere spectrum
• SU-bordism: structure results and geometric representatives
• On the rational motivic homotopy category
• Chow-Witt rings of classifying spaces for symplectic and special linear groups
• Towards the dual motivic Steenrod algebra in positive characteristic
• The unit map of the algebraic special linear cobordism spectrum
• Thom isomorphisms in triangulated motivic categories