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Generators of the algebraic symplectic bordism ring

Pietro Gigli

TL;DR

The paper advances the understanding of algebraic symplectic cobordism by analyzing the η-completed plus-part of the motivic spectrum ${\mathrm{MSp}}$ and relating it to ${\mathrm{MGL}}$ via an η-completed Adams spectral sequence. It develops a twisted Pontryagin–Thom framework using symplectic twists, proves cellularity and a motive decomposition for ${\mathrm{MSp}}$, and computes the mod-$\ell$ Adams spectral sequence to show the plus-part injects into the GL-oriented theory and is polynomial in even-degree generators. A key innovation is the construction of symplectic bordism classes from stable symplectic twists of real varieties and a Newton-class-based generator-detection criterion, enabling explicit generators ${[Y_{2d}]}_{\mathrm{MSp}}$ whose images control the whole ring after localization. Collectively, these results parallel the SL-case while addressing the unique symplectic geometry, quaternionic-Grassmannian structure, and motivic complications, providing a concrete path toward a full description of ${\mathrm{MSp}}^*$. The methods offer a robust toolkit for analyzing additive and multiplicative structures in algebraic cobordism with symplectic flavor and connect to the universality of ${\mathrm{MGL}}$ and its Lazard-type formal group laws.

Abstract

In this paper, we study the $η$-completed part of the motivic spectrum $\text{MSp}$ constructed by Panin and Walter, representing the universal $\text{Sp}$-oriented cohomology theory. In particular, we investigate the inclusion $(\text{MSp}^\wedge_η)^*\hookrightarrow \text{MGL}^*$ of the cofficient rings, by studying the motivic Adams spectral sequence associated to $\text{MSp}$, mimiking a strategy used by Levine,Yang, Zhao for $\text{MSL}^*$. In order to give a description of $(\text{MSp}^\wedge_η)^*$, we refine the Pontryagin-Thom construction in a way that allows one to obtain symplectic bordism classes from a large family of varieties that carry a certain "symplectic twist", and we prove a criterion to select generators among these classes.

Generators of the algebraic symplectic bordism ring

TL;DR

The paper advances the understanding of algebraic symplectic cobordism by analyzing the η-completed plus-part of the motivic spectrum and relating it to via an η-completed Adams spectral sequence. It develops a twisted Pontryagin–Thom framework using symplectic twists, proves cellularity and a motive decomposition for , and computes the mod- Adams spectral sequence to show the plus-part injects into the GL-oriented theory and is polynomial in even-degree generators. A key innovation is the construction of symplectic bordism classes from stable symplectic twists of real varieties and a Newton-class-based generator-detection criterion, enabling explicit generators whose images control the whole ring after localization. Collectively, these results parallel the SL-case while addressing the unique symplectic geometry, quaternionic-Grassmannian structure, and motivic complications, providing a concrete path toward a full description of . The methods offer a robust toolkit for analyzing additive and multiplicative structures in algebraic cobordism with symplectic flavor and connect to the universality of and its Lazard-type formal group laws.

Abstract

In this paper, we study the -completed part of the motivic spectrum constructed by Panin and Walter, representing the universal -oriented cohomology theory. In particular, we investigate the inclusion of the cofficient rings, by studying the motivic Adams spectral sequence associated to , mimiking a strategy used by Levine,Yang, Zhao for . In order to give a description of , we refine the Pontryagin-Thom construction in a way that allows one to obtain symplectic bordism classes from a large family of varieties that carry a certain "symplectic twist", and we prove a criterion to select generators among these classes.

Paper Structure

This paper contains 19 sections, 66 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on ${\mathbb A}^1$-homotopy theory
  4. The six functors
  5. ${\operatorname{GL}}$-oriented theories
  6. Algebraic cobordism
  7. ${\operatorname{Sp}}$-oriented theories
  8. Symplectic cobordism
  9. A twisted degree map
  10. Cellular structure of ${\operatorname{MSp}}$
  11. Vector bundles over quaternionic Grassmannians
  12. The motive of ${\operatorname{MSp}}$
  13. The Adams spectral sequence for ${\operatorname{MSp}}$
  14. The motivic Steenrod algebra
  15. The $\text{Ext}_{A^{**}}$-algebra of $H^{*,*}({\operatorname{MSp}})$
  16. ...and 4 more sections

Key Result

Theorem 1

Let $p$ be the exponential characteristic of $k$, namely $p={\operatorname{char}} k$ if ${\operatorname{char}} k>0$ and $p=1$ otherwise. Then we have an isomorphism of graded ring with $x_i$ in degree $-i$, and ${\operatorname{MGL}}^*[1/p]$ is generated by classes $[X]_{\operatorname{MGL}}$, $X$ a smooth and proper $k$-scheme.

Theorems & Definitions (164)

  • Theorem 1
  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3: voe:homotopy_theory, Proposition 5.4, Theorem 5.6
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8: MazWei:lectures, Theorem 19.1, Theorem 19.3, Corollary 4.2
  • ...and 154 more