On $\eta$-periodic Formal Ternary Laws

Technical Summary of "ON η-PERIODIC FORMAL TERNARY LAWS" by Tao Huang

Problem Statement
The paper addresses the algebraic classification of symplectic ($Sp$) orientations in the $\eta$-periodic motivic stable homotopy category $SH(k)[\eta^{-1}]$. While Quillen's theorem establishes that complex cobordism ($MU$) is classified by the Lazard ring via formal group laws (FGLs), the symplectic analogue remains elusive. In the symplectic setting, the tensor product of two symplectic bundles is orthogonal, but the tensor product of three is symplectic. Consequently, the natural geometric operation is ternary rather than binary, leading to the concept of a formal ternary law (FTL).

The central problem is whether the geometric formal ternary law (GFTL) derived from the Borel classes of symplectic bundles fully classifies $Sp$-orientations integrally. Previous work by Panin and Walter established the geometric framework, but it was unclear if the coefficients of the universal GFTL generate the entire coefficient ring of the symplectic cobordism spectrum $MSp[\eta^{-1}]$.

Methodology
The author employs a combination of motivic homotopy theory, algebraic geometry, and formal power series techniques:

1. Motivic Framework: The study is conducted over a Dedekind domain $k$ within the $\eta$-periodic category, where the motivic Hopf map $\eta$ is inverted. This setting relies on structural results by Bachmann and Hopkins regarding the coefficient rings of $MSp[\eta^{-1}]$ and $MSL[\eta^{-1}]$ as polynomial algebras over the Witt ring $W(k)$.
2. Geometric Analysis: The paper analyzes the GFTL associated with the triple tensor product of tautological rank-2 symplectic bundles on $(HP^\infty)^{\times 3}$. The Hurewicz map $h: MSp[\eta^{-1}] \to HW \wedge MSp$ is used to compute the image of the GFTL coefficients in unramified Witt cohomology.
3. Framed Involutions: To capture missing information, the author introduces framed involutions. This structure arises from the dualization of the universal symplectic bundle over $HP^\infty$. A framed involution is defined as a pair $(\iota, \Psi)$ satisfying specific axioms, where $\Psi$ trivializes the involution $\iota$.
4. Universal Algebra Construction: The paper constructs the $\eta$-periodic Walter ring ($W_\eta$), the universal ring representing formal ternary laws equipped with a framed involution. It also defines the universal ring of framed involutions ($R_{fr}$).
5. Comparison with FGLs: By inverting 2, the author constructs natural transformations between formal group laws (FGLs) and formal ternary laws (FTLs), demonstrating that the ternary theory collapses to the classical binary theory in this regime.

Key Contributions and Results

• Failure of Integral Classification by GFTL: The paper proves that the subring $\Lambda \subset (MSp[\eta^{-1}])_*$ generated by the coefficients of the universal geometric formal ternary law is a proper subring of the full coefficient ring ($\Lambda \subsetneq (MSp[\eta^{-1}])_*$). However, after inverting 2, they become isomorphic ($\Lambda[1/2] = (MSp[\eta^{-1}])_*[1/2]$). This indicates that the failure of the GFTL to classify $Sp$-orientations integrally is purely 2-primary.
• Framed Involutions and Splitting: The author shows that $MSp[\eta^{-1}]$ carries a canonical framed involution. This induces a Quillen-type idempotent endomorphism $\xi: MSp[\eta^{-1}] \to MSp[\eta^{-1}]$ whose telescope is homotopy equivalent to $MSL[\eta^{-1}]$. Consequently, the coefficient ring splits canonically:
  $$(MSp[\eta^{-1}])_* \cong R_{fr} \otimes_{\mathbb{Z}} (MSL[\eta^{-1}])_*$$
  This splitting isolates the missing 2-primary data into the ring of framed involutions.
• The $\eta$-Periodic Walter Ring ($W_\eta$): The paper axiomatizes FTLs in the $\eta$-periodic setting and constructs the universal ring $W_\eta$. It is shown that $W_\eta \cong R_{fr} \otimes_{\mathbb{Z}} \widehat{W}_\eta$, where $\widehat{W}_\eta$ is the normalized Walter ring (representing FTLs with trivial involution).
• Lazard-type Theorem: After inverting 2, the natural transformations between FGLs and FTLs establish an isomorphism:
  $$W_\eta[1/2] \cong L[1/2]$$
  where $L$ is the classical Lazard ring. This confirms that the ternary theory is equivalent to the classical binary theory when 2 is invertible.
• Quillen-type Theorem: Assuming $W(k) \cong \mathbb{Z}$ (e.g., $k=\mathbb{Z}$ or $\mathbb{R}$), the universal GFTL and the canonical framed involution induce a classifying map:
  $$\phi: W_\eta \to (MSp[\eta^{-1}])_*$$
  This map is injective and becomes an isomorphism after inverting 2. However, integrally, $\phi$ is not surjective. The remaining missing 2-primary information lies on the $MSL[\eta^{-1}]$ side and requires additional secondary power series not captured by the normalized FTL.

Significance
The paper provides a rigorous algebraic framework for understanding $Sp$-orientations in the $\eta$-periodic motivic context. It clarifies that while the geometric formal ternary law captures the "classical" part of the theory (recovering the Lazard ring after inverting 2), it is insufficient for an integral classification due to 2-primary phenomena.

The introduction of framed involutions is the primary methodological innovation, offering a mechanism to decompose the symplectic cobordism ring into a part governed by the Walter ring and a part governed by the special linear cobordism ring. The paper concludes that a complete integral classification of $Sp$-orientations requires data beyond the standard formal ternary law, specifically secondary power series encoding the 2-torsion structure, which are left for future investigation. The work serves as a motivic analogue to Quillen's theorem, identifying the precise algebraic obstructions to a full symplectic analogue of the Lazard ring.