On $\eta$-periodic Formal Ternary Laws

We study the algebraic structure underlying Sp-orientations in the $\eta$-periodic motivic stable homotopy category $SH(k)[\eta^{-1}]$.

Borel classes determine a geometric formal ternary law, but the HW-Hurewicz map shows that its universal coefficients generate a proper subring $\Lambda\subsetneq (MSp[\eta^{-1}])_*$, although $\Lambda[1/2]=(MSp[\eta^{-1}])_*[1/2]$.

Thus the failure of classification is purely 2-primary.

To capture part of the missing information, we introduce framed involutions.

The spectrum $MSp[\eta^{-1}]$ carries a canonical framed involution, yielding a Quillen-type idempotent with telescope $MSL[\eta^{-1}]$ and a canonical splitting $(MSp[\eta^{-1}])_* \cong \mathcal{R}_{fr} \otimes_{\mathbb{Z}} (MSL[\eta^{-1}])_*$, where $\mathcal{R}_{fr}$ is the universal ring of framed involutions.

We then axiomatize formal ternary laws, construct the universal Walter ring $\mathcal{W}^{\eta}$, and prove that $\mathcal{W}^{\eta}$ is isomorphic to the Lazard ring L after inverting 2.

If $W(k)\cong \mathbb{Z}$, the universal geometric formal ternary law together with the canonical framed involution induces a classifying map $\phi:\mathcal{W}^{\eta}\to (MSp[\eta^{-1}])_*$ that is injective and becomes an isomorphism after inverting 2.

Integrally, however, additional secondary power series are needed to recover the full orientation data.