On $η$-periodic Formal Ternary Laws

Abstract

We study the algebraic structure underlying Sp-orientations in the $η$-periodic motivic stable homotopy category $SH(k)[η^{-1}]$. Borel classes determine a geometric formal ternary law, but the HW-Hurewicz map shows that its universal coefficients generate a proper subring $Λ\subsetneq (MSp[η^{-1}])_*$, although $Λ[1/2]=(MSp[η^{-1}])_*[1/2]$. Thus the failure of classification is purely 2-primary. To capture part of the missing information, we introduce framed involutions. The spectrum $MSp[η^{-1}]$ carries a canonical framed involution, yielding a Quillen-type idempotent with telescope $MSL[η^{-1}]$ and a canonical splitting $(MSp[η^{-1}])_* \cong \mathcal{R}_{fr} \otimes_{\mathbb{Z}} (MSL[η^{-1}])_*$, where $\mathcal{R}_{fr}$ is the universal ring of framed involutions. We then axiomatize formal ternary laws, construct the universal Walter ring $\mathcal{W}^η$, and prove that $\mathcal{W}^η$ 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 $ϕ:\mathcal{W}^η\to (MSp[η^{-1}])_*$ that is injective and becomes an isomorphism after inverting 2. Integrally, however, additional secondary power series are needed to recover the full orientation data.