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 W(k)-subalgebra Λsubsetneq (MSp[η⁻¹])_*, although Λ[1/2]=(MSp[η⁻¹])_*[1/2]. Thus the failure of generation 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])_* ≅ ℛ_fr ⊗_ℤ (MSL[η⁻¹])_*, where ℛ_fr is the universal ring of framed involutions. We then axiomatize formal ternary laws, construct the universal Walter ring 𝒲^η, and prove that 𝒲^η is isomorphic to the Lazard ring L after inverting 2. If W(k)≅ ℤ, the universal geometric formal ternary law together with the canonical framed involution induces a classifying map ϕ:𝒲^η→ (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 cobordism coefficient ring.

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BibTeX RIS

BibTeX

@misc{huang2026ηperiodic,
  title = {{On $η$-periodic Formal Ternary Laws}},
  author = {Tao Huang},
  howpublished = {HAL},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2607.06795},
  eprint = {2607.06795},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05687153v2},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On η-periodic Formal Ternary Laws
AU  - Tao Huang
PY  - 2026
DA  - 2026/07/07
PB  - HAL
DO  - 10.48550/arxiv.2607.06795
UR  - https://hal.science/hal-05687153v2
AB  - 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 W(k)-subalgebra Λsubsetneq (MSp[η^−1])_*, although Λ[1/2]=(MSp[η^−1])_*[1/2]. Thus the failure of generation 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])_* ≅ ℛ_fr ⊗_ℤ (MSL[η^−1])_*, where ℛ_fr is the universal ring of framed involutions. We then axiomatize formal ternary laws, construct the universal Walter ring 𝒲^η, and prove that 𝒲^η is isomorphic to the Lazard ring L after inverting 2. If W(k)≅ ℤ, the universal geometric formal ternary law together with the canonical framed involution induces a classifying map ϕ:𝒲^η→ (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 cobordism coefficient ring.
ER  -

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