Logarithm of the Universal Two-Valued Formal Group
Abstract
We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm B(x)=x+∑n≥1bnxn+1, obtained from the universal formal group of complex cobordism by the modulus square construction: bn=Cn/dn, dn=(n+1)/2lcm(1,…,2n+2), where Cn∈ΩU−4n is primitive and undecomposable. We prove that Cn belongs to the coefficient ring Λ of the universal two-valued law and to the subring ΛSt generated by quotient Stong manifolds, and give an explicit integral Stong manifold formula for it. The rational lift of bn to quaternionic cobordism modulo torsion has exact denominator 2dn. Consequently, Cn has no integral quaternionic lift, whereas 2Cn does. Every Chern number of Cn is divisible by dn, and c2n(Cn)=dn. We introduce odd genera BcN. For N≥ n, BcN detects the odd part of dn. Its restriction to the Stong ring is integral exactly for N≤3, while the universal odd genus Bc∞ takes values there with only odd denominators. For every n≥5, we construct undecomposable classes in Λ−4n with equal Ochanine genera and top Chern numbers but distinct Bc3 values. Finally, the oriented extension of Bc3 is integral on closed spin manifolds of real dimension below $24$ and on closed string manifolds of real dimension at most $24$. The spin bound is sharp: an Anderson-Brown-Peterson spin $24$-manifold has nonintegral Bc3 genus.
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BibTeX
@misc{buchstaber2026logarithm,
title = {{Logarithm of the Universal Two-Valued Formal Group}},
author = {Victor Buchstaber and Mikhail Kornev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15878},
eprint = {2609.15878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15878},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logarithm of the Universal Two-Valued Formal Group
AU - Victor Buchstaber
AU - Mikhail Kornev
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15878
UR - https://arxiv.org/abs/2609.15878
AB - We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm B(x)=x+∑_n≥1b_nx^n+1, obtained from the universal formal group of complex cobordism by the modulus square construction: b_n=C_n/d_n, d_n=(n+1)/2lcm(1,…,2n+2), where C_n∈Ω_U^−4n is primitive and undecomposable. We prove that C_n belongs to the coefficient ring Λ of the universal two-valued law and to the subring Λ_St generated by quotient Stong manifolds, and give an explicit integral Stong manifold formula for it. The rational lift of b_n to quaternionic cobordism modulo torsion has exact denominator 2d_n. Consequently, C_n has no integral quaternionic lift, whereas 2C_n does. Every Chern number of C_n is divisible by d_n, and c_2n(C_n)=d_n. We introduce odd genera Bc_N. For N≥ n, Bc_N detects the odd part of d_n. Its restriction to the Stong ring is integral exactly for N≤3, while the universal odd genus Bc_∞ takes values there with only odd denominators. For every n≥5, we construct undecomposable classes in Λ^−4n with equal Ochanine genera and top Chern numbers but distinct Bc_3 values. Finally, the oriented extension of Bc_3 is integral on closed spin manifolds of real dimension below $24$ and on closed string manifolds of real dimension at most $24$. The spin bound is sharp: an Anderson-Brown-Peterson spin $24$-manifold has nonintegral Bc_3 genus.
ER -