万有二值形式群的对数
Logarithm of the Universal Two-Valued Formal Group
AI总结：
本文确定了万有二值形式群对数的系数分母，证明其属于 Stong 环并给出显式公式，引入奇数亏格 Bc_N 并研究其整性，构造了具有不同 Bc_3 值的不可分解类。
AI中文摘要：
我们解决了长期存在的问题，即确定由复配边的万有形式群通过模长平方构造得到的对数 $B(x)=x+\sum_{n\geq1}b_nx^{n+1}$ 的系数的精确分母：$$ b_n=\frac{C_n}{d_n},\qquad d_n=\frac{n+1}{2}\operatorname{lcm}(1,\ldots,2n+2), $$ 其中 $C_n\in\Omega_{\mathrm U}^{-4n}$ 是本原且不可分解的。我们证明了 $C_n$ 属于万有二值律的系数环 $\Lambda$，并且属于由商 Stong 流形生成的子环 $\Lambda_{\mathrm{St}}$，并给出了它的一个显式积分 Stong 流形公式。$b_n$ 到四元数配边模挠的有理提升具有精确分母 $2d_n$。因此，$C_n$ 没有积分四元数提升，而 $2C_n$ 有。$C_n$ 的每个 Chern 数都被 $d_n$ 整除，且 $c_{2n}(C_n)=d_n$。我们引入了奇数亏格 $\mathrm{Bc}_N$。对于 $N\geq n$，$\mathrm{Bc}_N$ 检测 $d_n$ 的奇数部分。它对 Stong 环的限制在 $N\leq3$ 时恰好是整值的，而万有奇数亏格 $\mathrm{Bc}_{\infty}$ 在该环上取值时仅出现奇数分母。对于每个 $n\geq5$，我们在 $\Lambda^{-4n}$ 中构造了具有相同 Ochanine 亏格和顶 Chern 数但 $\mathrm{Bc}_3$ 值不同的不可分解类。最后，$\mathrm{Bc}_3$ 的有向扩展在实维数低于 $24$ 的闭自旋流形上以及在实维数至多 $24$ 的闭弦流形上是整值的。自旋界是精确的：一个 Anderson-Brown-Peterson 自旋 $24$ 流形具有非整值的 $\mathrm{Bc}_3$ 亏格。
英文摘要：
We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm $B(x)=x+\sum_{n\geq1}b_nx^{n+1}$, obtained from the universal formal group of complex cobordism by the modulus square construction: $$ b_n=\frac{C_n}{d_n},\qquad d_n=\frac{n+1}{2}\operatorname{lcm}(1,\ldots,2n+2), $$ where $C_n\inΩ_{\mathrm U}^{-4n}$ 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 $Λ_{\mathrm{St}}$ 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 $\mathrm{Bc}_N$. For $N\geq n$, $\mathrm{Bc}_N$ detects the odd part of $d_n$. Its restriction to the Stong ring is integral exactly for $N\leq3$, while the universal odd genus $\mathrm{Bc}_{\infty}$ takes values there with only odd denominators. For every $n\geq5$, we construct undecomposable classes in $Λ^{-4n}$ with equal Ochanine genera and top Chern numbers but distinct $\mathrm{Bc}_3$ values. Finally, the oriented extension of $\mathrm{Bc}_3$ 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 $\mathrm{Bc}_3$ genus.
