标题： 严格交换复数定向理论 显示英文标题

标题： Strictly commutative complex orientation theory

摘要： 对于乘法上同调理论E，复数定向与从复数 bordism 上同调MU到E的乘法自然变换之间存在双射对应关系。 如果E由具有高度结构化乘法的谱表示，我们给出一种迭代过程，用于将一个定向MU→E提升为尊重这种额外结构的映射，这基于Arone-Lesh的工作。 严格交换定向的空间是参数化部分提升的逆塔的极限；第1阶段对应于普通的复数定向，从第(m-1)阶段提升到第m阶段由在固定基空间F_m上一族E模的定向的存在性所决定。 当E是p局部时，我们可以得出更多结论。 我们发现这个塔仅在m是p的幂时发生变化，如果E是E(n)局部的，则塔在阶段p^n之后保持恒定。 此外，如果系数环E^*是p-挠自由的，从阶段1提升到阶段p的能力等价于由Ando证明的关于相关形式群定律的条件。 显示英文摘要

摘要： For a multiplicative cohomology theory E, complex orientations are in bijective correspondence with multiplicative natural transformations to E from complex bordism cohomology MU. If E is represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientation MU -> E to a map respecting this extra structure, based on work of Arone-Lesh. The space of strictly commutative orientations is the limit of an inverse tower of spaces parametrizing partial lifts; stage 1 corresponds to ordinary complex orientations, and lifting from stage (m-1) to stage m is governed by the existence of a orientation for a family of E-modules over a fixed base space F_m. When E is p-local, we can say more. We find that this tower only changes when m is a power of p, and if E is E(n)-local the tower is constant after stage p^n. Moreover, if the coefficient ring E^* is p-torsion free, the ability to lift from stage 1 to stage p is equivalent to a condition on the associated formal group law that was shown necessary by Ando.