标题： 严格交换复向量丛定向理论 显示英文标题

标题： Strictly commutative complex orientation theory

摘要： 对于一个乘法上同调理论 \(E\)，复定向与从复配边上同调 \(MU\) 到 \(E\) 的乘法自然变换之间存在双射对应关系。 如果 \(E\) 被一个具有高度结构化乘法的谱表示，我们基于 Arone-Lesh 的工作，给出了将一个定向 \(MU \to E\) 提升到尊重这种额外结构的映射的迭代过程。 严格交换定向的空间是参数化部分提升的逆塔空间的极限；第 1 阶段对应于普通的复定向，并且从阶段 \((m-1)\) 到阶段 \(m\) 的提升由固定基空间 \(F_m\) 上的 \(E\)-模族的定向存在性所控制。 当 \(E\) 是 \(p\)-局部时，我们可以说得更多。 我们发现这个塔只有当 \(m\) 是 \(p\) 的幂时才会发生变化，并且如果 \(E\) 是 \(E(n)\)-局部的，则塔在阶段 \(p^n\) 后保持恒定。 此外，如果系数环 \(E^*\) 是 \(p\)-无挠的，则从阶段 1 提升到阶段 \(p\) 的能力等价于一个条件，该条件由 Ando 显示为必要条件。 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.