Strictly commutative complex orientation theory
Strictly commutative complex orientation theory
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DOI：
10.1007/s00209-017-2009-6
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发表时间：
2016-02
影响因子：
0.8
通讯作者：
M. Hopkins;T. Lawson
中科院分区：
文献类型：
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作者：
M. Hopkins;T. Lawson
对于乘法上同调理论 E，复数方向与从复数边函数上同调 MU 到 E 的乘法自然变换是双射对应的。如果 E 由具有高度结构化乘法的谱表示，我们基于 Arone-Lesh 的工作给出了一个迭代过程，用于将方向提升到尊重这种额外结构的地图。严格交换方向的空间是参数化部分升力的空间逆塔的极限；第 1 阶段对应于普通的复杂方向，从一个阶段到另一个阶段的提升由固定基础空间上一系列 E 模块的方向的存在来控制。当Eisp-local时，我们可以说得更多。我们发现这个塔只有当m是p的幂时才会改变，并且如果EisE(n)-局部，那么塔在阶段之后是恒定的。此外，如果系数无环扭转，则从阶段 1 提升到阶段 p 的能力相当于安藤证明必要的相关形式群定律的条件。
For a multiplicative cohomology theoryE, complex orientations are in bijective correspondence with multiplicative natural transformations toEfrom complex bordism cohomologyMU. IfEis represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientationto 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 stageto stagemis governed by the existence of an orientation for a family ofE-modules over a fixed base space. WhenEisp-local, we can say more. We find that this tower only changes whenmis a power ofp, and ifEisE(n)-local the tower is constant after stage. Moreover, if the coefficient ringisp-torsion free, the ability to lift from stage 1 to stagepis equivalent to a condition on the associated formal group law that was shown necessary by Ando.