Periodicity Phenomena at the Chromatic Edge, the Chromatic Splitting Conjecture, and the Chromatic Segal Conjecture

9971850Sadofsky Mike Hopkins的“色分裂猜想”涉及局部化相对于局部化的(n-1)stJohnson-Wilson理论相对于(有限CW-复形的)nthMorava K理论的合成。这是试图理解稳定同伦理论的有用的一步，因为分解是用更简单的局部化来表示的。特别是，这个猜想将概括已知的0级、1级和2级色度之间非常美丽的关系。同样，“色西格尔猜想”(由Mike Hopkins、Mark Mahowald和Doug Ravenel组合而成的各种形式)涉及有限CW-复形的局部化的泰特上同调相对于第n个Morava K-理论的分解。同样，这个猜想以一种美丽的方式将不同的色级缠绕在一起；它们一起给出了一些从较低的色级归纳确定较高色级的信息的技术(类比同伦理论和K-理论)。这两个猜想还提供了不同Morava稳定剂基团的上同调如何相关的数据。用来攻击这些猜想的工具是基于Goerss工作的思想的体谱序列，该序列根据BP-运算的Hopfagebroid上的余模的逆极限的派生函子来计算涉及到这一计算的各种谱的BP-同调。令人惊讶的是，至少在第一个有趣的案例中，较高的反向极限似乎是容易处理的。这项研究关注于理解函数的同伦类。通过考虑纽结理论，给出了这类问题的一个简单例子。把空间中的“结”看作是以某种方式嵌入到空间中的一圈弦。人们并不是真的对嵌入字符串循环的所有方法感兴趣；如果只需将字符串稍微移动一点就可以从一种方法嵌入到另一种方法，则认为这两种嵌入是等价的。类似地，如果查看所有将字符串循环放在去除原点的平面上的方法，如果字符串的循环绕原点缠绕相同次数，则认为两种方法是相同的。正如第二个例子所表明的那样，理解映射到的空间中的“洞”以及映射的空间如何与这些洞交互作用是至关重要的。然而，有不同种类的洞；圆形有一个可以穿过东西的洞，而(中空的)球体有不同类型的洞。与地图(或空间)相关的“洞”可以分为不同的类型(通常称为不同的色分量)。这些分量对于空间和基于几何考虑的地图似乎并不独立，目标是理解这些相关性。这类问题的解决方案可能与几何问题有关，但同时，令人惊讶的是，与数论中出现的对象有关，这项工作比单独考虑这两个方面的工作更重要。*

9971850Sadofsky The ``chromatic splitting conjecture'' of Mike Hopkins concerns adecomposition of the localization with respect to the (n-1)stJohnson-Wilson theory of the localization with respect to the nthMorava K-theory (of a finite CW-complex). This is a useful step intrying to understand stable homotopy theory, since the decomposition isexpressed in terms of simpler localizations. In particular, theconjecture would generalize the very beautiful relationships knownbetween the 0th, 1st, and 2nd chromatic levels. Similarly, the``chromatic Segal conjecture'' (due in various forms to combinations ofMike Hopkins, Mark Mahowald, and Doug Ravenel) deals with adecomposition of the Tate cohomology of the localization of a finiteCW-complex with respect to the nth Morava K-theory. Again, thisconjecture intertwines different chromatic levels in a beautiful way;together they give some techniques for inductively determininginformation about higher chromatic levels from lower ones (likerational homotopy theory and K-theory). These conjectures both alsogive data about how the cohomology of the different Morava stabilizergroups are related. The tool used to attack these conjectures is aspectral sequence based on ideas from work of Goerss that computes theBP-homology of various spectra involved in this calculation in termsof the derived functors of inverse limits of comodules over the Hopfalgebroid of BP-operations. Surprisingly, at least in the firstinteresting cases, the higher inverse limits seem to be tractable. This research is concerned with understanding ``homotopy classes offunctions.'' A simple example of the sort of problem studied is givenby considering knot theory. Take a ``knot'' in space to be a loop ofstring embedded in space in some way. One is not really interested in*all* ways of embedding that loop of string; if one can get from oneway of embedding the loop to another just by moving the string alittle bit, one considers those two embeddings to be equivalent.Similarly, if one looks at all ways of putting a loop of string in theplane with the origin removed, one considers two ways the same if theloop of string is wound around the origin the same number of times.As the second example makes clear, it is critical to understand``holes'' in the space to which one is mapping, and how the space fromwhich one is mapping interacts with those holes. There are differentsorts of holes, though; a circle has a hole one can pass something through,while a (hollow) sphere has a different sort of hole. The ``holes''associated to a map (or to a space) can be sorted into differentflavors (usually referred to as different chromatic components).These components do not appear to be independent for spaces and mapsarising from geometric considerations, and the goal is to understandthese dependencies. Solutions to this sort of problem are potentiallyrelated to questions from geometry, but at the same time, surprisingly,to objects arising in number theory, something that gives this work fargreater importance than would either aspect alone.***