Mathematical Sciences: Comparing Different Periodicities in Homotopy Theory

9401404 Sadofsky首席研究员将研究Mahowald的根不变量（通过研究群Z/pZ的分类空间的胞结构形成）与同伦理论中的vn-周期性之间的关系（这里取为Johnson-Wilson上同伦理论E(n)中可以使用初等和高级运算检测到的同伦理论）。特别地，他想了解利用E(n)到被E(n)砸碎的Lin逆系统的逆极限的映射所形成的chen字符的类比，该映射给出了一个v(n-1)-周期同伦理论。他还将继续努力理解在同调理论E(n)下，用空间的Bousfield局部化对Lin逆系统求逆极限的结果。后一个逆系统似乎通过Hopf不变量的类比将v（n-1）周期同伦类和v（n-1）周期同伦类联系起来。拓扑学是对“空间”的研究，如果一个空间可以变形为另一个空间，则两个空间被认为是相同的。(就目前的目的而言，一个空间可以被认为是某个n维欧几里得空间的子集，即平面的子集，或三维欧几里得空间的子集，四维欧几里得空间的子集等。尽管从我们的日常经验来看，需要超过3个维度来描述的空间并不熟悉，但它们在数学、物理学和工程应用的大多数分支中经常出现。同伦理论使用代数技术来研究这些空间中“洞”的数量和种类（例如，将圆上的洞与空心球体上的洞进行对比）。尽管后一种信息相当粗糙（它无法区分拓扑上不同的一些空间），但它仍然描述了空间的大部分几何形状，并且足以满足许多应用程序。此外，对于每个非负整数n，由同伦理论提供的代数信息可以被划分为所谓的“n周期”类型。一次检查一个n周期类型的同伦会导致一些简化，并且比一次检查关于空间的所有同伦理论信息更容易处理。本课题研究不同n周期同伦信息与(n+1)-周期同伦信息之间的相互作用。这些看起来有些不相关，但是一个空间的n周期信息限制了相同空间的(n+1)-周期信息的可能性。这是一个非常有用的关系，因为0周期的信息很容易理解，1周期的信息也很容易理解，所以人们可能希望用这个关系用k周期的同伦来描述n周期的同伦

9401404 Sadofsky The principal investigator will examine the relationship between Mahowald's root invariant (formed by studying the cell structure of the classifying space of the group Z/pZ), and vn-periodicity in homotopy theory (which is taken here to be the homotopy theory that can be detected using primary and higher operations in the Johnson-Wilson cohomology theory E(n)). In particular, he would like to understand the analog of the Chern character formed by using the map from E(n) to the inverse limit of the Lin inverse system smashed with E(n) which gives a v(n-1)-periodic homotopy theory . He will also continue his efforts to understand the result of taking the inverse limit of the Lin inverse system smashed with the Bousfield localization of a space with respect to the homology theory E(n). This latter inverse system seems to relate vn-periodic and v(n-1)-periodic homotopy classes via an analog of the Hopf invariant. Topology is the study of "spaces" where two spaces are considered the same if one is deformable to the other. (For present purposes, a space can be thought of as a subset of some n-dimensional Euclidean space; i.e. a subset of the plane, or a subset of 3-dimensional Euclidean space, a subset of 4-dimensional Euclidean space, etc. Although the spaces requiring more than 3 dimensions to describe are not familiar from our ordinary experience, they arise frequently in most branches of mathematics, in physics, and in engineering applications.) Homotopy theory uses algebraic techniques to study the number and kind of "holes" in these spaces (contrast, for example, the sort of hole in a circle with the sort of hole in a hollow sphere). Although this latter information is rather coarse (it fails to distinguish between some spaces that are topologically distinct), it still describes much of the geometry of a space, and suffices for many applications. Furthermore, it turns out that the algebraic information provided by homotopy theory can be divided up into what are called "n-periodic" types, for each non-negative integer n. Examining one n-periodic type of homotopy at a time leads to some simplifications, and is more tractable than examining all the homotopy theoretic information about a space at once. This project is concerned with understanding the interaction between different n-periodic homotopy information and (n+1)-periodic homotopy information. These appear somewhat unrelated, but the n-periodic information for a space restricts the possibilities for the (n+1)-periodic information for the same space. This is ultimately a very useful relationship to understand, for the 0-periodic information is quite accessible, and the 1-periodic information is also quite well understood, so one might hope to use this relationship to describe much of the n-periodic homotopy in terms of the k-periodic homotopy for k n. ***