Mathematical Sciences: Comparing Different Periodicities in Homotopy Theory

小行星9401404 主要研究者将研究Mahowald的根不变量（通过研究群Z/pZ的分类空间的细胞结构形成）和同伦理论中的vn-周期性（这里被认为是可以使用Johnson-Wilson上同调理论E（n）中的初等和高等运算检测的同伦理论）之间的关系。 特别是，他想了解的模拟陈德铭字符形成的使用地图从E（n）的逆极限林逆系统粉碎与E（n）这给了一个v（n-1）-周期同伦理论。 他还将继续努力，以了解结果采取的逆极限林逆系统粉碎与Bousfield本地化的空间就同源理论E（n）。 后一个逆系统似乎通过类似的Hopf不变量将vn-周期和v（n-1）-周期同伦类联系起来。 拓扑学是对“空间”的研究，其中两个空间被认为是相同的，如果一个是可变形的。 (For在本发明的目的中，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3维欧几里得空间的子集，4维欧几里得空间的子集，等等。虽然从我们的普通经验来看，需要多于3维来描述的空间并不熟悉，但它们经常出现在数学的大多数分支中，in physics物理，and in engineering工程application应用.） 同伦理论使用代数技巧来研究这些空间中的“洞”的数量和种类（例如，将圆中的洞与空心球中的洞进行对比）。 虽然后一种信息相当粗糙（它不能区分拓扑上不同的空间），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。 此外，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，对于每个非负整数n。 一次检查一个n周期同伦类型会导致一些简化，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。 这个项目关注的是理解不同的n-周期同伦信息和（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，但空间的n周期信息限制了同一空间的（n+1）周期信息的可能性。 这最终是一个非常有用的关系来理解，因为0-周期的信息是相当容易获得的，1-周期的信息也是相当好理解的，所以人们可能希望使用这个关系来描述大部分的n-周期同伦的k-周期同伦n。 ***

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. ***