Mathematical Sciences: Comparing Different Periodicities in Homotopy Theory

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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. ***