Groups, Manifolds, and Stable Homotopy Theory

Johns Hopkins University, Baltimore MD

Investigators

Abstract

Spaces, like a plane or a sphere, are easy to visualize, but for high-dimensional objects like the shape of a large data set with many parameters, this is impossible. Algebraic topology associates algebraic invariants to spaces, moving the problem of understanding spaces (up to continuous deformation), to the more tractable world of algebra. At the heart of this project is the problem of understanding some of these invariants when the space in question is considered with the additional data of its symmetries. For example, instead of just considering a sphere, we can try to build in the additional data of the symmetry which maps each point to its antipodal point; or instead of considering a plane, we can consider a plane together with all the rigid motions of the plane that fix a square centered at its origin. The particular invariants considered in this project are algebraic K-theory and A-theory: deep invariants that have very significant connections to problems in number theory and geometry. Their versatility, however, requires sophisticated topological and categorical constructions. Part of this project is concerned with completing some necessary foundational work that allows these constructions to encode the extra data of symmetries, and part of it is concerned with using newly developed tools to rigorously define and study these theories, while keeping track of symmetries. There is also an educational component to the proposal-the PI plans to start a Directed Reading Program at the Johns Hopkins University, modeled on the one at the University of Chicago. The goal is to provide mentorship to undergraduates by pairing them with graduate students for reading courses, and in the process encourage interested students to pursue graduate work in mathematics. Building on the PI's previous work on equivariant algebraic K-theory, the PI will develop an equivariant version of A-theory that is related to equivariant pseudo-isotopies and h-cobordisms of G-manifolds. Th project fits into a long-term research program of the project team aimed at establishing a chain of homotopy-theoretic constructions that relate the geometric behavior of compact G-manifolds to that of their underlying equivariant homotopy types. In current work, the project team has defined an equivariant A-theory spectrum, and the next step is to show that it fits into an equivariant stable parametrized h-cobordism theorem. In a related project with collaborators, the PI plans to study an equivariant version of the additivity theorem for Waldhausen G-categories. Nonequivariantly, K-theory is universal with respect to the additivity theorem, and equivariantly one can hope for a similar characterization. Progress has been slow in equivariant K and A-theory partly because adequate foundations were not in place. Equivariant infinite loop space theory, the machinery which turns suitable categories with G-action into G-spectra, has seen a lot of development in the last few years, and, motivated by the need for a solid foundation for equivariant algebraic K and A-theory, the PI, together with collaborators, is completing a series of projects on equivariant infinite loop space theory. This work not only produces new results (some of which have been anticipated for about 35 years), but they also consolidate work in important areas where the literature is fragmented and in need of clarification. They should provide some of the tools for the projects proposed above, as well as some related ones. In a different direction, the PI and collaborators have defined and are studying a spectral character map that generalizes the classical character map from representation theory. Lastly, the PI plans to study motivic Galois extensions.