CAREER: Applications of equivariant homotopy theory to manifolds

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

A powerful idea in many areas of mathematics has been that of studying a geometric object such as a polygon by associating it with an algebraic invariant, for example a number that does not change under continuous deformation. An early instance of this is Euler who noted that for a polygonal surface the number of vertices minus edges plus faces is always two, which allowed him to prove that there are only five platonic solids. In the 1920s, Emmy Noether shifted the focus from numerical invariants to the algebraic structures underlying them, such as sets with algebraic operations. In Homotopy Theory, the focus is extended further to more advanced mathematical notions of infinite-dimensional spaces whose number of connected components recover the underlying numerical invariant. The projects funded by this CAREER award aim to lift classical algebraic invariants of geometric objects to the higher homotopical world and study them from this richer perspective. The educational part of the project consists in expanding access to mathematics not only within the academic world via conferences, but also beyond the usual academic settings to incarcerated individuals: the PI will introduce and teach an inquiry-based learning college level mathematics elective in the B.A. program offered at South Woods Prison. The primary research goal of the project is to use new results in equivariant stable homotopy theory to study manifolds with group actions and their equivariant diffeomorphisms. Stable homotopy theory, the study of spectra, has played a crucial role in the longstanding program to classify manifolds; however, the classification of G-manifolds for a group G is far less understood than the classical nonequivariant story. Building in part on prior work of the PI on equivariant algebraic K-theory and her collaborative foundational work on equivariant infinite loop space theory, the machinery that turns suitable G-categories into G-spectra, the PI and Malkiewich were have built an equivariant generalization of Waldhausen's algebraic K-theory of spaces. Inspired by Goodwillie’s vision, they conjecture that, for a compact smooth G-manifold M, equivariant A-theory splits off a stable space of equivariant h-cobordisms. This project involves a significant extension of these efforts focused on proving this conjecture and exploiting it to further our understanding of equivariant h-cobordisms and diffeomorphisms of G-manifolds. The second research goal is to apply stable homotopy techniques to study classical invariants by deriving these, i.e., lifting them to spectra. The PI will pursue two projects of this flavor. First, together with Malkiewich and Moi, the PI will study a derived version of the classical character map from representation theory that they have defined. The second, which is a project that the PI is co-leading with Rovi for the Women in Topology Workshop, is to apply and extend recent work of Zakharevich and Campbell on scissors congruence K-theory spectra to manifolds. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.