Metastable Pseudoisotopy, G-Manifolds, and Functor Calculus

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AbstractAward: DMS 1608259, Principal Investigator: Thomas G. GoodwillieThis award provides support for principal investigator's research in algebraic topology, which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to certain equivalence. Manifolds are topological spaces that locally resemble Euclidean spaces. The first part of the project holds the promise of new invariants for families of manifolds and new connections between higher-dimensional geometric topology and string topology. The second part should lead to other new invariants for families of manifolds with a group action. The third part will continue the study of "functor calculus", a theory developed by the PI previously, from a geometric point of view. Graduate students will be involved and trained in the project.The planned research has three components. The first involves systematically detecting that about manifolds which is invisible to surgery and algebraic K-theory, beginning with the homotopy fiber of the Hatcher suspension map for pseudoisotopy spaces. Though the appropriate analogue of K-theory is elusive, the corresponding analogue of topological cyclic homology and the cyclotomic trace seems to be within reach. The second is concerned with actions of finite groups on manifolds; the goal is to develop G versions of various constructions related to spaces of manifolds, and especially of spaces of h-cobordisms. The third is about the "geometric" view of homotopy functor calculus, and especially about some fundamental questions about jets and differential operators in that context.