Factorization homology and the topology of manifolds

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Factorization homology is a homology theory for topological n-manifolds, constructed as a topological analogue of the homology of a factorization algebra in the algebro-geometric sense of Beilinson and Drinfeld. The coefficient system for such a theory is provided by a choice of n-disk algebra, a structure arising from n-fold loop spaces, Lie algebras, and certain quantum field theories, among other sources. This project aims to further develop this relatively new theory, to both bring new techniques to bear on previously studied invariants and to find new manifold invariants. A particular focus is 3-manifold topology, where one goal is to express quantum knot and 3-manifold invariants, such as Reshetikhin-Turaev invariants, Vassiliev knot invariants, and Chern-Simons invariants, in a uniform way in terms of factorization homology, and a second goal is to construct new knot homology theories via factorization homology. Another focus is the relation of factorization homology to the surgery classification of higher-dimensional topological manifolds.This project lies in topology, which studies abstract notions of space, motivated by mathematical physics - an intersection which had led to a great deal of work and cross-pollination. The problems are concerned with what possible global geometry may exist in a theoretical model for physical space or space-time and how this global geometry can be detected. It is interesting problem to try to describe the global geometry of such a candidate model for space-time by local observations. This project is one approach to this problem. Namely, if one were allowed to make local observations of some kind at different points throughout space-time, then compare them, how could one then reconstruct the global geometry from these compatible collections of observations? Factorization homology, the focus of this project, provides one avenue toward addressing this question.