Factorization Homology and Low-Dimensional Topology

Recent decades have seen an explosion of research developments at the interface of low-dimensional topology and mathematical physics (particularly quantum field theory). These developments largely center around topological invariants: assignments of algebraic objects to topological objects. For example, each link (i.e. knotted curve) in Euclidean 3-space can be assigned a polynomial called its Jones polynomial, as well as a vector space called its Khovanov homology.

As another example, each 3-manifold (i.e. 3-dimensional space) can be assigned a vector space called its Heegaard Floer homology. Typically, these invariants are defined by first making an arbitrary choice (e.g. a link diagram representing the chosen link in Euclidean 3-space, or a Heegaard diagram representing the chosen 3-manifold); in order to have a topological invariant, one must then show that the result is independent of this choice.

Factorization homology is a tool for defining topological invariants that does not require any arbitrary choices, which brings a number of advantages as discussed below. The present project seeks to further develop the theory of factorization homology, and to apply it to study invariants in low-dimensional topology of physical interest as also discussed below.

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Factorization homology is a categorified form of integration: whereas ordinary integration sums numbers, factorization homology sums vector spaces or chain complexes. More precisely, factorization homology integrates (enriched) n-categories over n-manifolds, giving an invariant of n-manifolds that adheres to the local-to-global principles appearing in TQFT.

Given its well-definedness and homotopy-coherent functoriality (including a continuous action of diffeomorphisms), it plays an analogous role for state-sum TQFT as singular homology plays for cellular homology. Moreover, a slight modification is expected to give an invariant of links in 3-manifolds, e.g. the Witten--Reshetikhin--Turaev invariant (valued in the skein module of the 3-manifold).

The present project also seeks to apply factorization homology to the study of Khovanov homology and other link homology theories. Beyond making them manifestly well-defined, such a connection would further endow them with extensions to links in arbitrary 3-manifolds and with homotopy-coherent functoriality for cobordisms. The symmetries that would result lead to conjectural relationships with arithmetic objects such as cyclotomic spectra and topological Cartier modules, which the present project also seeks to explore.

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.