Structured singular manifolds and factorization homology

Abstract

We provide a framework for the study of structured manifolds with singularities and their locally determined invariants. This generalizes factorization homology, or topological chiral homology, to the setting of singular manifolds equipped with various tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Factorization homology theories for singular manifolds are characterized by a generalization of the Eilenberg-Steenrod axioms. Using these axioms, we extend the nonabelian Poincar\'e duality of Salvatore and Lurie to the setting of singular manifolds -- this is a nonabelian version of the Poincar\'e duality given by intersection homology. We pay special attention to the simple case of singular manifolds whose singularity datum is a properly embedded submanifold and give a further simplified algebraic characterization of these homology theories. In the case of 3-manifolds with 1-dimensional submanifolds, this structure gives rise to knot and link homology theories akin to Khovanov homology.