Factorization homology of stratified spaces
David Ayala, John Francis, Hiro Lee Tanaka
TL;DR
The paper develops factorization homology for stratified spaces with tangential structures, unifying concepts such as intersection homology and knot/link invariants within an ∞-categorical framework. It proves a generalization of the Eilenberg–Steenrod axioms, establishing an equivalence between Disk(B)-algebras in a symmetric monoidal ∞-category V and V-valued homology theories on B-manifolds, with a relative version capturing spacetime dependence. A stratified nonabelian Poincaré duality is proved, linking factorization homology to compactly supported section spaces under connectivity hypotheses, and a pushforward/Fubini theorem for constructible bundles is developed. The framework yields concrete examples, including stratified 1-manifolds, intersection homology, and knot/link theories via D_{d⊂n}^{fr}-algebras, suggesting new algebraic invariants and connections to perturbative quantum field theory observables and Chern–Simons-type constructions.
Abstract
This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg--Steenrod axioms; it can also be viewed as an analogue of the Baez--Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topological quantum field theory. Using these axioms, we extend the nonabelian Poincaré duality of Salvatore and Lurie to the setting of stratified spaces -- this is a nonabelian version of the Poincaré 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, these structure gives rise to knot and link homology theories.