Factorization homology of topological manifolds

Factorization homology theories of topological manifolds, after Beilinson, Drinfeld, and Lurie, are homology‐type theories for topological n ‐manifolds whose coefficient systems are n ‐disk algebras or n ‐disk stacks. In this work, we prove a precise formulation of this idea, giving an axiomatic characterization of factorization homology with coefficients in n ‐disk algebras in terms of a generalization of the Eilenberg–Steenrod axioms for singular homology. Each such theory gives rise to a kind of topological quantum field theory, for which observables can be defined on general n ‐manifolds and not only closed n ‐manifolds. For n ‐disk algebra coefficients, these field theories are characterized by the condition that global observables are determined by local observables in a strong sense. Our axiomatic point of view has a number of applications. In particular, we give a concise proof of the non‐abelian Poincaré duality of Salvatore, Segal, and Lurie. We present some essential classes of calculations of factorization homology, such as for free n ‐disk algebras and enveloping algebras of Lie algebras, several of which have a conceptual meaning in terms of Koszul duality.