Factorization homology of topological manifolds. (16th October 2015)

• Record Type: Journal Article
• Authors: Ayala, David; Francis, John
• Is Part Of: Journal of topology. Volume 8:Part 4(2015)
• Page Start: 1045
• Page End: 1084
• Publication Date: 2015-10-16
• Subjects: Topology -- Periodicals; 514.05
• DOI: 10.1112/jtopol/jtv028
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract : Factorization homology theories of topological manifolds, after Beilinson, Drinfeld, and Lurie, are homology‐type theories for topologicaln ‐manifolds whose coefficient systems aren ‐disk algebras orn ‐disk stacks. In this work, we prove a precise formulation of this idea, giving an axiomatic characterization of factorization homology with coefficients inn ‐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 generaln ‐manifolds and not only closedn ‐manifolds. Forn ‐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 freen ‐disk algebras and enveloping algebras of Lie algebras, several of which have a conceptual meaning in terms of Koszul duality.