Factorization homology of topological manifolds

TL;DR

This paper provides an axiomatic framework for factorization homology of topological manifolds, linking it to topological quantum field theories and demonstrating applications like nonabelian Poincaré duality and explicit calculations.

Contribution

It offers a precise axiomatic formulation of factorization homology with $n$-disk algebra coefficients, generalizing classical homology axioms and connecting to quantum field theories.

Findings

Axiomatic characterization of factorization homology.

Proof of nonabelian Poincaré duality.

Calculations for free $n$-disk algebras and Lie algebra enveloping algebras.

Abstract

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 nonabelian Poincar\'e 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.