TY - JOUR

T1 - Factorization homology of topological manifolds

AU - Ayala, David

AU - Francis, John

N1 - Publisher Copyright:
© London Mathematical Society.

PY - 2014/6/5

Y1 - 2014/6/5

N2 - 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.

AB - 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.

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U2 - 10.1112/jtopol/jtv028

DO - 10.1112/jtopol/jtv028

M3 - Article

AN - SCOPUS:84950110104

SN - 1753-8416

VL - 8

SP - 1045

EP - 1084

JO - Journal of Topology

JF - Journal of Topology

IS - 4

ER -