Factorization homology
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General Introduction
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Search results
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Online References
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1206.5164 Structured singular manifolds and factorization homology from arXiv Front: math.AT by David Ayala, John Francis, Hiro Lee Tanaka We provide a framework for the study of structured manifolds with singularities and their locally determined invariants. This generalizes factorization homology, or topological chiral homology, to the setting of singular manifolds equipped with various tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Factorization homology theories for singular manifolds are characterized by a generalization of the Eilenberg-Steenrod axioms. Using these axioms, we extend the nonabelian Poincaré duality of Salvatore and Lurie to the setting of singular manifolds -- 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, this structure gives rise to knot and link homology theories akin to Khovanov homology.
arXiv:1206.5522 Factorization homology of topological manifolds from arXiv Front: math.AT by John Francis The factorization homology of topological manifolds, after Beilinson & Drinfeld and Lurie, is a homology-type theory for topological n-manifolds whose coefficient systems are n-disk algebras. In this work we prove a precise formulation of this idea, giving an axiomatic characterization of factorization homology in terms of a generalization of the Eilenberg-Steenrod axioms for singular homology. These homology theories give rise to a specific kind of topological quantum field theory, which can be characterized by a condition that observables can be naturally defined on general n-manifolds, not only closed n-manifolds, and that global observables are determined by local observables. This axiomatic point of view has a number of applications, some in joint work with David Ayala & Hiro Tanaka, surveyed in the remainder of this article. In particular, we discuss the nonabelian Poincaré duality of Salvatore and Lurie; the relation of Koszul duality of n-disk algebras with factorization homology; calculations of factorization homology for free n-disk algebras and enveloping algebras of Lie algebras.
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