Factorization homology of topological manifolds
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2012
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| _version_ | 1866916297833447424 |
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| author | Ayala, David Francis, John |
| author_facet | Ayala, David Francis, John |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1206_5522 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Factorization homology of topological manifolds Ayala, David Francis, John Algebraic Topology Geometric Topology Quantum Algebra 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 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. |
| title | Factorization homology of topological manifolds |
| topic | Algebraic Topology Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/1206.5522 |