Factorization homology of topological manifolds

By David Ayala, J. Francis. 2012

228 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dc4d0973d2945d14156523f2711e61a0a1db7072

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In Geometry and Topology

1. A. Oancea
2. Kai Cieliebak
3. David Ayala
4. J. Francis
5. V. Hinich
6. Ben Knudsen
7. Gijs Heuts
8. D. Jordan

In Mathematical Physics

1. A. Oancea
2. Kai Cieliebak
3. V. Hinich
4. David Ayala
5. J. Francis
6. Ben Knudsen
7. Gijs Heuts
8. D. Jordan

Citations in the corpus

• Factorization Algebras in Quantum Field Theory: Volume 1 (2016, background): “) In further work Ayala et al. (n.d.b), they have developed a generalization of factorization techniques where the coefficients are (∞, n)categories. One payoff of their results is a proof of the Cobordism Hypothesis, and hence an explanation of functorial topological field theories via factorizatio”
• On the equivalence between Lurie's model and the dendroidal model for infinity-operads (2013, background): “It plays an important role in current advances in derived algebraic geometry, specifically chiral homology [17, 28], geometric representation theory [1, 5, 6, 18, 19, 20] and mirror symmetry [32].”
• The finiteness conjecture for skein modules (2019, background): “F(SkAlgint TL( ∗)) ∼= ∫ [n]∈TL HomSkCatTL( ∗)(P([n]), 1) ⊗ F([1])⊗n ∈ Vect.”
• Betti numbers and stability for configuration spaces via factorization homology (2014, background): “ments. 6 Ben Knudsen 2 Factorization Homology 2.1 Homology theories In this section, we review the basic notions of factorization homology, alias topological chiral homology. The primary reference is [AF15]. As there, our point of view is that factorization homology is a natural theory of homology f”
• Symplectic homology and the Eilenberg-Steenrod axioms (2015, background): “It is interesting to note at this point a formal similarity with the recent development of factorisation homology, see the paper [7] by Ayala and Francis as well as the references therein.”
• The cobordism hypothesis (2017, methodology): “or factorization homology is then the sheaf homology over this Ran space. A special, somewhat degenerate, form of the cobordism hypothesis is provable using this form of factorization homology theory [AF1]. In the case of a smooth 4-manifold M, moduli of smoothly embedded surfaces in M, not merely fi”