Factorization homology of topological manifolds

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.

• The paper establishes an axiomatic framework that maps n-disk algebras to corresponding homology theories on topological manifolds.
• It adapts the classical Eilenberg-Steenrod axioms to a local-to-global paradigm essential in topological quantum field theory.
• The work demonstrates key equivalences, including the link between factorization and Hochschild homology, advancing insights in algebraic topology and quantum theory.

Factorization Homology of Topological Manifolds: A Detailed Exploration

The paper "Factorization Homology of Topological Manifolds" by David Ayala and John Francis provides a comprehensive formulation of factorization homology with a focus on topological manifolds. The research extends the foundational work in this domain by offering an axiomatic characterization of factorization homology with coefficients in $n$-disk algebras, following from the Beilinson-Drinfeld framework of algebraic geometry. This work is pivotal in knitting together numerous mathematical standpoints like the Eilenberg-Steenrod axioms in the context of topological quantum field theory and Koszul duality.

Core Contributions

Factorization homology, as the paper expounds, maps a given algebraic input—namely, $n$-disk algebra or a stack over $n$-disk algebras—to a corresponding homology-type theory specifically tailored for $n$-manifolds. Notably, this mapping maintains adherence to an adapted form of the Eilenberg-Steenrod axioms, traditionally applied within singular homology. It delineates a scenario where global observables are decipherable through local observables, a paradigm aligning closely with fields governed by quantum field theory's perturbative approaches.

The authors provide an elegant synthesis of several areas, notably highlighting the interplay between the axiomatic approach and practical applications. They showcase the adaptability of factorization homology in proving dualities, such as the nonabelian duality by Salvatore, Segal, and the enveloping algebra dualities within the field of Lie algebras.

Theoretical Implications and Future Investigations

From a theoretical perspective, the paper describes significant advancements in homology theories tailored for manifolds rather than arbitrarily defined spaces. By advancing from the standard Eilenberg-Steenrod framework to a manifold-specific model, this work enhances the ability to distinguish topologically equivalent, yet homeomorphically distinct, manifold structures.

The implications of these characterizations open avenues for exploring broader quantum field theories utilizing topological invariants provided by factorization homology, establishing a fertile ground for exploring manifold topology through computational and geometrical invariants. Furthermore, the exploration into $n$-disk algebras, especially through the mention of Koszul duality, suggests deep structural insights into these algebraic entities that may have implications extending into algebraic topology and categorical factorization models.

Numerical Results and Claims

One of the numerical highlights from the results is the equivalence results concerning factorization homology of circle manifolds leading to Hochschild homology—an intriguing result that speaks to the power of factorization homology in redefining known homological constructs. Additionally, the paper claims equivalences for homology theories specific to various topological structures, fundamentally altering the established landscape of manifold homology.

Conclusion and Speculations in AI

Ayala and Francis' work is a meticulous example of reshaping homology theories within a topological context, providing a catalyst for advancing manifold theory and its applications across mathematics and theoretical physics. The foundations laid here cater to advanced developments such as generalized manifold calculus. As AI continues to explore mathematical research, the adaptation of topological quantum field theories' robustness—accentuated through the illustrated axiomatic approaches—could influence machine learning models validating theoretical physics, alongside broadening AI's capacity for mathematical dataset analysis. Overall, the study enhances mathematical elegance and facilitates future exploitation of factorization homology in computational models and theoretical constructs.