Factorization Homology: A Local-to-Global Approach

• Factorization homology is a homology-type invariant that integrates local disk algebra data over structured spaces, including singular manifolds.
• It leverages operadic extensions and excision via collar-gluing to compute global invariants and generalizes classical theories like Hochschild and intersection homology.
• The framework offers computational tools through push-forward formulas and non-abelian Poincaré duality, impacting topology and mathematical physics.

Factorization homology is a homology-type invariant that systematically “integrates” local algebraic data—modeled by (generalized) disk algebras—over highly structured spaces such as manifolds, stratified spaces, or singular spaces equipped with tangential structures. This theory generalizes classical homology and topological chiral homology, encompasses and recovers familiar invariants such as Hochschild homology and intersection homology, and provides a flexible framework for constructing invariants sensitive to singularity data, knot/link complements, and beyond.

1. Local-to-Global Principle: Disk Algebras and Operadic Extension

The core conceptual framework is to assign local invariants to “basics”—the geometric building blocks of singular manifolds (e.g., products of Euclidean spaces with cones)—and extend this data globally via operadic left Kan extension. Formally, for a symmetric monoidal quasi-category $C$ that admits appropriate colimits and a Disk$(B)$-algebra $A$ in $C$, one produces a functor

$X \mapsto \int_X A$

from the quasi-category $\mathrm{Mfld}(B)$ of singular $B$-manifolds (including, for example, smooth, pseudomanifolds, and those with embedded defects) to $C$ (Definition 3.15).

This assignment is completely determined by the local values (“values on basics”) and is functorially extended using a colimit: $$\int_X A \simeq \operatorname{colim}_{U \to X \in \mathrm{Disk}(B)/X} A(U)$$ where the colimit is over all basic open embeddings into $X$.

The main structural theorem (Theorem 3.36) states that symmetric monoidal functors on $\mathrm{Mfld}(B)$ that satisfy the corresponding excision axiom are equivalent, as a quasi-category, to Disk$(B)$-algebras in $C$. Therefore, every homology-type theory on singular manifolds that is locally determined and satisfies excision is classified by its values on the local models.

2. Excision via Collar-Gluing and Generalized Eilenberg–Steenrod Axioms

Excision—a “local-to-global” property analogously generalizing the Eilenberg–Steenrod axioms—is formulated using collar-gluing decompositions of singular manifolds. Specifically, if $X = X_{-} \cup_{R^{+}V} X_{+}$ (gluing along a collar neighborhood $R^+V$), then for any Disk$(B)$-algebra $A$

$$\int_X A \simeq \int_{X_{-}} A \otimes_{\int_{R^+ V} A} \int_{X_{+}} A$$

(Theorem 3.33).

This excision property substitutes collar-gluing for arbitrary pushouts, in recognition of the fact that the relevant categories of singular manifolds may not admit arbitrary (homotopy) pushouts. Thus, excision is tailored to the geometry of singularities and stratifications.

The uniqueness theorem (Theorem 3.36) ensures that any homology theory with these properties is determined up to equivalence by its Disk$(B)$-algebra structure, cementing the classification-by-local-data paradigm.

3. Push-Forward, Functoriality, and Computability

A significant technical innovation is the operadic push-forward formula (Theorem 3.28), which underlies a robust base-change mechanism: If $F$ is a $B_0$-family of $B$-manifolds parametrized by $P$, then the push-forward $F_* A$ of a Disk$(B)$-algebra $A$ along $F$ satisfies

$\int_P (F_*A) \simeq \int_P A$

(see Definition 3.27).

This “base-change” property provides computational leverage: factorization homology on total spaces of families can be reduced to homology over the base. In practice, this results in efficient calculations and conceptual simplifications akin to Fubini’s theorem in classical integration.

4. Non-Abelian Poincaré Duality in the Singular Context

Extending previous dualities (Salvatore, Segal, Lurie), a version of non-abelian Poincaré duality is proven for singular manifolds (Theorem 4.12). For “connective” coefficient systems—corresponding to group-like Disk$(B)$-algebras—the factorization homology of a finite $B$-manifold $X$ is explicitly identified with a mapping space: $\int_X A_E \simeq \mathrm{Map}_{*}(X, T_E)$ where $T_E$ is a space derived from the dualizing (co)sheaf associated to the coefficient system.

For $B = D_{fr}$ (framed, no singularities), this recovers Lurie’s non-abelian Poincaré duality. For singular $B$ (i.e., with stratification), it yields a “non-abelian” generalization of intersection cohomology duality, incorporating contributions from both smooth and singular strata.

5. Canonical Examples: Hochschild, Intersection, and Knot/Link Invariants

Factorization homology specializes to classical invariants and permits the construction of new, sensitive invariants for singular spaces:

• Hochschild Homology: For $C$ the category of chain complexes over a commutative ring $k$ and $A$ an associative algebra, one recovers

$\int_{S^1} A \simeq \mathrm{HH}_*(A)$

(Proposition 5.1), using a decomposition of the circle via excision.

• Marked Points, Singularities, and Bimodules: For 1-manifolds with a marked point, the corresponding Disk-algebra involves an associative algebra with a distinguished bimodule, and the factorization homology realizes variants of Hochschild homology with coefficients in the bimodule (Proposition 5.3).
• Intersection Homology: On higher-dimensional (stratified) pseudomanifolds, factorization homology—using the appropriate Disk$(B)$-algebra—recovers generalized intersection homology theories, serving as a non-abelian analog that reduces to classical intersection homology in abelian cases.
• Knot and Link Invariants: For 3-manifolds with embedded 1-dimensional submanifolds (links), the structure of the free Disk$_{3,1}$-algebra recovers invariants that encode the homotopy type of the link complement. For example, factorization homology can distinguish between the unknot and the trefoil, as it is sensitive to configuration spaces of points in the complement (Proposition 5.13). These constructions admit simplified algebraic characterizations when the singularities are properly embedded and support novel knot and link homology theories akin to Khovanov homology.

6. Differential Topology of Singular Manifolds

A comprehensive account of the differential topology of singular manifolds is developed (Section 6). Singular manifolds are locally modeled on spaces of the form $\mathbb{R}^{n-k} \times C(Y)$, with $C(Y)$ a cone over a compact stratified space $Y$. Key results include:

• The automorphism space of a basic is an $\infty$-groupoid compatible with the cone structure (Lemma 6.10).
• The tangent bundle $TX \to X$ may fail to be a locally trivial fiber bundle; for instance, at a node in a nodal surface, the tangent “fiber” is a cone, not $\mathbb{R}^2$. However, on smooth strata $TX$ retains standard local vector bundle structure.
• Functoriality is ensured using “conically smooth” maps, accommodating the singularities.

This machinery establishes the necessary foundation to rigorously define and manipulate the singular categories and their morphisms required in factorization homology, ensuring soundness even in the presence of high-codimension singular strata.

7. Impact and Computational Power

The unified framework described in the paper systematically generalizes homological invariants and duality phenomena for arbitrary singularity types. By encoding all global invariants in the Disk$(B)$-algebra structure and enforcing excision via collar-gluing, the authors create a setting in which classical invariants—such as Hochschild and intersection homology—become specific instances of a universal construction, and new invariants (e.g., those of links, knots, or stratified mapping spaces) arise naturally as factorization homologies for different algebraic inputs.

Additionally, the base-change formula and explicit push-forward computations furnish computational tools that render the resulting invariants tractable, both conceptually and algorithmically. This approach also extends directly to applications in topology and topological field theory, where the “local-to-global” gluing principle of factorization homology mirrors fundamental locality axioms for quantum field theories.

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This rigorous development of factorization homology for singular manifolds establishes a general, computable, and functorial “integration” mechanism of local algebraic structures, synthesizing excision, duality, and stratification sensitivity within a single operadic and categorical theory. The excision property and the full local-to-global determination via Disk$(B)$-algebra structure are the cornerstones by which homology-type invariants for singular spaces are constructed, classified, and applied across topology, algebra, and field theory.