ScalingStacks

Factorization homology takes an algebraic input, either an $n$-disk algebra or more generally a stack over $n$-disk algebras, and outputs a homology-type theory for $n$-dimensional manifolds. In the case where the input coefficients are an $n$-disk algebra, this factorization homology – the topological chiral homology introduced by Lurie [Lu2] – satisfies an analogue of the axioms of Eilenberg and Steenrod. This work proves this statement and some consequences afforded by this point of view.

While the subject of factorization homology is new, at least in name, it has important roots and antecedents. Firstly, it derives from the factorization algebras of Beilinson and Drinfeld [BD], a profound and elegant algebro-geometric elaboration on the role of configuration space integrals in conformal field theory. Our work is in essence a topological version of theirs, although the topological setting allows for arguments and conclusions ostensibly unavailable in the algebraic geometry. Secondly, it has an antecedent in the labeled configuration space models of mapping spaces dating to the 1970s; it is closest to the models of Salvatore [Sa] and Segal [Se3], but see also [Ka], [Bö], [Mc], [Ma], and [Se1]. Factorization homology thus lies at the broad nexus of Segal’s ideas on conformal field theory [Se2] and his ideas on mapping spaces articulated in [Se1] and [Se3].

In keeping with these two directions, we offer two primary motivations for the study of factorization homology. Returning to our first point, factorization homology with coefficients in $n$-disk algebras are homology theories for topological manifolds satisfying a generalization of the Eilenberg–Steenrod axioms for ordinary homology; as such, it generalizes ordinary homology in a way that is only defined on $n$-manifolds and not necessarily on arbitrary topological spaces. Second, these homology theories define topological quantum field theories. Following the vision of Costello–Gwilliam [CG], factorization homology with coefficients in $n$-disk stacks offers an a algebraic model for the observables in a general topological quantum field theory. The special case of $n$-disk algebra coefficients corresponds to $n$-dimensional field theories whose global observables are determined by the local observables, as in a perturbative quantum field theory.

We first elaborate on the homology theory motivation, which serves as the main artery running through this work. One can pose the following question: what can a homology theory for topological manifolds be? Singular homology, of course, provides one answer. One might not want it to be the only answer, since singular homology is not specific to manifolds and can be equally well defined for all topological spaces; likewise, it is functorial with respect to all maps of spaces, not just those that are specifically meaningful for manifolds (such as embeddings or submersions). One could thus ask for a homology theory which is specific to manifolds, not defined on all spaces, and which might thereby distinguish manifolds which are homotopic but not homeomorphic and distinguish embeddings that are homotopic but not isotopic through embeddings. That is, one could ask for a homology theory for manifolds that can detect the more refined and interesting aspects of manifold topology. The question then becomes, do such homology theories exist?

One might address this question by first making it more precise, by defining exactly what one means by a homology theory for manifolds; since we know what a homology theory for spaces constitutes, by the Eilenberg–Steenrod axioms, one might simply modify those axioms as little as possible, but so as to work only for manifolds and with the maximum possibility that new theories might arise.

Reformulating the Eilenberg–Steenrod axioms slightly, one can think of an ordinary homology theory $\mathcal{F}$ as a functor $\Space^{\sf fin}\rightarrow{\sf Ch}$ from the topological category of spaces homotopy equivalent to finite CW complexes to the topological category of projective chain complexes¹¹ 1 There is the standard functor $|\Map_{\sf Ch}\bigl(\mathsf{C}_{\ast}(\Delta^{\bullet}),-\bigr)|\colon{\sf Ch}\to\Top$ to topological spaces given by the Dold–Kan correspondence. While this functor does not send tensor product to Cartesian product, it does so up to a coherent natural transformation. The internal hom-objects of ${\sf Ch}$ thus give a topological enrichment. satisfying two conditions:

• •

  The canonical morphism $\bigoplus_{J}\mathcal{F}(X_{\alpha})\rightarrow\mathcal{F}(\coprod_{J}X_{\alpha})$ is an equivalence for any finite set $J$;

• •

  Excision: for any diagram of cofibrations of spaces $X^{\prime}\hookleftarrow X\hookrightarrow X^{\prime\prime}$, the resulting map of chain complexes

  $$\mathcal{F}(X^{\prime})\underset{\mathcal{F}(X)}{\oplus}\mathcal{F}(X^{\prime\prime})\longrightarrow\mathcal{F}\Bigl(X^{\prime}\underset{X}{\sqcup}X^{\prime\prime}\Bigr)$$

  is a quasi-isomorphism.

The first condition can be restated as saying that the functor $\mathcal{F}$ is symmetric monoidal with respect to the disjoint union and direct sum. Let $\mathbf{H}(\spaces,{\sf Ch}^{\oplus})$ stand for the collection of such functors. The Eilenberg–Steenrod axioms for ordinary homology can then be reformulated as follows.

In particular, each homology theory $\mathcal{F}$ is equivalent to the functor of singular chains with coefficients in the chain complex $\mathcal{F}(\ast)$, which is the value of $\mathcal{F}$ on the singleton: $\mathsf{C}_{\ast}(-,\mathcal{F}(\ast))\simeq\mathcal{F}.$ Further, every natural transformation of homology theories $\mathcal{F}\rightarrow\mathcal{F}^{\prime}$ is determined by the map $\mathcal{F}(\ast)\rightarrow\mathcal{F}^{\prime}(\ast)$.

To adapt this definition to manifolds, we make two substitutions:

1. (1)

   We replace $\spaces^{\sf fin}$ with $\mfld_{n}$, the collection of topological $n$-manifolds, not necessarily closed but with suitably finite covers, with embeddings as morphisms.

2. (2)

   We replace the target ${\sf Ch}^{\oplus}$ by a general symmetric monoidal $\oo$-category $\mathcal{V}$.

As so, we define $\mathbf{H}(\mfld_{n},\mathcal{V})$ to be the collection of all symmetric monoidal functors from $\mfld_{n}$ to $\mathcal{V}$ that satisfy a monoidal version of excision. We arrive at the following analogue of the Eilenberg–Steenrod axioms, for $\mathcal{V}$ a symmetric monoidal $\oo$-category satisfying a technical condition (Definition 3.4).

The $n$-disk algebras appearing in the theorem are equivalent to the $\mathcal{E}_{n}$-algebras of Boardman and Vogt [BV] together with an extra compatible action of the group of automorphisms of $\mathbb{R}^{n}$. Thus, at first glance, this result appears different and more complicated than the Eilenberg–Steenrod axioms because the characterization as $\mathcal{V}$ alone has been replaced by $n$-disk algebras in $\mathcal{V}$. This is for two reasons, each of substance. For one, the object $\mathbb{R}^{n}$ has more structure than a singleton; for instance, automorphisms of $\mathbb{R}^{n}$ form an interesting and noncontractible space, whereas the automorphisms of a singleton is just a point. Secondly, the symmetric monoidal structure of $\mathcal{V}$ is not necessarily the coproduct, so there need not be a canonical map $V\otimes V\to V$ for each object $V\in\mathcal{V}$; this allows for a multiplicative form of homology. Nonetheless, our result does specialize to Eilenberg and Steenrod’s, as we shall see in Example 3.28. In particular, in the example $\mathcal{V}={\sf Ch}^{\oplus}$, these $n$-disk algebras are equivalent to just chain complexes with an action of the automorphisms of $\mathbb{R}^{n}$, and these homology theories are then just ordinary homology twisted by the tangent bundle.

This result has a number of immediate applications and leads to new proofs of known results. For instance, it gives a one line proof that factorization homology of the circle, in the case $\mathcal{V}$ is chain complexes with tensor product, is Hochschild homology. It also gives a new and short proof of the nonabelian Poincaré duality of Salvatore [Sa], Segal [Se3], and Lurie [Lu2]. We give further results and computations in Section 5.

The second motivation for factorization homology comes from mathematical physics. In the various axiomatics for topological quantum field theory after Segal [Se2], one restricts to compact manifolds, possibly with boundary; the locality of a quantum field theory is then reflected in the functoriality of gluing cobordisms. However, it is frequently possible to define a quantum field theory on noncompact manifolds, such as Euclidean space. Since there are more embeddings between general noncompact manifolds than between closed manifolds, one might then expect the axiomatics of this situation to be slightly different were one to account for the structure in which one can restrict a field, or extend an observable, along an open embedding $M\hookrightarrow M^{\prime}$.

Our characterization of homology theories can thereby be seen as an analogue of the cobordism hypothesis for the observables in a topological quantum field theory where one allows for noncompact manifolds and a strong locality principle by which local observables determine global observables.²² 2 There is a slight, but interesting, difference in that the homology theory characterization applies successfully to topological manifolds (as well as piecewise linear or smooth), whereas the cobordism hypothesis requires that the manifolds involved have at least a piecewise linear structure; the absence of triangulations for nonsmoothable topological 4-manifolds appears as a genuine obstruction. Not all topological field theories come from such homology theories, and the question of which do and do not is an interesting one. Costello and Gwilliam use closely related ideas in studying more general quantum field theories which are not topologically invariant, and the setting of their work suggests that perturbative quantum field theories are exactly those amenable to this characterization; see [CG] and [Gw]. The structure of observables of a topological quantum field theory which is not perturbative is better described by a generalization of factorization homology with coefficients given by a stack over $n$-disk algebras [Fra1], e.g., an algebraic variety $X$ whose ring of functions $\mathcal{O}_{X}$ is enhanced to have a compatible structure of an $n$-disk algebra.

Factorization homology and related ideas have recently become the subject of closer study; in addition to Lurie’s originating work, see [Lu2] and [Lu3], and Costello and Gwiliam [CG], see also [An], [GTZ2], [Gw], and [MW]. We expect the theory of factorization homology to be a source of interesting future mathematics and to carry many important manifold invariants. Especially fertile ground lies in low-dimensional topology, in the study of 3-manifold and knot invariants, where invariants stemming from the homology of configuration spaces are already prevalent. This is a source of work joint with Tanaka in [AFT2], where we construct factorization knot homology theories from a $3$-disk algebra with extra structure.

It is a compelling general question as to how much of manifold topology can be captured by factorization homology; this question is closely related to the Goodwillie–Weiss manifold calculus [We]. Factorization homology of $M$ is determined by an object $\mathbb{E}_{M}$, the presheaf of spaces on $n$-disks determined by embeddings into $M$, which is an a priori weaker invariant of $M$ than $M$ itself. $\mathbb{E}_{M}$ encodes the homotopy type of $M$, the homotopy type of all higher configuration spaces ${\conf}_{j}(M)$ of $M$, and the tangent bundles $T\conf_{i}(M)$, as well as coherence data relating these, and it would be very interesting to know when this is sufficient to reconstruct $M$.