Subsection 3.5. (0N62) · ScalingStacks

3.5. Homology theories

We give the following characterization, à la Eilenberg–Steenrod, for factorization homology; this is the central conceptual result of this paper.

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Theorem 3.24. For $\mathcal{V}$ a symmetric monoidal $\oo$ -category which is $\otimes$ -presentable, there is an equivalence

between $\disk^{B}_{n}$ -algebras in $\mathcal{V}$ and homology theories of $B$ -framed $n$ -manifolds with coefficients in $\mathcal{V}$ . This equivalence is implemented by the factorization homology functor $\int$ and the functor of evaluation on $\mathbb{R}^{n}$ .

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Proof. Proposition 3.7 recognizes factorization homology as a symmetric monoidal left Kan extensions, thereby implementing the left adjoint in an adjunction

i_{!}\colon\Alg_{\disk_{n}^{B}}(\mathcal{V})\rightleftarrows\Fun^{\otimes}\bigl(\mfld_{n}^{B},\mathcal{V}\bigr)\colon i^{\ast}~.

The unit of this adjunction is an equivalence because $\disk_{n}^{B}\rightarrow\mfld_{n}^{B}$ is fully faithful, and the Kan extension along a fully faithful functor restricts as the original functor. The counit of this adjunction evaluates on a symmetric monoidal functor $\mathcal{F}$ as a morphism $\int\!A\rightarrow\mathcal{F}$ , where $A=\mathcal{F}_{|\mathbb{R}^{n}}$ is the $\disk^{B}_{n}$ -algebra defined by the values of $\mathcal{F}$ on disjoint unions of $B$ -framed Euclidean $n$ -spaces. It remains to verify that this counit is an equivalence.

Since both $\mathcal{F}$ and $\int\!A$ are symmetric monoidal and agree on $\mathbb{R}^{n}$ , the map $\int_{M}A\rightarrow\mathcal{F}(M)$ is an equivalence for $M$ isomorphic to a disjoint union of Euclidean spaces, $M\cong\bigsqcup_{I}\mathbb{R}^{n}$ . Using induction, we will now see that the values of $\mathcal{F}$ and $\int\!A$ agree on thickened spheres $S^{k}\times\mathbb{R}^{n-k}$ , the base case of $k=0$ just having been shown. In the inductive step, assume the result for $S^{i-1}\times\mathbb{R}^{n-i+1}$ . Choose a standard collar-gluing $S^{i}\xrightarrow{f}[-1,1]$ with $S^{i-1}=f^{-1}(0)\subset S^{i}$ an equator. There results a collar-gluing of $S^{i}\times\mathbb{R}^{n-i}$ . For $\mathcal{F}$ a homology theory, we obtain the equivalence $\int_{S^{i}\times\mathbb{R}^{n-i}}A\simeq\mathcal{F}(S^{i}\times\mathbb{R}^{n-i})$ via the intermediate equvialences

\underset{S^{i}\times\mathbb{R}^{n-i}}{\int}\!A\simeq\underset{\mathbb{R}_{-1}^{i}\times\mathbb{R}^{n-i}}{\int}\!A\underset{\underset{S^{i-1}\times\mathbb{R}^{n-i+1}}{\int}\negthinspace A}{\bigotimes}\ \underset{\mathbb{R}_{+1}^{i}\times\mathbb{R}^{n-i}}{\int}\!A\ \simeq\ \mathcal{F}(\mathbb{R}_{-1}^{i}\times\mathbb{R}^{n-i})\underset{\mathcal{F}(S^{i-1}\times\mathbb{R}^{n-i+1})}{\bigotimes}\mathcal{F}(\mathbb{R}_{+1}^{i}\times\mathbb{R}^{n-i})\simeq\mathcal{F}(S^{i}\times\mathbb{R}^{j})

where the first equivalence is by the $\otimes$ -excision property of factorization homology, the last equivalence is by the assumption that $\mathcal{F}$ is a homology theory, and the middle equivalence is by induction.

We now restrict to the case of $B$ -framed $n$ -manifolds where $n$ is not equal to 4. By the handlebody theory for topological manifolds ([KS] for $n>5$ , [Qu] for $n=5$ , and [Mo] for $n=3$ ) all such manifolds admits a handle decomposition. We now prove the result outside dimension 4 by induction on the handle decomposition. The base case is assured. To verify the inductive step, let $M$ be obtained from $M_{0}$ by adding a handle of index $q+1$ . Therefore $M$ can be expressed as a collar-gluing $M\cong M_{0}\underset{S^{q}\times\mathbb{R}^{n-q}}{\bigcup}\mathbb{R}^{n}$ , where $\mathbb{R}^{n}$ is an open neighborhood of the $(q+1)$ -handle in $M$ . The values $\mathcal{F}$ and $\int\!A$ agree on the three constituent submanifolds of $M$ , and they both satisfy $\otimes$ -excision, so the values $\mathcal{F}(M)\simeq\int_{M}A$ are equivalent.

This leaves the case of topological 4-manifolds, which do not admit handle decompositions in general. Because both $\mathcal{F}$ and $\int_{\!}A$ are symmetric monoidal, we can reduce to the case that $M$ is connected. Now, any connected topological 4-manifold $M$ admits a smooth structure on the complement $M\smallsetminus\{x\}$ of a point $x\in M$ , [Qu]. Consequently, $M\smallsetminus\{x\}$ admits a handle decomposition, which can be constructed from any Morse function on $M\smallsetminus\{x\}$ , and the preceding argument thereby implies the equivalence $\mathcal{F}(M\smallsetminus\{x\})\simeq\int_{M\smallsetminus\{x\}}A$ . Applying the $\otimes$ -excision property to the collar-gluing $M\smallsetminus\{x\}\underset{S^{n-1}\times\mathbb{R}}{\bigcup}\mathbb{R}^{n}\cong M$ , since $\mathcal{F}$ and $\int\!A$ agree on the constituent submanifolds, we obtain the equivalence $\mathcal{F}(M)\simeq\int_{M}A$ . Therefore every homology theory $\mathcal{F}$ for $n$ -manifolds is equivalent to factorization homology with coefficients in $\mathcal{F}(\mathbb{R}^{n})$ .

∎

We record here this technical comparison of factorization homology with different target $\oo$ -categories. (This result is also Proposition 5.5.2.17 of [Lu2].)

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Lemma 3.25. For $G:\mathcal{V}\rightarrow\mathcal{V}^{\prime}$ a symmetric monoidal functor between $\otimes$ -presentable $\infty$ -categories whose restriction to underlying $\infty$ -categories preserves geometric realizations, there is a canonical equivalence $\int\circ G\xrightarrow{\simeq}G\circ\int$ of functors $\Alg_{\disk_{n}^{B}}(\mathcal{V})\rightarrow\Fun^{\otimes}(\mfld_{n}^{B},\mathcal{V}^{\prime})$ .

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Proof. The two functors agree on disjoint unions of Euclidean spaces, so it suffices to check that $G\int A$ is a homology theory with values in $\mathcal{V}^{\prime}$ . This is immediate by the assumption on $G$ . ∎

A result identical to Theorem 3.24 holds for topological $n$ -manifolds with boundary.

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Theorem 3.26. For $\mathcal{V}$ a symmetric monoidal $\oo$ -category which is $\otimes$ -presentable, and for $B\to\BTop(n)$ a map of spaces, there is an equivalence between $\infty$ -categories

from $\disk^{\partial,B}_{n}$ -algebras in $\mathcal{V}$ and homology theories of $B$ -framed $n$ -manifolds with coefficients in $\mathcal{V}$ . This equivalence is implemented by the factorization homology functor $\int$ and evaluation on $B$ -framed Euclidean $n$ -spaces and half-spaces.

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Proof. After Proposition 3.9 it is enough to consider the case where $B$ is equivalent to $\BTop(n)$ , and so we omit it from the notation and discussion. Let $\mathcal{F}:\mfld_{n}^{\partial}\rightarrow\mathcal{V}$ be a symmetric monoidal functor satisfying the $\otimes$ -excision condition, and let $A$ be the restriction of $\mathcal{F}$ to $\disk_{n}^{\partial}$ . For a manifold with boundary $\overline{M}$ , we prove that the canonical morphism $\int_{\overline{M}}A\rightarrow\mathcal{F}(\overline{M})$ is an equivalence. By $\otimes$ -excision applied to the collar-gluing $\partial\overline{M}\times[0,1)\underset{\partial\overline{M}\times\mathbb{R}}{\bigcup}M\cong\overline{M}$ , where $M$ is the interior of $\overline{M}$ , we obtain a diagram in $\mathcal{V}$

in which the vertical morphisms are equivalences. It now suffices to show that the top horizontal morphism is an equivalence. The equivalences of $\int_{M}A\rightarrow\mathcal{F}(M)$ and $\int_{\partial\overline{M}\times\mathbb{R}}A\rightarrow\mathcal{F}(\partial\overline{M}\times\mathbb{R})$ are given by Theorem 3.24; the last equivalence $\int_{\partial\overline{M}\times[0,1)}A\rightarrow\mathcal{F}({\partial\overline{M}\times[0,1)})$ by follows by Theorem 3.24 and Proposition 2.16. ∎

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Remark 3.27. There is an analogous theorem available for stratified spaces proved in §2 of [AFT2], where embeddings are conically smooth and preserve the stratifications. The previous theorems hold if the $\otimes$ -presentable condition is weakened to the condition that the monoidal structure distributes over sifted colimits.

The following example describes how factorization homology specializes to usual homology.

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Example 3.28. Let $\mathcal{V}^{\oplus}$ be either the $\oo$ -category of chain complexes or of spectra equipped with the direct sum monoidal structure. Since every object $V$ therein has an essentially unique morphism $V\oplus V\rightarrow V$ , there is an equivalence $\Alg_{\disk_{n}^{\sf fr}}(\mathcal{V}^{\oplus})\simeq\mathcal{V}$ . The factorization homology of a framed $n$ -manifold $M$ with coefficients in $V$ is then equivalent to $\int_{M}V\simeq\mathsf{C}_{\ast}(M,V)$ , or $\Sigma^{\infty}_{\ast}M\otimes V$ for spectra, the stabilization of $M$ smashed with $V$ . There is a natural functor $\varinjlim\mfld_{n}^{\fr}\rightarrow\spaces^{\sf fin}$ , and this functor is an equivalence because: it is fully faithful since $\varinjlim_{k}\Emb^{\fr}(M\times\mathbb{R}^{k},N\times\mathbb{R}^{k})\rightarrow\Map(M\times\mathbb{R}^{\infty},N\times\mathbb{R}^{\infty})\simeq\Map(M,N)$ is a weak homotopy equivalence for every $M$ and $N$ ; it is essentially surjective since every finite CW complex $X$ can be embedded into $\mathbb{R}^{m}$ for $m$ sufficiently large, and thus it is homotopy equivalent to a framed $n$ -manifold, namely an open regular neighborhood of the embedding. Theorem 3.24 thereby specializes to the formulation of the Eilenberg–Steenrod axioms given in the introduction. If one sets $\mathcal{V}$ to be the opposite ${\sf Ch}^{\op}$ , then one likewise recovers the Eilenberg–Steenrod axioms for cohomology.

To this point, we have worked with topological manifolds and embeddings with the compact-open topology, but other choices could have been made, for instance, to work with smooth manifolds, or to have regarded the embeddings spaces as discrete. We next remark on as to how these alternate choices play out.

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Remark 3.29. One could replace the $\oo$ -category of topological $n$ -manifolds and embeddings with that of smooth $n$ -manifolds and smooth embeddings, $\mfld^{\sm}_{n}$ , or piecewise linear $n$ -manifolds and piecewise linear embeddings, $\mfld^{\sf PL}_{n}$ , and Theorem 3.24 is still valid. The proof, in fact, is even simpler, as the existence of handlebody structures is much easier than in the case of topological manifolds. However, a consequence of smoothing theory [KS] is an equivalence

\mfld^{\sm}_{n}\simeq\mfld_{n}^{{\sf BO}(n)}

between smooth $n$ -manifolds and $\BO(n)$ -framed topological $n$ -manifolds, so long as $n$ is not equal $4$ , so nothing new is obtained by considering smooth or piecewise linear manifolds rather than $B$ -framed topological manifolds. In the case of smooth 4-manifolds, there is still an equivalence $\disk_{4}^{\sm}\simeq\disk_{4}^{{\sf BO}(4)}$ , and so combining the smooth and topological versions of Theorem 3.24 gives an equivalence

\mathbf{H}(\mfld_{n}^{\sm},\mathcal{V})\simeq\mathbf{H}(\mfld_{4}^{{\sf BO}(4)},\mathcal{V})

between homology theories for smooth 4-manifolds and homology theories for ${\sf BO}(4)$ -framed topological 4-manifolds. Since the ${\sf BO}(4)$ -framing on the tangent microbundle is only a very weak measure of a smooth structure in dimension 4 (e.g., there is a single ${\sf BO}(4)$ -framing of $\mathbb{R}^{4}$ , in contrast to the uncountably many smooth structures), this form of factorization homology is not a refined invariant of smooth 4-manifolds.

In the subsequent sections, we will be solely concerned with the homology theories of Definition 3.15. There do, however, exist very interesting functors in $\Fun^{\otimes}(\mfld_{n},\mathcal{V})$ which do not satisfy the $\otimes$ -excision property. In [BFN] the authors were particularly concerned with one such construction: given a stack $X$ over $k$ , one can define a functor $\mfld_{n}\rightarrow{\sf Stacks}\rightarrow\m_{k}$ given by sending a manifold $M$ to the cotensor with $X$ , $M\rightsquigarrow X^{M}$ , and then taking sheaf cohomology of the structure sheaf of this stack. In the case of the circle, $M=S^{1}$ , this gives the Hochschild homology of $X$ : $\mathcal{O}(X^{S^{1}})\simeq\hh_{*}(X)$ . As soon as $X$ is nonaffine, this construction will generically fail to satisfy $\otimes$ -excision. While the cotensor only depends on the homotopy type of $M$ , as we shall see in Proposition 5.1, it has a more refined generalization taking as input a derived stack defined over $n$ -disk algebras, rather than commutative algebras, as in [Fra1].

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Definition 3.30. Let $\mathcal{V}$ be a symmtric monoidal $\infty$ -category which is $\otimes$ -presentable. For a $B$ -framed $n$ -manifold $M$ and a functor $X:\Alg_{\disk^{B}_{n}}(\mathcal{V})\rightarrow\spaces$ , the factorization homology of $M$ with coefficients in $X$ is the object in $\mathcal{V}$

\int_{M}X:=\limit_{A\in{\sf Aff}^{\op}_{/X}}\int_{M}A

where ${\sf Aff}\simeq\Alg_{\disk^{B}_{n}}(\mathcal{V})^{\op}$ is the image of the Yoneda embedding in $\Fun\bigl(\Alg_{\disk_{n}^{B}}(\mathcal{V}),\spaces\bigr)$ .

Intuitively, the object $\int_{M}X$ is $\Gamma(X,\int_{M}\mathcal{O})$ , the global sections of the presheaf on $X$ obtained by applying factorization homology of $M$ to the structure sheaf of $X$ . From the vantage offered by Costello and Gwilliam in [CG], this generalization of factorization homology serves as a candidate for the structure of observables in a topological quantum field theory which is not necessarily perturbative, a direction we will pursue in future work.

David Ayala, John Francis

Original source: arXiv:1206.5522v6