Factorization Homology and Manifold Invariants

Factorization homology integrates a disk algebra over a manifold by a homotopy colimit over all embedded disk configurations. Its defining computational principle is tensor excision across collars. For an $E_1$ algebra $A$, integration over the circle gives Hochschild homology. The result is generally a chain complex or higher-categorical object, not a number, an ordinary singular-homology group, or a path integral.

Required background. Locally constant factorization algebras and $E_n$ algebras supplies disk coefficients, and Weiss descent supplies the local-to-global mechanism.

Helpful background. Topological and boundary sectors motivates stratified coefficients, while subfactor index and sector information provides invariants with a different analytic origin.

Integration over disks

Fix a tangential structure $B\to B\operatorname{Top}(n)$ or its smooth analogue, and let $\operatorname{Disk}_n^B$ be the symmetric monoidal category of finite disjoint unions of $B$-framed $n$-disks. A disk algebra is a symmetric monoidal functor

$$A:\operatorname{Disk}_n^B\longrightarrow\mathcal V,$$

where $\mathcal V$ is a symmetric monoidal $\infty$-category admitting the required colimits and whose tensor product preserves them separately. For a $B$-framed manifold $M$,

$$\int_M A =\underset{(U\hookrightarrow M)\in \operatorname{Disk}_n^B/M}{\operatorname{colim}} A(U).$$

The indexing category remembers embeddings and their isotopies, so the colimit is homotopy-coherent. It sums local observables over every finite disk configuration and identifies refinements. This is the symmetric monoidal left Kan extension of $A$ from disks to manifolds, not an integral of a differential form. Ayala and Francis 2015, §§2–3 gives the construction and its universal characterization.

Two normalization checks follow immediately. On the standard framed disk, the over-category has a final object up to homotopy, so $\int_{\mathbb R^n}A\simeq A$. On a disjoint union, symmetric monoidality gives $\int_{M\sqcup N}A\simeq(\int_MA)\otimes(\int_NA)$. Any proposed model that fails either check is not computing the stated left Kan extension. For noncompact $M$, one ordinarily uses an exhaustion by relatively compact opens and verifies that the target colimit agrees with the disk-indexed definition.

Tensor excision

Suppose a manifold is cut along a collared $(n-1)$-manifold,

$$M=M_-\cup_{N\times\mathbb R}M_+.$$

The cylinder $N\times\mathbb R$ has an $E_1$ multiplication in the normal direction. Its factorization homology therefore acts on the two pieces from the appropriate sides, and excision gives

$$\int_M A\simeq \left(\int_{M_-}A\right) \underset{\int_{N\times\mathbb R}A}{\otimes^{\mathbb L}} \left(\int_{M_+}A\right).$$

The tensor product must be derived: ordinary quotienting loses Tor terms that record higher gluing relations. Subject to $\otimes$-presentability, factorization homology is characterized among symmetric monoidal manifold invariants by continuity and this excision property Ayala and Francis 2015, Theorems 3.24 and 3.26.

Tangential structure cannot be added afterward. Integrating a framed algebra over an oriented but unframed manifold requires compatible rotation data or a chosen framing. Boundary components likewise require disk-stratified coefficients, often modules or bimodules, rather than the bulk algebra alone.

The circle and Hochschild homology

Let $A$ be an associative dg algebra, regarded as an $E_1$ algebra. Cut $S^1$ at two points into intervals. The two boundary directions give the enveloping algebra $A^e=A\otimes A^{\mathrm{op}}$, and excision yields

$$\int_{S^1}A \simeq A\underset{A^e}{\otimes^{\mathbb L}}A \simeq HH_\bullet(A).$$

A bar resolution makes the equivalence explicit. Its degree-$k$ term is $A^{\otimes(k+1)}$, with differential that multiplies adjacent factors and a cyclic last face multiplying the final and initial factors. The cyclic face is precisely the information produced by closing an interval into a circle. This calculation is Ayala and Francis 2015, Theorem 3.19.

For $A=M_r(\mathbb C)$, Morita invariance gives $HH_0(A)\cong\mathbb C$ and higher Hochschild homology vanishes. This is a useful check: the circle invariant does not count the $r^2$ matrix entries as independent topological degrees of freedom. For a nonsemisimple algebra, higher groups can survive and record derived self-intersections.

More generally, coefficients can carry defects or boundary conditions. A marked point on a one-manifold is labeled by an $A$-bimodule $M$; cutting at that point replaces the cyclic trace by Hochschild homology with coefficients. This already shows why bulk $E_n$ data alone cannot determine every stratified observable: each new stratum needs a compatible module-type label and gluing action.

The physical owner What Is a Topological Field Theory? explains when a disk algebra comes from observables of a topological theory. The calculation here does not assert that $\int_M A$ is its partition function; state-sum, dualizability, trace, and finiteness data may still be required.

Failure boundaries

Replacing $\otimes^{\mathbb L}$ by an ordinary tensor product can erase higher homology and give the wrong gluing result. Treating a noncompact manifold as one finite collar decomposition can miss compact-support or exhaustion conditions. Calling the output a numerical invariant is false unless a trace or dimension functor has been supplied. Finally, factorization homology is sensitive to smooth or tangential data in its disk category and is not, for arbitrary coefficients, determined only by the weak homotopy type of $M$.

Exercises

Compute $HH_0(A)$ from the cyclic bar differential.

Solution

In degrees one and zero, the differential sends $a_0\otimes a_1$ to $a_0a_1-a_1a_0$. Hence $HH_0(A)=A/[A,A]$, the quotient by the span of commutators.

Why is the algebra on the collar $N\times\mathbb R$ associative?

Solution

Disjoint subcollars can be ordered along $\mathbb R$. Inserting them into a larger collar gives an $E_1$ multiplication, and isotopies between bracketings provide associativity up to coherent homotopy.

References

• Ayala, David, and John Francis. “Factorization Homology of Topological Manifolds.” Journal of Topology 8 (2015): 1045–1084. doi:10.1112/jtopol/jtv028.
• Lurie, Jacob. Higher Algebra. 2017. Author PDF.