ScalingStacks

We consider now partially oriented $1$-dimensional spaces. We build the theory so that the unoriented part is a manifold, and morphisms are injective on the unoriented part.

We put $Z_{u}=Z-Z_{o}$. Note that $\partial Z_{u}=Z_{u}\cap\overline{Z_{o}}$. Given $z\in Z-Z_{exc}$, we have $|C(z)|=2$ and we define $\iota$ as the unique non-trivial automorphism of $C(z)$.

We denote by $Z^{\operatorname{opp}\nolimits}$ the opposite curve to $Z$ all of whose data coincides with that of $Z$, except for $Z_{o}-Z_{exc}$, whose orientation is reversed.

Fix $n\geq 1$. The $1$-dimensional space $Z=\operatorname{St}\nolimits(2n)$ (cf §7.1.1) can be endowed with a structure of curve by giving ${\mathbf{R}}e^{i\pi r/n}$ the orientation of ${\mathbf{R}}$ for $0\leq r<n$ and setting $Z_{o}=Z$. The involution $\iota$ is defined by $\iota({\mathbf{R}}_{>0}e^{i\pi r/n})={\mathbf{R}}_{<0}e^{i\pi r/n}$.

Note that a composition of morphisms of curves is a morphism of curves. Let $f:Z\to Z^{\prime}$ be a morphism of curves. We have the following statements.

We say that $f$ is strict if $f(Z_{u})$ is closed in $Z^{\prime}_{u}$ and $f(Z_{o})\subset Z^{\prime}_{o}$. Note that this implies that $f(Z_{u})$ is also open in $Z^{\prime}_{u}$.

Let $Z$ be a curve.

If $X$ is a subcurve of $Z$, then $X$ is a curve with $X_{o}=X\cap Z_{o}$, $X_{exc}\subset Z_{exc}$ and $\iota$ is defined on $C_{X}(z)$ as the restriction of $\iota$ on $C_{Z}(z)$, for $z\in X_{exc}$. Note that $X_{u}$ is open in $Z_{u}$.

Equivalently, a subspace $X$ of $Z$ is a subcurve if it is a curve, $X_{o}=X\cap Z_{o}$ and the inclusion map $X\to Z$ is a morphism of curves.

We define an equivalence relation on connected components of $Z-Z_{exc}$: it is the relation generated by $T\sim T^{\prime}$ if there is $z\in Z_{exc}\cap\overline{T}\cap\overline{T^{\prime}}$, $U$ a small open neighbourhood of $z$ and $L\in\pi_{0}(U-\{z\})$ such that $L\subset T$ and $\iota(L)\subset T^{\prime}$.

Let ${\mathcal{E}}$ be the set of equivalence classes of connected components of $Z-Z_{exc}$. Given $E\in{\mathcal{E}}$, let $Z_{E}=\bigcup_{T\in E}\overline{T}$. The subspaces $Z_{E}$ of $Z$ are called the components of $Z$.

A curve has only finitely many components, each of which is a closed subcurve.

If $Z$ is non-singular, then its components are its connected components.

The local structure of a curve is described as follows. Let $z\in Z$. There is an open neighbourhood $U$ of $z$ that is a subcurve of $Z$ and an isomorphism of curves $U\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}X,\ z\mapsto 0$, where $X\subset{\mathbf{C}}$ is one of the following:

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  ${\mathbf{R}}$ viewed as an unoriented manifold, if $z\in Z_{u}-\partial Z_{u}$

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  ${\mathbf{R}}$ where ${\mathbf{R}}_{\geq 0}$ is unoriented and ${\mathbf{R}}_{<0}$ has either of its two orientations, if $z\in\partial Z_{u}$

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  ${\mathbf{R}}$ viewed as an oriented manifold, if $z\in Z_{o}-Z_{exc}$

• •

  $\mathrm{St}(n_{z})$ if $z\in Z_{exc}$.

Let $(\tilde{Z},\tilde{Z}_{o},\tilde{\iota})$ be a curve.

Consider a finite relation $\sim$ on $\tilde{Z}$. We define a curve structure on the $1$-dimensional space $Z=\tilde{Z}/\!\!\sim$.

Let $q:\tilde{Z}\to Z$ be the quotient map. We have $Z_{exc}=q(\tilde{Z}_{exc})\cup\{z\in Z|\ |q^{-1}(z)|>1\}$ (cf §7.1.3). Let $Z_{o}=q(\tilde{Z}_{o})$. The map $q_{|\tilde{Z}_{o}-q^{-1}(Z_{exc})}:\tilde{Z}_{o}-q^{-1}(Z_{exc})\to Z_{o}-Z_{exc}$ is a homeomorphism and we provide $Z_{o}-Z_{exc}$ with the orientation coming from $\tilde{Z}_{o}-q^{-1}(Z_{exc})$. Let $z\in Z_{exc}$. We define $\iota$ on $C(z)$ to make the canonical bijection $\coprod_{\tilde{z}\in q^{-1}(z)}C(\tilde{z})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C(z)$ $\iota$-equivariant. This makes $q$ into a strict morphism of curves.

We define the category of non-singular curves with a finite relation as the category with objects pairs $(Z,\sim)$ where $Z$ is a non-singular curve and $\sim$ is a finite relation on $Z$, and where $\operatorname{Hom}\nolimits((Z,\sim),(Z^{\prime},\sim^{\prime}))$ is the set of morphisms of curves $f:Z\to Z^{\prime}$ such that if $z_{1}\sim z_{2}$, then $f(z_{1})\sim^{\prime}f(z_{2})$.

The next proposition shows that curves can be viewed as non-singular curves with a finite relation.

Note that $Z_{exc}=Z_{q}$ where $q:\hat{Z}\to Z$ is the canonical map. Proposition 7.2.8 shows that non-singular covers exist and are unique up to a unique isomorphism. The following proposition makes this more precise.

We describe here the relation between singular curves and chord (or arc) diagrams.

We define a chord diagram to be to be a triple $({\mathcal{Z}},{\mathbf{a}})$ where

• •

  ${\mathcal{Z}}$ is a closed oriented $1$-dimensional manifold (i.e., a finite disjoint union of copies of $S^{1}$ and $[0,1]$)

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  ${\mathbf{a}}$ is a finite set of pairs of points of $\mathring{{\mathcal{Z}}}$, all of which are distinct.

A chord diagram gives rise to a smooth oriented curve $\tilde{Z}=\mathring{{\mathcal{Z}}}$ with the following relation: given $z\neq z^{\prime}$, we have $z\sim z^{\prime}$ if $\{z,z^{\prime}\}\in{\mathbf{a}}$. We obtain an oriented curve $Z=\tilde{Z}/\!\sim$ and a map $\mu:\bigcup_{\{z,z^{\prime}\}\in{\mathbf{a}}}\{z,z^{\prime}\}\to Z_{exc}$ inducing a bijection ${\mathbf{a}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}Z_{exc}$.

Up to suitable isomorphism, this defines a bijection from chord diagrams to oriented singular curves with $n_{z}\in\{2,4\}$ for all $z$.

When ${\mathcal{Z}}$ is a union of intervals, we recover the notion of (possibly degenerate) arc diagram due to Zarev [Za, Definition 2.1] (compare Example 7.2.11 and [Za, Figures 3 and 4]).

The chord diagrams such that the singular curve $Z$ is connected and $k>0$ correspond to the chord diagrams of [AnChePeReiSu].

Zarev’s definition generalizes that of pointed matched circles due to Lipshitz, Ozsváth and Thurston [LiOzTh1, §3.2]: they correspond to the case where ${\mathcal{Z}}$ is a single interval ($\mathring{{\mathcal{Z}}}$ is obtained from the circle considered in [LiOzTh1] by removing its basepoint).

We define a sutured surface to be a quadruple $(F,\Lambda,S^{+},S^{-})$ where $F$ is a compact oriented surface, $\Lambda$ is a finite subset of $\partial{F}$, and $S^{+}$ and $S^{-}$ are unions of components of $\partial{F}-\Lambda$ such that $\Lambda=\overline{S^{+}}\cap\overline{S^{-}}$ and $\partial{F}-\Lambda=S^{+}\cup S^{-}$ (this is [Za, Definition 1.2] without the topological restrictions). Note that $(F,\Lambda,S^{+},S^{-})$ is determined by the data of $(F,S^{+})$: we have $\Lambda=\overline{S}^{+}-S^{+}$ and $S^{-}=\partial F-\overline{S}^{+}$. A sutured surface is representable by a chord diagram (as we define it) if and only if each component of $F$ (not $\partial{F}$) intersects $S^{+}$ and $S^{-}$ nontrivially.

Let $({\mathcal{Z}},{\mathbf{a}})$ be a chord diagram. We define a sutured surface $(F,\Lambda,S^{+},S^{-})$:

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  the oriented surface $F$ is obtained from ${\mathcal{Z}}\times[0,1]$ by adding $1$-handles at $\{(z,0),(z^{\prime},0)\}$ for all pairs $\{z,z^{\prime}\}$ in ${\mathbf{a}}$

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  $S^{+}=\bigl({\mathcal{Z}}\times\{1\}\bigr)\cup\bigl(\partial{\mathcal{Z}}\times(\frac{1}{2},1]\bigr)$

When ${\mathcal{Z}}$ is a union of intervals, this is Zarev’s construction [Za, §2.1].

Let $Z$ be a singular curve giving rise to $({\mathcal{Z}},{\mathbf{a}})$. The oriented surface $F$ can be identified with $Z\times[0,1]$ and paths in $Z$ give rise to paths in $F$.

The sutured surface $F$ also comes with an arc decomposition: for each $\{z,z^{\prime}\}$ in ${\mathbf{a}}$, we have an arc $\omega_{\{z,z^{\prime}\}}$ with set of end points $\{(z,1),(z^{\prime},1)\}$ in $S^{+}$ corresponding to the $1$-handle added at $\{(z,0),(z^{\prime},0)\}$.

A sutured surface can be viewed as a morphism in the 2d open-closed cobordism category with empty source; if $(F,\Lambda,S^{+},S^{-})$ is a sutured surface, the corresponding open-closed cobordism has target given by $S^{+}$ and non-gluing boundary given by $S^{-}$. See the bottom row of the figure in Example 7.2.13; the targets of these open-closed cobordisms are shown in green and the non-gluing boundary is shown in black.

Let us consider how the end-to-end gluings of chord diagrams covered by our results in §8 can be viewed in terms of open-closed cobordisms. When gluing two distinct intervals of a chord diagram end-to-end, the corresponding sutured surface gets glued as in the top-left picture below: the two intervals marked in blue are glued together to form the top-middle picture. However, we can also consider the top-middle picture as arising from the top-right picture; in this latter case the gluing is an instance of composition (with an open pair of pants) in the open-closed cobordism category. Similarly, when self-gluing the two endpoints of an interval of a chord diagram, the sutured surface gets glued as in the bottom-left picture below, producing the bottom-middle picture; we can also think of the bottom-middle picture as arising from the bottom-right picture, which is another instance of composition in the open-closed category.

One could try to view our constructions as giving part of the structure of an open-closed 2d TQFT valued in a category whose objects are dg 2-categories and whose morphisms are certain dg 2-functors. In particular, this hypothetical open-closed TQFT would assign a dg 2-category of 2-representations of ${\mathcal{U}}$ to an interval. To an open-closed cobordism with empty source, the open-closed TQFT would assign an object of the dg 2-category of the target, encoding the data of a lax multi-2-action of ${\mathcal{U}}$ for the interval components of the target. Our approach doesn’t quite realize that. We associate 2-representations of ${\mathcal{U}}$ to chord diagrams or singular curves rather than directly to surfaces.

One can also consider the extent to which such a theory would extend to a point. Things are considerably simpler for the decategorified version of the theory, where one sees many relationships with other work on 3d TQFTs; this will be addressed in more detail in a follow-up paper [ArMa].