Legendrian non-squeezing via microsheaves

In Section 2 we review some basics of contact geometry, and collect a few facts about fronts of generic Legendrians. Section 3 is devoted to a (brief) proof of three dimensional Legendrian non-squeezing using our approach, intended to serve as a summary of the strategy unburdened by the additional technicality of the microsheaf formalism. In the process we introduce some new variants of rulings for Legendrian knots beyond the $\mathbb{R}$ valued jet.

Section 4 is devoted to collecting algebraic and categorical preliminaries for our discussion of microsheaves, which are introduced in Section 5. In Section 6 and Section 7 we introduce the relevant forms of microsheaves that we consider in order to approach Legendrian non-squeezing in our setting, and collect some local computations useful for getting a handle on objects. In Section 8 we prove Theorem 1.2 as sketched above. Finally, in Section 9 we use the results of the previous section to prove our main result, Theorem 1.1.

Acknowledgments The author would like to thank Yasha Eliashberg for introducing him to the work of Chekanov–Pushkar, and many helpful discussions, and David Nadler for suggesting the use of microsheaves to go beyond dimension 3. The author also thanks Maya Sankar for showing him how to use Tikz to generate most of the figures that appear in this paper.

Let $(W,d\lambda)$ be a Liouville manifold. Its pre-quantization is the contact manifold

It carries a standard contact form $\alpha=dt+\lambda$ whose Reeb flow is rotation in the $S^{1}$ factor, and is equipped with a canonical projection $\varpi:\widehat{W}\rightarrow W$ which forgets the $S^{1}$ factor.

If $\iota:L\rightarrow W$ is a smooth Lagrangian embedding, such that $\iota^{*}\lambda\in H^{1}(L;\mathbb{Z}),$ then for any choice of $l_{0}\in L$ the map

is a smooth Legendrian embedding such that $\varpi\circ\tilde{\iota}=\iota.$ In this case, we say that $\Lambda:=\tilde{\iota}(L)$ is a Legendrian lift of $L.$ Note that lifts of such $L$ are unique up to Legendrian isotopy: all are related by some (global) rotation of the $S^{1}$ factor, which is a Reeb flow as described above. Moreover, if $L^{\prime}$ is Hamiltonian isotopic to $L,$ then any lift $\Lambda^{\prime}$ of $L^{\prime}$ is Legendrian isotopic to a (thus any) lift $\Lambda$ of $L.$

In the case $W=T^{*}M,$ a cotangent bundle equipped with its tautological 1-form, there is a canonical contactomorphism

between the pre-quantization of $W,$ and the circle-valued 1-jet space of $M.$

We will habitually identify $\widehat{\mathbb{C}}^{n}$ with $\widehat{T^{*}\mathbb{R}^{n}}\cong J^{1}(\mathbb{R}^{n};S^{1}).$ This is justified by an application of Gray’s theorem: Any Legendrian isotopy of a closed Legendrian can be realized by a compactly supported contactomorphism. The two contact forms induced by Liouville homotopic Liouville forms are isotopic through contact forms, and thus yield contactomorphic contact manifolds on any compact subset.

Let $(Y,\xi)$ be a contact manifold. A Legendrian fibration is a smooth submersion $p:Y\rightarrow F$ whose fibers are Legendrian submanifolds. We will refer to the base $F$ as the front space and the map $p$ as the front projection. When $Y$ is a 1-jet, $Y=J^{1}(X;Z)$ for some $Z\in\left\{\mathbb{R},\S^{1}\right\},$ and $p:Y\rightarrow X\times Z$ is the standard Legendrian fibration by cotangent fibers, there is a further projection $\Pi:Y\rightarrow\mathfrak{B}:=X.$ We refer to this as the bifurcation projection, and to the base $\mathfrak{B}$ as the bifurcation space. The intermediate projection from the front to bifurcation space is denoted $\pi:F\rightarrow\mathfrak{B}$. While fixing further notation (as in the rest of this section) we will be careful to distinguish between $\Lambda$ and its image under the front or bifurcation projection, but later on we may conflate these, and denote all by $\Lambda$ when it is clear where we wish to view it from context.

A subspace $A\subset X$ is called smoothly stratified if it is equipped with a filtration $A_{0}\subset A_{1}\subset\ldots\subset A_{N}=A$ such that $A_{i}\setminus A_{i-1}$ is a smooth manifold for every $i.$

Let $Y=J^{1}(X;Z)$ be a standard Legendrian fibered jet space, $p,F,\Pi,\mathfrak{B}$ as in Section 2.2. The frontal singular locus $\tilde{\mathbb{X}}_{\Lambda}$ of $p(\Lambda)$ is defined to be the set of points $x\in F$ where either $Dp$ drops rank, or over which $p$ fails to be injective. By standard jet transversality, and the abundance of Legendrian perturbations, $\tilde{\mathbb{X}}_{\Lambda}$ is a smoothly stratified, codimension at least 2 subspace of $F,$ for generic $\Lambda.$ Observe that $\tilde{\mathbb{X}}_{\Lambda}$ is automatically transverse (in the sense of smoothly stratified spaces) to the fibers of $\pi,$ since $\Pi(\Lambda)$ is on the complement of $\tilde{\mathbb{X}}_{\Lambda}$.

The bifurcating locus, $\mathbb{X}_{\Lambda},$ of $\Lambda$ is defined to be $\pi(\tilde{\mathbb{X}}).$ Keeping in mind the automatic transversality of $\tilde{\mathbb{X}}_{\Lambda},$ we conclude that, after further generic perturbation, $\mathbb{X}_{\Lambda}$ is also smooth, stratified, and now codimension 1. Note that this stratification is not quite the same as that on $\tilde{\mathbb{X}}_{\Lambda},$ but is induced by the projection of a refinement of this stratification, where the additional strata are precisely the pre-images of the subset where $\pi\!\!\mid_{\tilde{\mathbb{X}}_{\Lambda}}$ fails to be injective, which are of positive codimension in $\mathbb{X}_{\Lambda}$. Compatibility of these stratifications induces a partition into the pointlike, and XX loci, the latter defined to be the sub-stratified sub-space of $\mathbb{X}_{\Lambda}$ over which $\pi$ is not injective, with the former comprising the rest.

The following lemma will be useful later:

We discuss here the geometric aspects of a polarization for a contact manifold, and corresponding notion of (twisted) Maslov potentials. We use the word twisted to emphasize that we allow ourselves work with respect to a non-standard polarization. We will revisit these objects in greater detail (and generality) in Section 5.2.

A (co-oriented) contact manifold $(Y^{2n+1},\xi)$ carries a canonical conformal symplectic structure on the contact distribution $\xi\subset TY.$ In particular, $\xi$ can be regarded as a rank $2n$ real symplectic vector bundle over $Y.$ To such a vector bundle there are two associated fiber bundles, $\operatorname{LGr}(Y),\operatorname{LGr}^{+}(Y),$ with fibers $\operatorname{LGr}_{n}$ and $\operatorname{LGr}^{+}_{n},$ respectively the ordinary and oriented Lagrangian Grassmannians of the symplectic vector space of real dimension $2n$. The latter double covers the former by construction, and so induces a double covering on the level of the Lagrangian–Grassmann bundles. A polarization (resp. orientable polarization) of $Y$ consists of a section of $\operatorname{LGr}(Y)$ ($\operatorname{LGr}^{+}(Y)$), i.e. a smoothly varying choice of (oriented) Lagrangian subspace of $\xi$ over each point of $Y.$ When $(Y,\xi)$ admits an (orientable) polarization it is said to be (orientably) polarizable.

Now suppose $(Y,\xi)$ is polarizable, and fix a polarization $\tau:Y\rightarrow\operatorname{LGr}(Y).$ A Legendrian submanifold $\Lambda\subset Y$ comes with a canonical Gauss map, $G_{\Lambda}:\Lambda\rightarrow\operatorname{LGr}(Y)$ given by its tangent space. If $\Lambda$ is orientable, then a choice of orientation yields a lift of $G_{\Lambda}$ to $\operatorname{LGr}^{+}(Y).$ In this case, fixing a basepoint $x_{0}\in\Lambda$ (assume, without loss of generality, that $G_{\Lambda}(x_{0})\pitchfork\tau(x_{0})$) we obtain a map $\pi_{1}(\Lambda,x_{0})\rightarrow\mathbb{Z},$ defined by evaluating $\mu_{\xi}$ on the image of $(G_{\Lambda})_{*}\pi(\Lambda,x_{0})$ under the Hurewicz map. Since $G_{\Lambda}$ admits a lift to the double cover, the image of this map lands in $2\mathbb{Z},$ and so can be characterized completely by an even integer $n^{\tau}_{\Lambda},$ the Maslov number of $\Lambda,$ which generates this image.

Now if $x\in\Lambda$ is any other point, lifting $\mu_{\xi}$ in the long exact sequence of the pair $(\operatorname{LGr}(\Lambda),G_{\Lambda}(x_{0})\cup G_{\Lambda}(x))$ over $\mathbb{Z}/n^{\tau}_{\Lambda}\mathbb{Z},$ and evaluating on any class in $H_{1}(\Lambda,x_{0}\cup x;\mathbb{Z})$ induced by a path $x_{0}\mapsto x$ yields a well defined element of $\mathbb{Z}/n^{\tau}_{\Lambda}\mathbb{Z}.$ We define the Maslov potential twisted by $\tau$ based at $x_{0}$ on $\Lambda$ to be the $\mathbb{Z}/n^{\tau}_{\Lambda}\mathbb{Z}$ valued function on $\Lambda$ obtained via this procedure, and denote it by $m_{(\Lambda,x_{0})}^{\tau}$. Note that, by Arnol’d’s geometric characterization of the universal Maslov class ([Arnold-maslov]) $m_{(\Lambda,x_{0})}^{\tau}$ is locally constant over the region where $G_{\Lambda}\pitchfork\tau$ (where the images are regarded as Lagrangian subspaces). We leave as an exercise to verify that when $Y=J^{1}(\mathbb{R};\mathbb{R}),$ $\tau$ the polarization induced by cotangent fibers, and $\Lambda\subset Y$ some front-generic Legendrian knot, $n^{\tau}_{\Lambda}$ recovers (up to sign, twice the) the classical rotation number of $\Lambda,$ and our notion of twisted Maslov potential reduces to the usual notion as defined in, say, [cp2005].

Note that, in the end, all relevant objects were pulled back from the stable Lagrangian–Grassmannian, so in the above description we could instead have asked only for a stable polarization, i.e. a section of the $U/O$ bundle obtained by the limit of stable trivializations of $\xi.$ Similarly, a Legendrian carries a stable Gauss map obtained by stabilizing $G_{\Lambda},$ and the remainder of the story is as before. Note that, in practice, one can always truncate at a finite stage in the limit defining $U/O,$ so the above “local constancy on the transverse locus” can still be made sense of directly. Indeed, we will use the fact that any stable polarization is already defined after stabilizing by a finite rank trivial bundle in our definition of microsheaves later on.

In dimension 3 the singularity stratification is particularly simple: generically $\mathbb{X}_{\Lambda}\subset\mathfrak{B}$ is a finite collection of points, each corresponding to a transverse crossing of two smooth strands, or a fold singularity. Moreover a generic Legendrian isotopy features only the following three pointlike codimension 2 front singularities, which are captured by Legendrian Reidemeister moves:

• •

  R1, the swallowtail singularity

• •

  R2, the intersection (in the front) of a fold with a smooth point

• •

  R3, a triple intersection of smooth strands.

The bifurcation singularities consist of these, alongside the coincidence (i.e. occurring in the same fiber of $\pi$) of two generic front singularities. These describe the XX locus, and consist of:

• •

  $\mathfrak{XX}$, the coincidence of two simple crossings.

• •

  $\mathfrak{XC}$, the coincidence of a simple crossing and a fold.

• •

  $\mathfrak{CC}$, the coincidence of two folds.

Since these are all the singularities which occur in a generic 1-parameter family, in order to check Legendrian isotopy invariance of an object defined in terms of a front, it suffices to verify that it persists under ambient fibered isotopy, and these six bifurcations. This is the strategy employed by Chekanov–Pushkar to show invariance of the count of normal rulings (i.e. positive, Maslov, pseudo-involutions) [cp2005]. This will also be our strategy in Section 3.

We begin with a review of rulings for Legendrians in $J^{1}(\mathbb{R};\mathbb{R}),$ i.e. upwardly co-oriented fronts in $\mathbb{R}\times\mathbb{R}.$ Let $\Lambda\subset J^{1}(\mathbb{R};\mathbb{R})$ be a front-generic Legendrian. Then, as described in Section 2.5, $\tilde{\mathbb{X}}_{\Lambda}$ is discrete, finite, and partitioned into $\Lambda_{\mathfrak{X}}\sqcup\Lambda_{\mathfrak{C}},$ the crossing and fold loci respectively. Denote the complement of $\tilde{\mathbb{X}}_{\Lambda}$ in $p(\Lambda)$ (i.e. the smooth points of the front) by $\Lambda_{\mathfrak{S}}.$

Now over each $x\in\mathfrak{x}$ the fiber of the bifurcation projection $\pi^{-1}(x)$ is a disjoint union of points.

This last condition imposes that the two smooth strands near a point of $\Lambda_{\mathfrak{X}}$ may not be exchanged by $\iota.$

Notice that $\mathbb{X}_{\Lambda}$ partitions $\mathfrak{B}\cong\mathbb{R}$ into a collection disjoint open intervals: $\mathfrak{B}\setminus\mathbb{X}_{\Lambda}=\sqcup_{i=1}^{N}I_{i},$ without loss of generality ordered such that $\overline{I}_{i}\cap\overline{I}_{j}=\emptyset$ unless $i=j\pm 1.$ So for any collection $\left\{x_{i}\right\}_{i=1}^{N}$ with $x_{i}\in I_{i}$, by continuity a pseudo-involution on $p(\Lambda)$ is determined by its action on the fibers $\pi^{-1}(x_{i}).$

Suppose $\mathfrak{x}\in\Lambda_{\mathfrak{X}}$ lies over the intersection $\overline{I}_{i}\cap\overline{I}_{i+1}.$ Let $\sigma_{i}^{\pm}$ denote the two strands of $\Lambda_{\mathfrak{S}}\cap\pi^{-1}(I_{i})$ whose closure contains $\mathfrak{x}$ (ordered such that the short path along the co-orientation direction near $\mathfrak{x}$ runs from $\sigma_{i}^{-}$ to $\sigma_{i}^{+}$), and let $\tau_{i}^{\pm}$ denote the component in $\Lambda_{\mathfrak{S}}$ of their respective images under $\iota.$ Similarly let $\sigma_{i+1}^{\pm}$ denote the strands over $I_{i+1}$ with common boundary $\mathfrak{x},$ ordered as before and $\tau_{i+1}^{\pm}$ their images. By continuity, either $\tau_{i}^{\pm}=\tau_{i+1}^{\pm}$ or $\tau_{i}^{\pm}=\tau_{i}^{\mp}.$ In the former case the crossing $\mathfrak{x}$ is called switching, and in the latter it is called non-switching.

Fix some $x_{j}\in I_{j},j\in\left\{i,i+1\right\}.$ Then

define a pair of embeddings of $S^{0}$ into $\pi^{-1}(x_{j})$ for each $j.$ Suppose $\mathfrak{x}$ is switching. Then we call $\mathfrak{x}$ positive if the $\mathbb{Z}/2$ linking numbers $\operatorname{lk}(S_{j}^{+},S_{j}^{-})\equiv 0$ for both $j\in\left\{i,i+1\right\}.$

Let $\mu$ be a $\mathbb{Z}/2$ valued Maslov potential¹¹1The notion of Maslov potential is unambiguous in this setting, since the jet is contractible. on $p(\Lambda).$ A switching crossing $\mathfrak{x}$ is called Maslov if $\mu(\sigma_{j}^{+})=\mu(\sigma_{j}^{-}).$ Note that if $\mathfrak{x}$ is Maslov for a particular Maslov potential, it is so for every such choice.

The main technical result of [cp2005] shows that the number of normal rulings of a front generic Legendrian link $\Lambda\subset J^{1}(\mathbb{R};\mathbb{R})$ is a Legendrian isotopy invariant. The proof proceeds via a case-by-case analysis of the codimension 2 bifurcation singularities that appear in generic 1-parametric Legendrian isotopies, showing that, given a normal ruling of the front before the bifurcation, there exists a unique normal ruling of the front after the bifurcation which preserves the difference between the number of switching crossings, and folds:

Now suppose $\Lambda\subset J^{1}(\mathbb{R};S^{1}),$ equipped with the fiber polarization. As before, we assume that $p(\Lambda)$ is front-generic, and so has only simple crossings and folds. The definition of (positive, Maslov) pseudo-involution, and thus normal ruling, above may be implemented verbatim. We call these objects normal disk rulings, and the set of all such $\mathrm{R}^{\mathbb{C}}(\Lambda).$ The count of these is invariant by repeating the arguments of [cp2005], and denoted by $\#\mathrm{R}^{\mathbb{C}}(\Lambda).$

There is another reasonable definition of ruling for Legendrians in the $S^{1}$-valued jet. Let $\iota$ be a weak pseudo-involution on $p(\Lambda).$

A circular ruling of $\Lambda$ is a weak pseudo-involution $\iota$ on $p(\Lambda),$ together with an admissible marking $\Gamma$.

If a circular ruling $(\iota,\Gamma)$ satisfies condition (iv) of Definition 3.1 at $x\in\Lambda_{\mathfrak{X}},$ then we can still categorize $x$ as either switching or non-switching as before. In fact, even for those $x\in\Lambda_{\mathfrak{X}}$ which are not in the image of $\iota$ we can still recover the notions of switching and non-switching using the marking data $\Gamma:$ A crossing in the front of a Legendrian link admits a unique (up to isotopy) horizontal smoothing. A crossing point is called switching if the marking data of $(\iota,\Gamma)$ away from the crossing extends over the Legendrian resolution of $\mathfrak{x},$ and non-switching otherwise.

As before, a ruling of $p(\Lambda)$ can only be guaranteed to persist under Legendrian isotopy under further assumptions on its behavior at switching points, i.e. if it satisfies some analog of the positivity condition described above. The correct notion here is a slight relaxation of positivity in the sense above, and can be thought of as “positivity on the universal cover:” Let $x\in\Lambda_{\mathfrak{X}}$ be a switching crossing of a circular ruling $(\iota,\Gamma),$ and suppose $x\in\operatorname{im}(\iota).$ Let $\sigma_{L}^{\pm},\sigma_{R}^{\pm}$ denote the left and right pairs of smooth strands of $p(\Lambda)$ limiting to $x,$ and $[\gamma_{\sigma_{L}^{\pm}}],[\gamma_{\sigma_{R}^{\pm}}]$ their markings. Fix a lift $\tilde{U}_{x}$ of a neighborhood $U_{x}$ of $x$ to the universal cover of the front $\tilde{F}.$ Denote by $[\tilde{\gamma}_{\sigma_{\bullet}^{\pm}}]$ ($\bullet\in\left\{L,R\right\}$) the lift of a marking path to the universal cover which is based in $\tilde{U}_{x}.$

By convention, if $x\not\in\operatorname{im}(\iota),$ we still call $x$ positive.

With this in hand we define a normal circular ruling as a pair $(\iota,\Gamma)$ of weak pseudo-involution and admissible marking which is positive and Maslov at switching crossings, and denote the set of all such by $\mathrm{R}^{\mathbb{S}}(\Lambda).$ Note that such a ruling has no switching points in the complement of the image of $\iota,$ since these would necessarily be non-Maslov.

With our two ruling counts in hand we are ready to tackle the non-squeezing problem. The main result of this section is the following elementary analog of Theorem 1.2:

The simplest example of a non-squeezable Legendrian knot is the lift of an embedded loop in $T^{*}\mathbb{R}$ enclosing area 1, a front for which is drawn below:

This admits exactly one disk ruling, which simply pairs the strands in each fiber, and no circular ruling, since any marking cannot be trivial at both ends. By passing to a cover, a lift of any loop enclosing integral area can be made to be of this form, and so we conclude:

This is the 3-dimensional case of Theorem 1.1. The remainder of this article is devoted the high dimensional version.

We briefly recall the essentials of dg-categories, and their homotopy theory. For details we refer to [Toen-lectures, Keller-dgcats, Toen-Morita]. A (small) dg-category $\mathcal{C}$ over $R$ consists of the data of

• •

  A set of objects $\operatorname{Ob}(\mathcal{C}).$

• •

  For each pair $x,y\in\operatorname{Ob}(\mathcal{C})$ an object of $C(R),$ $\hom(x,y).$

• •

  For each triple $x,y,z\in\operatorname{Ob}(\mathcal{C})$ a composition ($C(R)$-)morphism

  $$\mu_{x,y,z}:\hom(x,y)\otimes\hom(y,z)\rightarrow\hom(y,z)$$

• •

  For each $x\in\operatorname{Ob}(\mathcal{C})$ a unit morphism $e_{x}:R\rightarrow\hom(x,x).$

Such that the diagrams

and

commute for any $x,y,z,w\in\operatorname{Ob}(\mathcal{C}).$ In the sequel, we will leave the choice of ground ring implicit, and simply write dg-category.