AALGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS
LAURENT C ˆOT´E
Abstract.
Motivated rational homotopy theory, we consider a class of Liouville manifoldswhich we call algebraically hyperbolic . These Liouville manifolds are characterized by theproperty that (the pre-triangulated closure of) their wrapped Fukaya category has exponen-tial algebraic growth, in a sense which we define. On the one hand, we prove that the classof algebraically hyperbolic Liouville manifolds is closed under various natural geometric op-erations, which allows one to construct many examples. On the other hand, under certainnon-degeneracy hypotheses, we prove that Reeb flows on the ideal boundary of an algebraicallyhyperbolic Liouville manifold are always chaotic. This leads to various concrete applications,including new constructions of Weinstein manifolds which are not algebraic varieties and ofcontact manifolds whose Reeb flows have positive topological entropy. In dimension 3, thelatter property implies a strong form of the Weinstein conjecture. Finally, we consider alge-braically hyperbolic Weinstein manifolds from the perspective of their skeleton. In particular,we describe concrete criteria at the level of the skeleton which ensure that a Weinstein mani-fold is algebraically hyperbolic. This provides some connection between stratified-topologicalproperties of skeleta of Weinstein manifolds and Reeb dynamics on their ideal boundary. Introduction
Overview. A Liouville manifold ( X, λ ) is an open symplectic manifold which is modelednear infinity on the symplectization of a contact manifold ( ∂ ∞ X, ξ ∞ ). Liouville manifolds areimportant objects of study in symplectic topology and nearby fields.An important invariant of Liouville manifolds is their wrapped Fukaya category. Given aLiouville manifold ( X, λ ), its wrapped Fukaya category W ( X ) is an A ∞ category whose objectsare certain suitably decorated Lagrangian submanifolds. The A ∞ operations are defined viaFloer theory by counting solutions to elliptic partial differential equations. It is often morenatural to consider Tw W ( X ), the A ∞ category of twisted complexes over W ( X ), which is (amodel for) the pre-triangulated closure of W ( X ). In this paper, the wrapped Fukaya categorywill always be defined over a field k of characteristic zero.The purpose of this paper is to study a special class of Liouville manifolds which we call algebraically hyperbolic . To set the stage, let us say that a k -linear category C has exponentialalgebraic growth if there exists a finite collection of morphisms { x , . . . , x j } such that thedimension of the vector space spanned by composable words of length at most n grows atleast exponentially with n (see Definition 3.2). An A ∞ category is said to have exponentialalgebraic growth if its cohomology category does. A Liouville manifold ( X, λ ) is now definedto be algebraically hyperbolic if Tw W ( X ) has exponential algebraic growth.The class of algebraically hyperbolic Liouville manifolds contains the cotangent bundlesof all hyperbolic manifolds and the cotangent bundles of many (conjecturally all) rationallyhyperbolic spin manifolds. It is also closed under various natural geometric operations, includ-ing subcritical surgeries, taking products with any non-degenerate Liouville manifold havingnonzero symplectic cohomology, and certain critical surgeries and gluings. This allows one toconstruct a myriad of examples. Date : November 11, 2020. a r X i v : . [ m a t h . S G ] N ov LAURENT C ˆOT´E
It is however not the case that all Liouville manifolds are algebraically hyperbolic. Any Li-ouville manifold with vanishing wrapped Fukaya category is certainly not algebraically hyper-bolic. Cotangent bundles of rationally elliptic manifolds (such as spheres, complex projectivespaces, etc.) also fail to be algebraically hyperbolic. More interestingly, we will show thatsmooth complex affine varieties are never algebraically hyperbolic.An important goal in the study of Liouville manifolds is to clarify the interplay betweenthe global symplectic topology of a given manifold (
X, λ ), the contact topology of its idealboundary ( ∂ ∞ X, ξ ∞ ), and the stratified topology of its skeleton Skel(
X, λ ). Algebraicallyhyperbolic Liouville manifolds turn out to exhibit several distinguishing features both at thelevel of their contact boundary and of their skeleton.If (
X, λ ) is algebraically hyperbolic and
Weinstein up to deformation (or more generallyif W ( X ) is generated by cylindrical Lagrangians whose ideal boundary is diffeomorphic to asphere), we prove that any Reeb flow on ( ∂ ∞ X, ξ ∞ ) has positive topological entropy. Thetopological entropy is a well-known invariant of dynamical systems, and flows having positivetopological entropy are typically interpreted as being chaotic. In dimension 3, any flow withpositive topological entropy has the property that the number of geometrically distinct orbitsgrows exponentially in time. Thus, ideal boundaries of algebraically hyperbolic 4 manifoldssatisfy a strong version of the Weinstein conjecture.Turning now to skeleta, we describe certain geometric conditions on an arbitrary stratifiedset which imply that any Weinstein thickening (if it exists) must be algebraically hyperbolic.We also explain how various geometric operations which preserve algebraic hyperbolicity canbe interpreted at the level of the skeleton. This in some sense provides a recipe for constructing(some) skeleta of algebraically hyperbolic manifolds. Combining the above perpsectives shedssome light on the relationship between skeleta and Reeb dynamics. In particular, one canconstruct a large class of singular skeleta whose Weinstein thickenings have chaotic Reebdynamics.1.2. Rational homotopy theory.
Our original motivation for considering algebraically hy-perbolic Liouville manifolds comes from rational homotopy theory. Recall that a simply-connected, finite CW complex is said to be rationally hyperbolic if the sum of the loop spaceBetti numbers(1.1) b ≤ n (Ω ∗ X ) := (cid:88) i ≤ n dim H i (Ω ∗ X ; Q )grows at least exponentially in n .To make the connection to symplectic topology, recall that if M is a closed manifold, then acelebrated result of Abbondandolo and Schwarz [1] gives an isomorphism of associative algebras(1.2) HW • ( F p , F p ) = H −• (Ω p M ; L Q ) , where the left-hand side is the wrapped Floer cohomology of a fiber with Q -coefficients andthe right-hand side is the Pontryagin algebra twisted by a local system which is trivial if M is spin (see Section 7.1). Thus, the notion of rational hyperbolicity for spin manifolds canbe rephrased in purely symplectic terms. It is then natural to look for generalizations of thisnotion which make sense for arbitrary Liouville manifolds.A general Liouville manifold does not have a distinguished cotangent fiber. However, forWeinstein manifolds, it is known that the wrapped Fukaya category is generated by cocores The skeleton of a Liouville manifold is defined as the set of points which do not escape to the ideal boundaryunder the Liouville flow. It is sometimes less frighteningly called the core or the spine
LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 3 and Z -graded (under some mild topological assumptions), so we can consider the algebra(1.3) (cid:77) i,j ∈{ ,...,m } HW • ( L i , L j ) , where L , . . . , L m are the cocores. It is then tempting to define such a Weinstein manifold as“hyperbolic” if the rank of (1.3) grows exponentially in terms of the grading. However, thisdefinition is somewhat problematic, because it can happen that the rank of HW • ( L i , L j ) isinfinite in a single degree. A hint towards a possibly better definition, which also makes sense for arbitrary Liouvillemanifolds, is provided by a well-known conjecture in rational homotopy theory due indepen-dently to Avramov and F´elix (see [20, Sec. 39.4] or [19, Sec. 1]). This conjecture, which hasbeen verified for a wide range of examples, states that a simply connected, finite CW complex X is rationally hyperbolic if and only if its rational homotopy Lie algebra π • ( X ) ⊗ Q containsa free graded Lie algebra on two generators.By the Milnor–Moore theorem, the universal enveloping algebra of π • ( X ) ⊗ Q is isomorphicto the Pontryagin algebra H • (Ω ∗ X ; Q ). Using this, one can show (see Lemma 2.18) that theAvramov-F´elix conjecture implies that X is rationally hyperbolic if and only if the Pontryaginalgebra H • (Ω ∗ X ; Q ) has exponential algebraic growth in the sense we have defined. In particu-lar, if true, this conjecture would indicate that the exponential growth of the Betti numbers ofa rationally hyperbolic space should really be viewed as a manifestation of a more fundamentalfact, namely the exponential algebraic growth of the Pontryagin algebra H • (Ω ∗ X ; Q ).From this perspective, it becomes natural to define a Liouville manifold ( X, λ ) as “hyper-bolic” if W ( X ) has exponential algebraic growth. This is essentially the definition we haveadopted in this paper, except that we only demand that the pre-triangulated closure Tw W ( X )has exponential algebraic growth, a weaker condition which seems more suited to computa-tions. Remark . There is also a notion of rational ellipticity in rational homotopy theory: asimply-connected, finite CW complex is rationally elliptic if it is not rationally hyperbolic.Remarkably, this turns out to be equivalent to the property that the sum of the loop space Bettinumbers b ≤ n ( X ) grows at most polynomially. It could be interesting to consider generalizationsof rational ellipticity to Liouville manifolds, perhaps with the view of finding possible analogsof the elliptic vs. hyperbolic dichotomy in rational homotopy theory, but we do not pursue thisquestion in this paper.1.3. Constructions of algebraically hyperbolic Liouville manifolds.
The simplest ex-amples of algebraically hyperbolic Liouville manifolds are cotangent bundles of manifoldswhose based loop space homology (with suitably twisted coefficients) has exponential algebraicgrowth. Given a closed, connected manifold M with a basepoint ∗ , let w ( M ) ∈ H ( M ; Z / L be the Z -valued local system on Ω ∗ M whosemonodromy along a loop is defined by evaluating w ( M ) on the corresponding torus and set L k := L ⊗ Z k . Definition 1.2.
Let E be the class of all closed, connected manifolds M with the propertythat H −• (Ω ∗ M ; L k ) has exponential algebraic growth.We will see in Section 7.1 that the class E contains all orientable surfaces of genus at least2, all non-orientable surfaces of non-orientable genus at least 2, and many (conjecturally all)rationally hyperbolic spin manifolds. Indeed, this happens even for the cotangent fiber in T ∗ S ; any reasonable definition of “hyperbolicity”should presumably exclude this example! LAURENT C ˆOT´E
The following proposition is an immediate corollary of (1.2).
Proposition 1.3. If M ∈ E , then ( T ∗ M, λ ) is algebraically hyperbolic. Next, we consider the behavior of algebraically hyperbolic Liouville manifolds under prod-ucts. The following proposition, proved in Section 7.3, follows directly from the K¨unnethformula for wrapped Fukaya categories proved by Ganatra–Pardon–Shende [30, Thm. 1.5].
Proposition 1.4 (Products) . Let ( X , λ ) and ( X , λ ) be Liouville manifolds and let ( X, λ ) =( X , λ ) × ( X , λ ) . Suppose that the following two conditions hold: • ( X , λ ) or ( X , λ ) is algebraically hyperbolic; • W ( X ) and W ( X ) are both nonzero (i.e. not equivalent to the zero category).Then ( X, λ ) is algebraically hyperbolic. In case ( X , λ ) and ( X , λ ) are non-degenerate in the sense of [28, Def. 1.1] (e.g. Weinstein),it follows from work of Ganatra [28, Thm. 1.1] (and the fact that any non-zero category has non-vanishing Hochschild cohomology) that the second condition could equivalently be replaced bythe condition that SH • ( X ) (cid:54) = 0 and SH • ( X ) (cid:54) = 0. Example 1.5.
Proposition 1.4 can be used to exhibit algebraically hyperbolic Weinstein man-ifolds which do not contain a closed exact Lagrangian. For any n ≥
6, Abouzaid and Seidel[3, Thm. 1.3] constructed infinitely many examples of Weinstein manifolds of dimension 2 n whose symplectic cohomology with Z / Z -coefficients (and hence also with k -coefficients) is nonzero.Let X be such a Weinstein manifold and let Y be an algebraically hyperbolic Liouville man-ifold. Then Proposition 1.4 implies that X × Y is algebraically hyperbolic. On the other hand,the K¨unneth theorem for symplectic cohomology (see [42]) implies that SH • ( X × Y ; Z /
2) = SH • ( X ; Z / ⊗ SH • ( Y ; Z /
2) = 0 . It follows by Viterbo restriction for symplectic cohomologywith Z / X × Y does not contain a closed exact Lagrangian.The next two theorems are concerned with the behavior of algebraically hyperbolic Liouvillemanifolds under gluings. To set the stage, suppose that ( X , λ ) and ( X , λ ) are Liouvillemanifolds of dimension 2 n which each contain a copy of some fixed Weinstein hypersurface( F, λ ) in their ideal boundary. Then there is a procedure (described in detail in [18, Sec. 3.1]and reviewed in Section 2.3) for constructing a new Liouville manifold (
X, λ ) by gluing X and X along F . This is a very general procedure which can be used to produce a wealth ofconstructions of Liouville manifolds. In particular, it includes as special cases many familiaroperations in contact topology such as handle attachments and plumbings.Let us say that a gluing along a Weinstein hypersurface F (of dimension 2 n −
2) is subcriticalif the skeleton of F has dimension < n −
1. Otherwise, the gluing is critical. The followingtheorem implies that the class of algebraically hyperbolic Weinstein manifolds is preservedunder arbitrary subcritical gluings.
Theorem 1.6 (Subcritical gluing) . Let ( X , λ ) and ( X , λ ) be Liouville manifolds. Let ( X, λ ) be a Liouville manifold obtained by gluing ( X , λ ) and ( X , λ ) along a Weinsteinhypersurface F in the sense of Section 2.3. Suppose that one of the following two conditionsis satisfied. (i) F is subcritical ; (ii) F is the Weinstein thickening of a loose Legendrian sphere and ( X , λ ) is Weinsteinup to deformation.If ( X , λ ) is algebraically hyperbolic, then ( X, λ ) is also algebraically hyperbolic. LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 5
Figure 1.
The top picture represents the skeleton of a Liouville manifold(
X, λ ) obtained by gluing T ∗ T and T ∗ Σ along the Weinstein thickening ofa zero-sphere. It follows from Theorem 1.6 that ( X, λ ) is algebraically hy-perbolic. The bottom picture represents the skeleton of a Liouville manifold(
X, λ ) obtained by gluing T ∗ S and T ∗ Σ along the Weinstein thickening of aLegendrian knot Λ, where Λ is the conormal lift of a small loop in the zero sec-tion of each component of the gluing. It follows by combining Proposition 1.3,Corollary 2.12 and Theorem 1.7 that ( X, λ ) is algebraically hyperbolic.In particular, it follows from Theorem 1.6 that algebraic hyperbolicity is preserved undersubscritical handle attachment – however, the theorem is much more general, since it allowsone to glue entirely arbitrary Liouville manifolds along subcritical hypersurfaces.Next, we consider the effect of gluing Liouville manifolds along a critical hypersurface. Incontrast to the subcritical case, algebraic hyperbolicity is not preseved under arbitrary criticalgluings; see Example 1.8 below. However, the following theorem identifies a class of criticalgluings which do preserve algebraic hyperbolicity. (We refer the reader Section 2.2 for thedefinition of a Liouville sector.)
Theorem 1.7 (Critical gluing) . Let ( X , λ ) and ( X , λ ) be Liouville manifolds of dimension n . Let ( X, λ ) be obtained by gluing ( X , λ ) and ( X , λ ) along a Weinstein hypersurface F inthe sense of Section 2.3. Suppose that F is contained in the image of an embedding of Liouvillesectors ( T ∗ D n , λ std ) (cid:44) → ( X , λ ) . If ( X , λ ) is algebraically hyperbolic and Weinstein up todeformation, then ( X, λ ) is algebraically hyperbolic. As we will see in Corollary 2.12, any Weinstein manifold admits an abundance of sectorialembeddings of ( T ∗ D n , λ std ). In particular, given any pair of Weinstein manifolds, they canalways be glued along a critical hypersurface, and there is a large supply of possible gluings.In conclusion, Theorem 1.6 and Theorem 1.7 provide a large class of operations whichpreserve the class of algebraically hyperbolic Liouville manifolds and which may nonethelessact rather drastically on the smooth topology of the manifold. Example 1.8.
Let M be an n -dimensional manifold in the class E . As we will show inProposition 2.14, M embeds into the skeleton of a smooth complex affine variety ( X, λ ) ofreal dimension 2 n . This means that ( X, λ ) is obtained from T ∗ M by attaching a sequenceof handles. However, it follows from Theorem 1.10 below that ( X, λ ) is not algebraicallyhyperbolic. Thus the class of algebraically hyperbolic Weinstein manifolds is not closed under
LAURENT C ˆOT´E arbitrary critical handle attachments. This illustrates that the condition in Theorem 1.7 that F is contained in the image of an embedding of T ∗ D n is needed.1.4. Properties of algebraically hyperbolic Liouville manifolds.
We begin with thefollowing rather unsurprising theorem.
Theorem 1.9. If ( X, λ ) is an algebraically hyperbolic Liouville manifold, then ( X, λ ) is notLiouville equivalent to the cotangent bundle of a rationally elliptic spin manifold. An equivalence of Liouville manifolds is a diffeomorphism φ : ( X, λ ) → ( X (cid:48) , λ (cid:48) ) such that φ ∗ λ (cid:48) = λ + df for f a compactly-supported function. We remark that it immediately followsfrom (1.2) and the definition of rational ellipticity that HW • ( F p , F p ) does not have exponentialalgebraic growth if M is rationally elliptic and spin. However, we do not know how to showpurely algebraically that Tw W ( T ∗ M ) also does not have exponential algebraic growth. Ourproof of Theorem 1.9 is in fact geometric, and uses the length filtration of the based loop spaceof a manifold. Theorem 1.10 (Affine varieties) . Suppose that ( X, λ ) is an algebraically hyperbolic Liouvillemanifold. Then both of the following statements hold: • ( X, λ ) is not symplectomorphic to an affine variety. • ( ∂ ∞ X, ξ ∞ ) is not contactomorphic to the link of an isolated singularity. Our proof of Theorem 1.10 relies on work of McLean [38]. The basic idea is that the idealboundary of an affine variety admits a tame Reeb flow, due to the existence of a projectivecompactification with simple normal crossings. However, under the hypothesis that the rankof the wrapped Floer cohomology of a pair of Lagrangians grows sufficiently fast as a functionof action, McLean proves that such a Reeb flow cannot exist. In proving Theorem 1.10, thehypothesis that (
X, λ ) is algebraically hyperbolic is used only to show that McLean’s conditionon the action growth of wrapped Floer cohomology is satisfied for some pair of Lagrangians.The general problem of describing conditions under which a Weinstein manifold cannot besymplectomorphic to an affine variety has been studied by McLean [38, 39], building on earlierwork of Seidel [44, Sec. 4]. To our knowledge, the only previously known examples of Liou-ville manifolds which have been shown not be symplectomorphic to affine varieties are eithercotangent bundles or obtained by subcritical surgery on cotangent bundles. Theorem 1.10,combined with the constructions of Section 1.3, provides a large class of new examples (someof which in fact contain no exact Lagrangians by Example 1.5).
Theorem 1.11 (Topological entropy) . Suppose that ( X, λ ) is an algebraically hyperbolic Li-ouville manifold. If W ( X ) is split-generated by a collection of Lagrangians { L α } whose Leg-endrian boundary is diffeomorphic to a sphere (this holds e.g. if ( X, λ ) is Weinstein), then theReeb flow on the ideal contact boundary ( ∂ ∞ X, ξ ∞ ) has positive topological entropy for anycontact form. As explained above, the topological entropy is an important measure of complexity of adynamical system. A system with positive topological entropy can be interpreted as beingchaotic (see also Section 2.7). In general, a flow with positive topological entropy need notcontain a closed orbit. However, it follows from work of Lima and Sarig [35, Thm. 1.1] that anyflow with positive entropy on a 3 -manifold has the property that the number of geometricallydistinct simple closed orbits grows exponentially in time (a somewhat weaker statement alsofollows from earlier work of Katok [22, 23]). This leads to the following corollary, which canbe viewed as a strong version of the Weinstein conjecture.
LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 7
Corollary 1.12 (Strong Weinstein conjecture) . If ( X, λ ) is an algebraically hyperbolic Li-ouville manifold of dimension , then for any choice of contact form on the ideal boundary ( ∂ ∞ X, ξ ∞ ) , the number of geometrically distinct simple closed Reeb orbits grows exponentiallyas a function of their length. Our proof of Theorem 1.11 uses work of Alves and Meiwes [9]. More precisely, Alves andMeiwes give a condition in terms of the action growth of a pair of Lagrangians in a givenLiouville manifold (
X, λ ) which forces positivity of the topological entropy of any Reeb flowon the ideal boundary. We prove that this condition is satisfied for algebraically hyperbolicmanifolds.Topological entropy of Reeb flows of contact manifolds has been studied by various au-thors. Macarini and Schlenk [36] proved analogs of Theorem 1.11 and Corollary 1.12 for unitcotangent bundles of manifolds whose based loop space is sufficiently “complicated” (the mainexamples being hyperbolic and rationally hyperbolic manifolds). Their work was subsequentlygeneralized by Alves and Meiwes [9], who constructed contact structures whose Reeb flowshave positive topological entropy on various manifolds including high-dimensional spheres andideal boundaries of tree-plumbings of certain cotangent bundles. They also proved that anyexactly-fillable contact manifold admits a (possibly different) contact structure whose Reebflows have positive topological entropy.Concerning Corollary 1.12, we note that the Weinstein conjecture was proved by Taubes[46] for all contact 3-manifolds. More recently, is was shown by Colin–Dehornoy–Rechtman[15] that if ( Y , ξ ) is a contact 3-manifold which is not diffeomorphic to a sphere or lens space,then any non-degenerate contact form α on ( Y, ξ ) has infinitely many Reeb orbits. (Earlierwork of Cristofaro-Gardiner, Hutchings and Pomerleano [12] established the same result underthe additional assumption that c ( ξ ) is torsion.) Colin–Dehornoy–Rechtman also proved thatany Reeb flow on an irreducible, closed 3-manifold which is not a graph manifold has positivetopological entropy, which in particular implies Corollary 1.12 for this class of 3-manifolds.The constructions described in Section 1.3 allow one to construct a large class of new exam-ples of Liouville manifolds whose ideal boundaries satisfy the conclusions of Theorem 1.11 andCorollary 1.12. In particular, even in dimension three, one can build a wealth of new examplesby starting with the cotangent bundle of a manifold is the class E and repeatedly applyingTheorem 1.6 and Theorem 1.7. Remark . As we have explained, the proofs of Theorems 1.10 and 1.11 essentially consistin providing a lower bound for the action growth of wrapped Floer cohomology in terms ofthe internal algebraic structure of the wrapped Fukaya category. The author was led to thisidea after reading the beautiful paper of Alves and Meiwes [9] on topological entropy, whichwas already mentioned above. In this paper, Alves and Meiwes made the crucial observationthat one can bound the action growth of the Floer cohomology of a single Lagrangian L interms of the growth of HW • ( L, L ) as an associative algebra. The proofs of Theorems 1.10 and1.11 can be seen as categorical generalizations of Alves and Meiwes’ original observation. Thefull categorical structure turns out to be important for the purpose of applying the theory ofGanatra–Pardon–Shende [30] to construct algebraically hyperbolic Liouville manifolds.1.5.
Skeleta of Weinstein manifolds.
As explained above, if (
X, λ ) is the cotangent bundleof a closed manifold M , then it follows from (1.2) that ( X, λ ) is algebraically hyperbolic if andonly if Tw H −• (Ω ∗ M, L k ) has exponential algebraic growth. In particular, this is purely aquestion of algebraic topology. More generally, it is natural to ask whether the property of aWeinstein manifold being algebraically hyperbolic can be detected in terms of the stratifiedtopology of its skeleton. LAURENT C ˆOT´E
Figure 2.
On the left is an immersed surface of genus 2 with a single transversedouble point. On the right is the union of a sphere, a torus and a surface ofgenus 2, with each pair of components intersecting pairwise transversally in asingle point. (In particular, the double points occurring in these pictures shouldbe interpreted as transverse double points even though they cannot be drawnas such for dimension reasons). Neither of these stratified spaces can occur asthe skeleton of an algebraic variety light of Corollary 1.14 and Corollary 1.16.In Theorem 7.3, we make some progress in this direction by describing a condition on theskeleton of a Weinstein manifold which forces it to be algebraically hyperbolic. The statementrequires some preparations, so we state instead here some simple corollaries.
Corollary 1.14.
Suppose that ( X, λ ) is Weinstein and that Skel(
X, λ ) is the image of aLagrangian immersion f : M → ( X, λ ) with only simple double points. If M ∈ E , then ( X, λ ) is algebraically hyperbolic. Example 1.15.
The skeleton of the affine variety C − { xy = 1 } is the Whitney immersion ,which is an immersion of the 2-sphere with a single double point (see e.g. [16]). In contrast, itfollows from Corollary 1.14 and Theorem 1.10 that the skeleton of an algebraic variety cannotbe the image of an immersion of a (possibly non-orientable) surface of genus greater than2. We do not know whether there exists an algebraic variety whose skeleton is an immersedtorus or Klein bottle with at least one double point (of course, both T ∗ S and T ∗ T aresymplectomorphic to affine varieties). We remark that given any immersion M → R m with onlysimple double points, it is straightforward to construct a Weinstein manifold whose skeletonis the image of f . Corollary 1.16.
Suppose that ( X, λ ) is Weinstein and that Skel(
X, λ ) is a union of closedLagrangian submanifolds L , . . . , L l with the property that L i (cid:84) ( (cid:83) j (cid:54) = i L i ) ⊂ L i is contained in aball (this holds e.g. if X is a plumbing of cotangent bundles according to any graph). If L i ∈ E for some i , then ( X, λ ) is algebraically hyperbolic. In general, it is not known whether Weinstein manifolds are uniquely determined by theirskeleton. However, forthcoming work of Alvarez-Gavela–Eliashberg–Nadler [7] will show thatthis is indeed true provided that the singularities are in a relatively restricted class (in par-ticular, this includes so-called “arboreal” singularities, as well as simple normal crossings).More precisely, they define a class of locally-ringed topological spaces called arboreal spaces and prove that such spaces have a unique Weinstein thickening. One can therefore view thewrapped Fukaya category, and in particular the notion of algebraic hyperbolicity, as an invari-ant of arboreal spaces.Any arboreal space is homeomorphic to a finite CW complex. In light of the discussion ofSection 1.2, it is natural to ask whether the property of the skeleton being rationally hyperbolicas a topological space is related to its Weinstein thickening being algebraically hyperbolic. Tothis end, we remark that many smooth affine varieties have as their skeleton a wedge of
LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 9 spheres (arising e.g. from vanishing cycles near a singularity). Wedges of spheres are rationallyhyperbolic, and yet affine varieties are not algebraically hyperbolic. We do not know whetherthere exists an algebraically hyperbolic Weinstein manifold whose skeleton is rationally ellipticas a CW complex.1.6.
Acknowledgements.
The author has benefited from conversations, suggestions andcorrespondences with many mathematicians including Dani ´Alvarez-Gavela, Denis Auroux,Georgios Dimitroglou Rizell, Yasha Eliashberg, Yves F´elix, Kathryn Hess, Helmut Hofer,Yusuf Barı¸s Kartal, Thomas Massoni, John Pardon, Semon Rezchikov, Lisa Sauermann, VivekShende, Kyler Siegel, Zack Sylvan and Umut Varolgunes.Part of this research was carried out while the author was a graduate student at StanfordUniversity and supported by a Benchmark Graduate Fellowship. This material is based uponwork supported by the National Science Foundation under Grant No. DMS-1926686.2.
Background material
Unless otherwise specified, all manifolds in this paper and maps between them are smooth.Given a manifold M of dimension n , a ball B ⊂ M is the interior of the image of an embeddingof the closed unit ball in R n . Given a topological space X , a “neighborhood of infinity” meansthe complement of a precompact set.2.1. Linear and A ∞ categories. In this paper, we will often consider linear categories (i.e.categories whose morphisms sets are vector spaces and have bilinear composition) and A ∞ categories. Unless otherwise specified, all categories are defined over a field k of characteristiczero which is fixed throughout. Given a category C and an object L , we routinely abusenotation by writing L ∈ C . We let Mor( C ) denote the set of all morphisms in C (i.e. the unionover all pairs of objects of all sets of morphisms).All A ∞ categories considered in this paper are strictly unital. They are also assumed tocarry a Z /n -grading for some n ∈ N ≥ (where we define Z / Z ). However, gradings will beessentially irrelevant in this paper, so the reader is free to assume that n = 1 throughout. Afunctor C → D between A ∞ categories is said to be full (resp. faithful, fully faithful, essentiallysurjective, an equivalence etc.) if the induced functor on the cohomology category H • ( C ) → H • ( D ) has this property.Given a Z /n -graded A ∞ category C , we Σ C be the additive enlargment of C . There arevarious equivalent constructions of this category in the literature; for concreteness, we willalways use the construction described in [45, Rmk. 3.26] (i.e. the objects of Σ C are shiftedcopies of objects of C , with the shift taking values in Z /n ). We let Tw C denote the categoryof twisted complexes over C , constructed again as in [45, Rmk. 3.26]. There is a natural fullyfaithful functor C → Tw C which defines (a model for) the pre-triangulated closure of C . We let C →
Perf C denote the idempotent-completed pre-triangulated closure of C ; for concreteness,one can again freely set Perf( − ) = Π Tw( − ) as in [45, (4c)].If C is an A ∞ category and A ⊂ C is a full subcategory, we say that A generates C if thenatural functor Tw A → Tw C is an equivalence. We say that A split-generates if Perf A →
Perf C is an equivalence. If { L α } is a collection of objects of C and A is the full subcategory of C whose objects are the L α , we say that the collection (split-)generates if A (split-)generates.If C is a Z /n -graded A ∞ category, then for any m which divides n , there is a “forgetfulfunctor” C (cid:55)→ ( C ) m which collapses the Z /n -grading to a Z /m -grading. This “forgetful functor”commutes with passing to the cohomology category in the sense that H • ( C ) m = H • (( C ) m ).We also have a natural equivalence of categories (Tw C ) m → Tw( C ) m which collapses the Z /n -valued shifts on objects to a Z /m -valued shift. Liouville manifolds and sectors.Definition 2.1. A Liouville manifold ( X, λ ) is an exact symplectic manifold which admitsa proper embedding e : R × Y → X of the symplectization of some closed contact manifold( Y, λ Y ). This embedding is required to cover the complement of a compact subset of X andto preserve the Liouville structure (i.e. e ∗ λ = ˆ λ Y ).The contact manifold ( Y, ker λ Y ) in Definition 2.1 is unique up to contactomorphism and isusually called the ideal boundary of ( X, λ ). One writes ( Y, ker λ Y ) = ( ∂ ∞ X, ξ ∞ ). The 1-form λ is called the Liouville -form and the dual vector field V (i.e. the vector field satisfying i V dλ = λ ) is called the Liouville vector field . An equivalence of Liouville manifolds is adiffeomorphism φ : ( X, λ ) → ( X (cid:48) , λ (cid:48) ) such that φ ∗ λ (cid:48) = λ + df for f : X → R compactly-supported. Definition 2.2 (Def. 2.4 in [29]) . A Liouville sector ( X, λ ) is a Liouville manifold with bound-ary which admits a function I : ∂X → R which is linear at infinity (meaning i V dI = I ) andwhose Hamiltonian vector field X I is outward-pointing along the boundary. The function I iscalled a defining function . Example 2.3.
Let M be a compact manifold with boundary. Then ( T ∗ M, λ can ) is a Liouvillesector.
Definition 2.4.
A Liouville manifold (
X, λ ) is said to be
Weinstein if there exists a Morsefunction φ : X → R + which is gradient-like with respect to the Liouville vector field V ,meaning that(2.1) dφ ( V ) ≥ δ ( | V | + | dφ | )for some δ > X . A Liouville sector is said to be Weinsteinif its convex completion (see [29, Sec. 2.7]) is Weinstein.We emphasize that being Weinstein is a property of Liouville manifold and sectors in thispaper. However, it is often important in other contexts to consider Weinstein structures intheir own right. We also remark that there are many slightly weaker definitions of Weinsteinmanifolds which are often considered in the literature.2.3. Gluing Liouville sectors along a hypersurface.
In this section, we collect somematerial which is introduced in detail in [29, Sec. 2.5] and [18, Sec. 2]. Slightly generalizing[29, Def. 2.14], let us say that a sutured Liouville sector ( X , F ) consists of the following data: • a Liouville sector ( X , λ ); • a codimension 1 submanifold with boundary F ⊂ ∂ ∞ X ; • the germ of a contact structure α ∞ near F , whose restriction to F is denoted by λ F and makes ( F , λ F ) into a Liouville domain.It can be shown [18, Sec. 2] that Skel( F , λ ) is independent of the choice of α ∞ . We saythat the hypersurface F ⊂ ∂ ∞ X is subcritical if the skeleton is subcritical, meaning thatit is contained in the smooth image of a second countable manifold of dimension < n − F is said to be critical . Example 2.5.
Let (
X, λ ) be a Liouville sector and let Λ ⊂ ( ∂ ∞ X, ξ ∞ ) be a framed isotropicsubmanifold. Then Λ has a neighborhood of the form R × T ∗ Λ on which dt + λ can is a contactform for ξ ∞ . The Liouville domain T ∗≤ Λ ⊂ ∂ ∞ X is called the Weinstein thickening of Λ, andthe resulting sutured Liouville sector (
X, T ∗≤ Λ) well-defined up to homotopy. (See [29, Ex.2.15].)
LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 11
Construction 2.6 (Cutting) . Given a sutured Liouville sector ( X , F ), there is a procedurefor constructing a new Liouville sector ( X, λ ) by “cutting out” the positive symplectization ofa thickening F × [ − (cid:15), (cid:15) ]; see [29, Lem. 2.13]. The sector ( X, λ ) has an additional boundarycomponent H which comes equipped with a defining function I : H → R . The sector ( X, λ )and the function I are canonically defined up to homotopy, and depend on ( X , F ) only upto homotopy of sutured Liouville sectors. See [29, Sec. 2.5] for details.We now explain how to glue Liouville sectors along a common boundary component. Construction 2.7 (Gluing) . Let ( X , λ ) and ( X , λ ) be Liouville sectors. Let H ⊂ ∂X and H ⊂ ∂X be boundary components and suppose that there is a map j : H → H suchthat j ∗ λ = λ . Let us define X = X ∪ j X as a set and define H to be H ≡ H viewed as ahypersurface in X . To make X into a Liouville manifold, we will define a smooth collar near H and a Liouville form near H which extends to λ and λ on each side.Fix a defining function I : H → R . As argued in [29, Prop. 2.25], I induces a canonicalidentification ( X , λ ) = ( F × C Re ≥ , λ F + λ C + df ) which is valid in a cylindrical neighborhoodof H , where λ C := ydx . Let I = I ◦ j : H → R . Case I:
Suppose that − I : H → R is a defining function for H . Then by the sameargument as in the proof of [29, Prop. 2.25], I induces a canonical identification ( X , λ ) =( F × C Re ≤ , λ F + λ C + df ) in a cylindrical neighborhood of H , where F = ( I ◦ j ) − (0).Define a map ψ : N bd ( H ) ⊂ F × C Re ≤ → F × C by ( x, z ) (cid:55)→ ( j ( x ) , z ). Then ψ ∗ λ = λ F + λ C + ψ ∗ ( df ). Moreover, by the argument in [29, Prop. 2.25], ψ ∗ ( f ) = f ◦ ψ − = f outside a compact set. Let f : F × C be a smooth extension of f which agrees with ψ ∗ ( f )outside a compact subset of the image of ψ . Then λ := λ F + λ C + df is a Liouville form near H which extends to λ , λ . Note that the only choices we made are the choice of I and ofthe extension f ; both of these choices are contractible, from which it follows that ( X, λ ) iswell-defined up to homotopy.
Case II:
Suppose I : H → R is a defining function for H . Then I induces coordinatesnear H of the form ( F × C Re ≥ , λ F + λ C + df ). Let φ : F × C ≤ Re ≤ (cid:15) → F × C − (cid:15) ≤ Re ≤ by defined by φ ( x, z ) = ( x, − z ) for (cid:15) suitably small. Then ( F × C − (cid:15) ≤ Re ≤ , Φ ∗ λ ) is an openLiouville sector, since φ ∗ λ = λ F + λ C + φ ∗ ( df ) and f is independent of z for im( z ) largeenough. Moreover, − ( I ◦ j ◦ φ − ) = I (cid:48) is a defining function. Hence we can apply the sameconstruction as in Case I.Given a pair of sutured Liouville sectors ( X , F ) and ( X , F ), we can construct a newLiouville sector ( X, λ ) by cutting open the suture and then regluing the resulting sectors alongthe boundary component corresponding to the suture. Many natural operations (includinghandles attachments and plumbings) can be described in those terms.
Example 2.8.
Let ( X , λ ) be a Liouville sector of dimension 2 n and let Λ ⊂ ( ∂ ∞ X , ξ ∞ )be a framed isotropic submanifold of dimension k ≤ n −
1. Let ( F , λ F ) ⊂ ( ∂ ∞ X , ξ ∞ ) be aWeinstein thickening. Consider now ( X = C n , λ = (cid:80) i x i dy i − y i dx i ). Let Λ ⊂ ∂ ∞ X bedefined by the equation { x k +1 = · · · = x n = y = · · · = y n = 0 } and fix the normal framing { ∂ x k +1 , . . . , ∂ x n } . Let ( F , λ F ) ⊂ ( ∂ ∞ X , ξ ∞ ) be a Weinstein thickening.Then ( X , F ) and ( X , F ) are well-defined up to homotopy as a sutured Liouville sectors.We let ( X, λ ) be obtained by successively applying Construction 2.6 and Construction 2.7 tothe sutured Liouville sectors ( X , F ) and ( X , F ). The resulting Liouville sector ( X, λ ) is well-defined up to homotopy; we say that it is obtained from ( X , λ ) be attaching a k + 1-handlealong Λ .2.4. Skeleta of Weinstein manifolds.
Definition 2.9.
Let (
X, λ ) be a Liouville manifold and let ψ − ( − ) : R × X → X be theLiouville flow. The skeleton of ( X, λ ) is the set(2.2) Skel(
X, λ ) = X − e ( R × Y )for some choice of Liouville embedding e : R × Y → X as in Definition 2.1 (the skeleton isindependent of the choice of embedding e ). Example 2.10.
Given a closed manifold M , we have Skel( T ∗ M, λ can ) = 0 M , where 0 M ⊂ T ∗ M is the zero section.It is straightforward to see that the skeleton is compact and has Lebesgue measure zero,for our definition of a Liouville manifold. In general, the skeleton of a Liouville manifoldcan be quite a bad set (for instance, McDuff [37] famously constructed a Liouville manifoldwhose skeleton has codimension one). However, for Weinstein manifolds, it is known that theskeleton admits a stratification with isotropic strata [18, Sec. 1]. Recall that a stratification ofa topological space X is a finite filtration 0 = F ⊂ . . . ⊂ F n − ⊂ F n = X by closed subsets,such that F i − F i − is a smooth manifold (without boundary).Given a Weinstein manifold ( X, λ ), we define the
Lagrangian stratum as the union of allpoints p ∈ Skel(
X, λ ) such that Skel(
X, λ ) is an embedded Lagrangian submanifold near p .In particular, the Lagrangian stratum is itself a smooth (in general open and disconnected)Lagrangian submanifold of ( X, λ ). Proposition 2.11 (see [7] or Sec. 12.3 in [14]) . Let ( X, λ ) be a Weinstein manifold of dimen-sion n . Fix a Morse function φ : X → R + which is gradient-like for the Liouville flow of λ . Let M be a connected n -dimensional manifold, possibly with boundary, and let i : M → Skel(
X, λ ) be a smooth embedding into the Lagrangian stratum of the skeleton. Then given any openneighborhood U ⊂ X of the image of i , there exists a deformation of Liouville forms { λ t } t ∈ [0 , and Morse functions φ t such that the following holds: • λ = λ and λ t is independent of t in the complement of U ; • Skel(
X, λ t ) is independent of t ; • φ t is gradient-like for the Liouville vector field V t of λ t ; • there is an embedding (2.3) ˜ i : T ∗ M → X which covers i and such that ˜ i ∗ λ is the standard Liouville form on T ∗ M . We will mainly need the following obvious corollary of Proposition 2.11.
Corollary 2.12.
Let ( X, λ ) be a Weinstein manifold. Let M be a closed manifold (possibly withboundary) and suppose that i : M → Skel(
X, λ ) is a smooth embedding into the Lagrangianstratum. Then after possibly deforming λ in a compact set, i is covered by an inclusion ofLiouville sectors ˜ i ( T ∗ D n , λ std ) (cid:44) → ( X, λ ) . (cid:3) In particular, it follows from Corollary 2.12 that any point in the Lagrangian stratum is con-tained the image of some sectorial embedding ( T ∗ D n , λ std ) → ( X, λ ) after possibly deforming λ . Remark . The proof of Proposition 2.11 uses the assumption that (
X, λ ) is Weinstein in anessential way. It is unknown whether a similar result holds for arbitrary Liouville manifolds.Let us end this subsection by recording the following proposition, which was already men-tioned in the introduction. This proposition essentially states that any closed manifold can beembedded in the skeleton of some smooth affine variety.
LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 13
Proposition 2.14.
Let M be a closed manifold of real dimension n . For any N ≥ n + 1 ,there exists a smooth complex affine variety X ⊂ C N with the following properties: • ( X, λ | X ) is a Weinstein manifold, where λ = (cid:80) Ni =1 ( x i dy i − y i dx ) is the standard radialLiouville form; • M is diffeomorphic to a connected component of the skeleton Skel(
X, λ | X ) .Proof. According to a classical theorem of Nash (see [25, Thms. 2 and 3]), there exists acollection f , . . . , f k of real polynomials in N variables such that M is a connected componentof the zero set V R ( f , . . . , f k ) ⊂ R N . Viewing the f i as complex polynomials, we set X := V C ( f , . . . , f n ) ⊂ C n . After possibly perturbing the f i , we may assume that X is a smoothcomplex algebraic variety and that the function φ ( z , . . . , z n ) = | z | + · · · + | z n | is Morse.One can check that λ = d C φ , from which it follows that ( X, λ | X , φ ) is a Weinstein structure(see [14, Sec. 1.1]).Observe that complex conjugation sends λ (cid:55)→ − λ . It follows that the Liouville vector field v X on ( X, λ | X ) is preserved by complex conjugation. In particular, v X is tangent to the reallocus X R ⊂ X . Since M is a connected component of X R , it follows immediately that M is contained in the skeleton of ( X, λ | X ). It is similarly straightforward to check that X hascomplex dimension n : indeed, since T m X is also fixed by complex-conjugation for any x ∈ M ,it follows that dim T m X = 2 dim T m M . (cid:3) The wrapped Fukaya category.
Unless otherwise specified, all Fukaya categories con-sidered in this paper are k -linear and Z / k is a field of characteristic zero. Construction 2.15 (Linear setup; see [5]) . Let (
X, λ ) be a Liouville manifold of dimension atleast 4. Let { L α } α ∈ I be a countable collection of Lagrangian submanifolds which are requiredto be cylindrical at infinity (i.e. invariant under the Liouville flow) and equipped with a spinstructure. We may (and do) assume that this collection contains at least one representativefrom each isotopy classes of such Lagrangians (cf. [29, Sec. 3.5]), and that the ideal Legendrianboundaries of the Lagrangians are pairwise disjoint.Fix a standard decomposition X = X ∪ ([0 , ∞ ) × Y ), where ( X, λ ) is Liouville domain withcontact boundary ∂X = Y . After possibly isotoping the Lagrangians via the Liouville flow,we can assume that they are cylindrical in a neighborhood of [0 , ∞ ) × Y . Choose a strictlypositive Hamiltonian H : X → R + which agrees with e t on ( − (cid:15), ∞ ) for some small (cid:15) > H and the Lagrangians in the interior of ˆ X .Then work of Abouzaid–Seidel (see [5, Sec. 5a]) defines an A ∞ category W lin ( X ; H ). Given apair of Lagrangians L α , L β (cylindrical at infinity, equipped with spin structures) for α, β ∈ I ,we have(2.4) CW • ( L α , L β ) = ∞ (cid:77) w =1 CF • ( L α , L β ; wH ) , where q is a formal variable of degree − q = 0. This A ∞ category depends onall of the above data, as well as some additional auxiliary data (complex structures, strip-likeends, etc.), but we only record the dependence on H in our notation. Construction 2.16 (Localization setup; see [29]) . Let (
X, λ ) be a Liouville sector. Work ofGanatra–Pardon–Shende (see [29, Def. 3.36]) defines an A ∞ category W ( X ) whose objectsare Lagrangian submanifolds of X (cylindrical at infinity and equipped with a spin structure).The morphisms are constructed by a rather indirect categorical localization procedure firstintroduced by Abouzaid and Seidel [4]. This category depends only on the deformation class of ( X, λ ) up to quasi-equivalence. An inclusion of Liouville sectors i : ( X, λ ) (cid:44) → ( X (cid:48) , λ (cid:48) ) inducesa functor i ∗ : W ( X ) → W ( X (cid:48) ). Given a diagram { X σ } σ ∈ Σ of Liouville sectors indexed by aposet Σ, we can always assume that the induced diagram {W ( X σ ) } σ ∈ Σ is strictly commutingby choosing appropriate models for W ( X σ ) (cf. [29, Sec. 3.7]). Fact 2.17 (Folklore) . Let ( X, λ ) be a Liouville manifold. Then the wrapped Fukaya category W lin ( X ; H ) defined via Construction 2.15 is quasi-equivalent to the wrapped Fukaya category W ( X ) defined via Construction 2.16. In particular, W lin ( X ; H ) depends only on ( X, λ ) up toquasi-equivalence. In this paper, we need to consider wrapped Fukaya categories of Liouville sectors, so we takeConstruction 2.16 as our primary definition of the wrapped Fukaya category. However, for thepurposes of constructing an action filtration on the wrapped Fukaya category of a Liouvillemanifold in Section 5.1, it is more convenient to work with Construction 2.15. We expect thatone could work entirely within the framework of Construction 2.16, but this would come atthe cost of additional algebraic complications which seem orthogonal to the main goals of thispaper.2.6.
Rational homotopy theory.
In this subsection, we briefly collect some basic facts fromrational homotopy theory which will be needed later. We will only consider Z -gradings in thissubsection, and we also remind the reader that we work throughout this paper over a fixedfield k of characteristic zero. Standard texts on rational homotopy theory include [20] and [27].Given a simply-connected CW complex X , the graded vector space π • ( X ) ⊗ k can beequipped with the structure of graded Lie algebra via the Whitehead product (see [20, Sec.21(d)]). The resulting graded Lie algebra is usually called the homotopy Lie algebra of X andis denoted by L X .A fundamental theorem of Milnor and Moore (see [41] and [20, Thm. 21.5]) states thatthere is an isomorphism of (associative) graded algebras U L X ∼ −→ H • (Ω X ; k ), where U L X isthe universal enveloping algebra of L X (see [20, Sec. 21(a)]) and the right hand side is theusual Pontryagin algebra. Lemma 2.18.
Let ( X, ∗ ) be a pointed simply-connected finite CW complex and let L X be itshomotopy Lie algebra. Suppose that X admits a free graded subalgebra on two generators, i.e.there is an injective morphism of Lie algebras L V (cid:44) → L X , where V is a Z -graded vector space ofdimension two and L V denotes the free graded Lie algebra generated by V (see [20, Sec. 21(c)] ).Then H • (Ω ∗ X ; Q ) contains a free subalgebra on two generators. In particular, H • (Ω ∗ X ; Q ) hasexponential algebraic growth.Proof. The Lie algebra inclusion L V (cid:44) → L X induces a map of graded associative algebras U L V → U L X . Since we are working over a field, the Poincar´e–Birkoff–Witt theorem [20, Sec.21(a)] implies that this later map is also injective. But by the Milnor–Moore theorem statedabove, we have that U L X is quasi-isomorphic to H • (Ω ∗ X ; k ). (cid:3) We now state the definition of rationally hyperbolic and elliptic spaces.
Definition 2.19.
A simply connected, finite CW complex X is said to be rationally hyperbolic if(2.5) lim sup n b ≤ n (Ω ∗ X ) n = lim sup n (cid:80) i ≤ n dim H i (Ω ∗ X ; Q ) n > . Otherwise, X is said to be rationally elliptic . LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 15 If X is rationally elliptic, it is a remarkable fact from rational homotopy theory that b ≤ n ( X )grows at most polynomially in n . This phenomenon is often called the elliptic vs. hyperbolicdichotomy.There are various alternative characterizations of rationally hyperbolic and elliptic spaces.It particular, it is more customary to define a finite CW complex as being rationally hyperbolicif the rank of its rational homotopy groups grows exponentially, and rationally elliptic if therank of its rational homotopy groups is finite. We have adopted Definition 2.19 as our maindefinition because it can more naturally be related to wrapped Floer homology; we refer thereader to [20, Theorem 33.2] a proof that these definitions are equivalent. Example 2.20.
A “generic” simply-connected finite CW complex is rationally hyperbolic, andthere are many very strong criteria for checking that a particular CW complex is rationally hy-perbolic. Some concrete examples of rationally hyperbolic spaces include all simply-connected4-manifolds with second Betti number greater than six; all simply connected 5-manifolds withsecond Betti number greater than ten; any simply-connected finite CW complex with negativeEuler characteristic; etc.
Example 2.21.
Simply-connected compact homogeneous spaces (such as spheres, complexprojective spaces, compact Lie groups, etc.) are rationally elliptic; see [20, Sec. 32(e)]. Productsof rationally elliptic spaces are rationally elliptic.2.7.
Topological entropy.
The topological entropy is one of the simplest measures of thecomplexity of a dynamical system. Intuitively, a system with positive topological entropy canbe viewed as chaotic.Let (
X, d ) be a compact metric space. Let Φ = { φ t } be a one-parameter family of homeo-morphisms and consider the family of metrics(2.6) d Φ T ( x, y ) = max ≤ t ≤ T d ( φ t x, φ t y )depending on T >
0. Let B Φ ( x, (cid:15), T ) = { y ∈ X | d Φ T ( x, y ) } < (cid:15) be the ball of radius (cid:15) centeredat x .We now let S d (Φ , (cid:15), T ) ∈ N be defined as the smallest number of balls of radius (cid:15) needed tocover X . Set h d (Φ , (cid:15) ) := limsup T →∞ T log S d (Φ , (cid:15), T ). Definition 2.22.
The topological entropy of the flow Φ = { φ t } on the compact metric space( X, d ) is(2.7) h (Φ) = lim (cid:15) → h d (Φ , (cid:15) ) . It can be shown [24, Sec. 3.1] that this quantity only depends on Φ and the topology inducedby d . Example 2.23.
It follows from the definition that h (Φ) = 0 if φ t is an isometry for all t , orif it is periodic. More surprisingly, it can be shown [24, Sec. 3.3] that the topological entropyof any gradient flow on a compact manifold vanishes. Example 2.24.
The geodesic flow on a connected, orientable surface Σ endowed with a metricof constant curvature has positive topological entropy if g ≥ Algebraic growth of categories
Main definitions.
Definition 3.1.
Let C be a k -linear category. Given a finite set of morphisms Σ ⊂ Mor( C )and an integer n ≥
1, let W Σ ( n ) be the span of all composable words of length at most n inthe elements of Σ, that is(3.1) W Σ ( n ) = span k { a . . . a l | a i ∈ Σ , l ≤ n } ⊂ (cid:77) K,L ∈C hom( K, L ) . Given an object L ∈ C , let(3.2) W Σ ( n ; L ) = span k { a . . . a l | a i ∈ Σ , l ≤ n, a . . . a l ∈ hom( L, L ) } ⊂ hom( L, L )be the span of all composable words of length at most n which define a morphism from L toitself. Definition 3.2. A k -linear category C is said to have exponential algebraic growth if(3.3) limsup n n log dim k W Σ ( n ) > ⊂ Mor( C ). An A ∞ category A is said to have exponential algebraicgrowth if its cohomology category has exponential algebraic growth. Remark . There is a wide body of literature around the topic of growth rates of associative k -algebras (see e.g. [26]). One particularly well-studied growth invariant is the Gelfand–Kirillovdimension . Definition 3.2 can be viewed as a coarse generalization of this notion: if A is anassociative k -algebra viewed as a category with one object, then A has exponential growth ifand only if its Gelfand–Kirillov dimension is strictly positive. Lemma 3.4.
Let C (cid:44) → C (cid:48) be a faithful functor between k -linear categories. If C has exponentialalgebraic growth, then so does C (cid:48) . (cid:3) In particular, it follows from Lemma 3.4 that the exponential algebraic growth of a categoryis preserved under equivalence of categories.
Lemma 3.5.
Suppose that C is a category having exponential algebraic growth. Then thereexists an object L ∈ C such that (3.4) lim sup n n log dim k W Σ ( n ; L ) > for some finite set Σ ⊂ Mor( C ) .Proof. By hypothesis, there exists a finite set Σ ⊂ Mor( C ) with the property that(3.5) lim sup n n log dim W Σ ( n ) > . Note that there exists a full subcategory C (cid:48) of C , generated by a finite collection L , . . . , L m ofobjects of C , such that Σ ⊂ Mor( C (cid:48) ). We may as well assume that the identity e L i ∈ hom( L i , L i )is contained in Σ for all i = 1 , . . . , m .Let us now fix n ≥ x . . . x l be a word in the elements of Σ for some l ≤ n . Up toreordering the L i , we may assume that x ∈ hom( L , L α ) for some α ≥
1. Let 1 ≤ i ≤ l bethe greatest integer such that the subword x . . . x i defines an element of hom( L , L ), wherewe set i = 0 if no such subword exists. We thus have a decomposition(3.6) x . . . x l = ( x . . . x i ) x i +1 ( x i +2 . . . x l ) , where x j is defined as empty if j < j > l . LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 17
Assuming that the word ( x i +2 . . . x l ) is non-empty, then it is contained in the full subcate-gory generated by L , . . . , L m . We now repeat the same process with ( x i +2 . . . x l ) in place of( x . . . x l ). After q ≤ m − x . . . x l = ( x . . . x i ) x i +1 ( x i +2 . . . x i ) . . . ( x i q − +2 . . . x i q ) x i q +1 ( x i q +2 . . . x l )where each of the factors is either empty, or an element of Σ, or an element of hom( L i , L i ) forsome 1 ≤ i ≤ m . Since e L i ∈ Σ for all i , we may therefore write (non-uniquely)(3.8) x . . . x l = y z . . . y m − z m − y m for some z i ∈ Σ and y i ∈ (cid:83) mj =1 W Σ ( n ; L j ).By definition of W Σ ( n ), it follows that(3.9) dim k W Σ ( n ) ≤ [dim k ( ⊕ mi =1 W Σ ( n, L i ))] m · | Σ | m − , which implies that(3.10) log dim k W Σ ( n ) ≤ m m (cid:88) i =1 log(dim k W Σ ( n ; L i ) + 1) + ( m −
1) log | Σ | . (Here, we have used the fact that log( a + · · · + a k ) ≤ log( a + 1) + . . . log( a k + 1) for a i ≥ n n log dim k W Σ ( n ; L i ) > i ∈ { , . . . , m } , as desired. (cid:3) Definition 3.6.
Given a pair ( C , D ) of unital, k -linear categories, their tensor product C⊗D is aunital, k -linear category. The objects of C⊗D are the products L × L (cid:48) for L ∈ C and L (cid:48) ∈ D . Themorphisms are given by the formula hom C⊗D (( K, K (cid:48) ) , ( L, L (cid:48) )) = hom C ( K, L ) ⊗ hom D ( K (cid:48) , L (cid:48) ). Lemma 3.7.
Let C , D be unital k -linear categories, neither of which is equivalent to the zerocategory (i.e. the category having a single object ∗ and with hom( ∗ , ∗ ) = { } ). If either of thesecategories has exponential algebraic growth, then so does their tensor product C ⊗ D .Proof.
Since C , D are not equivalent to the zero category, it follows by unitality that the existfaithful functors k → C and k → D , where (by a standard abuse of notation) k denotes thecategory with one object ∗ and hom k ( ∗ , ∗ ) = k . We thus have faithful functors(3.12) D = k ⊗ D (cid:44) → C ⊗ D , and C = C ⊗ k (cid:44) → C ⊗ D . The claim now follows from Lemma 3.4. (cid:3)
Given an A ∞ category C , recall from Section 2.1 that Tw C denotes the A ∞ category oftwisted complexes over C and that Perf C denotes the idempotent completion of Tw C . Thereare canonical fully-faithful embeddings C → Tw C →
Perf C . Lemma 3.8.
Let C be an A ∞ category. Then Tw C has exponential algebraic growth if andonly if Perf C does.Proof. One direction is immediate from Lemma 3.4. For the other direction, choose a set ofmorphisms Σ := { x , . . . , x k } , where x i ∈ hom H • Perf C ( K i − , K i ) for some collection of (notnecessarily distinct) objects K , . . . , K k ∈ Perf C .By definition of the idempotent completion (see [45, (4c)]), there exists an object L i of Tw C and morphisms r i ∈ hom H Perf C ( K i , L i ) and k i ∈ hom H Perf C ( L i , K i ) such that r i k i is theidentity and k i r i is an idempotent. For each pair 1 ≤ α, β ≤ k , we define a map φ α,β : hom H • Perf C ( K α , K β ) → hom H • Perf C ( L α , L β )(3.13) y (cid:55)→ k α yr β . (3.14)It is clear that this map is injective since r α φ α,β ( y ) k β = r α k α yr β k β = y . The family of maps { φ α,β } are also compatible with composition, in the sense that φ α,β ( x ) φ γ,δ ( y ) = φ αδ ( xy ) if K β = K γ . Hence, letting ˜Σ := { φ , ( x ) , . . . , φ ( k − ,k ( x k ) } , we find that(3.15) dim W Σ ( n ) = dim W ˜Σ ( n ) , which clearly implies the desired claim. (cid:3) Algebraically hyperbolic Liouville manifolds.
All of the ingredients are now in placeto formally state the definition of an algebraically2 hyperbolic Liouville manifold.
Definition 3.9.
Let (
X, λ ) be a Liouville manifold and let W ( X ) be its wrapped Fukayacategory. We say that ( X, λ ) is algebraically hyperbolic if Tw W ( X ) has exponential algebraicgrowth.By Lemma 3.8, it would be equivalent in Definition 3.9 to require that Perf W ( X ) haveexponential algebraic growth. However, it is unclear to us whether it would also be equivalentto require that W ( X ) have exponential algebraic growth.It follows from the discussion in Section 2.1 that the notion of algebraic hyperbolicity isindependent of the grading on the wrapped Fukaya category. In contrast, we would obtainan a priori different notion if we defined the wrapped Fukaya category over a field of positivecharacteristic. 4. Filtrations on A ∞ categories Abstract filtrations on A ∞ categories.Definition 4.1. A filtration on an A ∞ category C , is the data for any pair of objects L, K ∈ C of an increasing and exhausting filtration by submodules(4.1) hom(
K, L ) a ⊂ hom( K, L )indexed by a ∈ R . For any collection of objects L , . . . , L k , real numbers a , . . . , a k , andmorphisms x i ∈ hom( L i − , L i ) a i , we require that(4.2) µ k ( x , . . . , x k ) ∈ hom( L , L k ) a + ··· + a k . In particular, (4.2) in the case k = 1 implies that (4.1) is a filtration by sub-complexes. An A ∞ category endowed with a filtration Φ is said to be a filtered A ∞ category and will be denotedby ( C , Φ).This notion of a filtered A ∞ category coincides mutatis mutandis with the notion of afiltered A ∞ category (with (cid:15) = 0) which was thoroughly developed by Biran–Cornea–Shelukhin[10, Sec. 2] in the more general setting of weakly-filtered A ∞ categories. Other related notionshave also been considered in the literature (e.g. in [21]).If ( C , Φ) is a filtered A ∞ category and A ⊂ C is a full subcategory, then the restriction of Φclearly defines a filtration on A . There is also a natural way to extend Φ to a filtration on theadditive enlargement Σ C . Namely, we set for any a ∈ R hom Σ C ( m (cid:77) i =1 S σ i K i , n (cid:77) j =1 S σ j L j ) a = ( (cid:77) i,j hom C ( K i , L j )[ σ i − σ j ]) a (4.3) := (cid:77) i,j hom C ( K i , L j ) a [ σ i − σ j ] . (4.4) LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 19
Here S σ is the σ -fold shift operator defined as in [45, (3d)]. It is immediate that this definitionindeed defines a filtration. In the sequel, we will not distinguish between a filtration Φ on an A ∞ category C , and its natural extension to Σ C . Definition 4.2.
Let ( C , Φ) be a filtered A ∞ category. Given any positive real number c >
0, let Tw C Φ ,c be the full subcategory of Tw C whose objects are twisted complexes( ⊕ mi =1 S σ i K i , ( a α,β )) with the property that(4.5) a α,β ∈ hom( S σ α K α , S σ β K β ) c . Note that Tw C Φ ,c is not in general pre-triangulated! Definition 4.3.
Let ( C , Φ) be a filtered A ∞ category and let c ∈ R be a real number. Wedefine a filtration Φ c on Tw C Φ ,c by assigning to any pair of objects(4.6) P = ( ⊕ mi =1 S σ i K i , δ P ) , and Q = ( ⊕ nj =1 S σ j L j , δ Q )the family of finite-rank submodules(4.7) hom Tw C ( P, Q ) a = (cid:77) i,j hom Σ C ( S σ i K i , S σ j L j ) a +( j − i ) c indexed by a ∈ R . The following lemma shows that this assignment indeed defines a filtrationon Tw C Φ ,c . Lemma 4.4.
Definition 4.3 satisfies the axioms for a filtration on Tw C Φ ,c .Proof. For i = 0 , . . . , k , let Q i = ( ⊕ α i l =1 ˜ K il , δ ( i ) ) be objects of Tw C Φ ,c , where each ˜ K il ∈ Σ C denotes a possibly shifted object of C . For j = 1 , . . . , k , choose a real number a j ∈ R and amorphism x j ∈ hom Tw C ( Q j − , Q j ) a j . We must show that(4.8) µ k Tw C ( x ⊗ · · · ⊗ x k ) ∈ hom Tw C ( Q , Q k ) a + ··· + a k . By linearity of µ k Tw C , we may assume that(4.9) x j ∈ hom Σ C ( ˜ K j − v j − , ˜ K jw j ) a j +( w j − v j − ) c ⊂ hom Tw C ( Q j − , Q j ) a j , for some 0 ≤ v j − ≤ α j − and 1 ≤ w j ≤ α j .Under this assumption, µ k Tw C ( x ⊗ · · · ⊗ x k ) is a finite sum of terms of the form(4.10) µ q C ( σ , . . . , σ l , x , σ , . . . , σ l , . . . , x k , σ k , . . . , σ kl k ) ∈ hom Σ C ( ˜ K α , ˜ K kβ ) ♦ for ♦ = (cid:80) kj =1 ( a j + ( w j − v j − ) c ) + (cid:80) ki =0 l i c , and where 1 ≤ α ≤ α , ≤ β ≤ α k , q = k + l + · · · + l k and the σ uv are coefficients of the twisted-complex differentials δ ( i ) .By the upper-triangular nature of the differentials δ ( i ) , we have v − α ≥ l , v j − w j ≥ l j for1 ≤ j ≤ k −
1, and β − w k ≥ l k . It follows that(4.11) k (cid:88) j =1 ( a j + ( w j − v j − ) c ) + k (cid:88) i =0 l i c ≤ ( a + · · · + a k ) + ( β − α ) c. In particular, we find that hom Σ C ( ˜ K α , ˜ K kβ ) ♦ ⊂ hom Tw C ( Q , Q k ) a + ··· + a k , which completesthe proof. (cid:3) Definition 4.5.
Let ( C , Φ) be a filtered A ∞ category. Given a pair ( L , L ) of objects of C ,we define a map i L ,L , Φ : R → Z ∪ {∞} by letting(4.12) i L ,L , Φ ( x ) = dim k im( H • hom( L , L ) x → H • ( L , L )) . Definition 4.6.
Let ( C , Φ) be a filtered A ∞ category. Given objects K, L ∈ C , we say thatthe pair (
K, L ) has exponential filtered growth (with respect to Φ) if(4.13) limsup x ∈ R + log i K,L, Φ ( x ) x = limsup n ∈ N + log i K,L, Φ ( n ) n > , where the equality in (4.13) is a consequence of the fact that i K,L, Φ ( x ) is monotonically in-creasing in x . Lemma 4.7.
Let ( C , Φ) be a filtered A ∞ category. Suppose that C has exponential algebraicgrowth. Then there exists an object L ∈ C such that the pair ( L, L ) has exponential filteredgrowth.Proof. By Lemma 3.5, there exists a finite set Σ = { x , . . . , x l } ⊂ Mor( H • ( C )) and an object L ∈ C with the property that(4.14) limsup n n log dim k W Σ ( n ; L ) > . Let ˜ x i ∈ hom( L i − , L i ) be a representative of x i ∈ H • ( L i − , L i ), where L , . . . , L l is asequence of objects of C (which may contain repetitions and contains L a fortiori ). Choose B > x i ∈ hom( L i − , L i ) B for all i . Fix n ≥
1. Given any composableword(4.15) x i . . . x i m ∈ H • ( L, L )for m ≤ n , Definition 4.1 implies that(4.16) µ (˜ x i , µ (˜ x i , . . . , µ (˜ x i m − , ˜ x i m ) . . . ))) ∈ hom( L, L ) mB ⊂ hom( L, L ) nB . But the image of the left-hand side of (4.16) under the map H • (hom C ( L, L ) nB ) → H • ( L, L )is precisely x i . . . x i m . It follows that i L,L, Φ ( nB ) ≥ dim k W Σ ( n ; L ).Hence limsup x ∈ R + x log i L,L, Φ ( x ) ≥ limsup n ∈ N + nB log i L,L, Φ ( nB )(4.17) ≥ B limsup n ∈ N + n log dim k W Σ ( n ; L )(4.18) > . (4.19) (cid:3) Lemma 4.8.
Let ( C , Φ) be a filtered A ∞ category. Let (4.20) Q := ( ⊕ mi =1 ˜ K i , ( a α,β ) ≤ α,β ≤ m ) be an object of (Tw C Φ ,c , Φ c ) for some c > , where each ˜ K i denotes a possibly shifted copy ofan object K i ∈ C . Then (4.21) i K j ,K k , Φ ( x + ( m − c ) ≥ m +1 i Q,Q, Φ c ( x ) , for some pair ( j, k ) with ≤ j, k ≤ m .Proof. For i = 1 , . . . , m , let us consider the twisted complexes(4.22) Q i := ( ⊕ ml ≥ i ˜ K l , ( a α,β ) l ≤ α,β ≤ m ) , which are evidently objects of Tw C Φ ,c . (Note by definition that we have Q = Q ).For any pair ( j, k ) and x ∈ R , we have a diagram of chain maps with exact rows LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 21 (4.23) 0 hom( Q j , Q k +1 ) x − c hom( Q j , Q k ) x hom( Q j , ˜ K k ) x
00 hom( Q j , Q k +1 ) hom( Q j , Q k ) hom( Q j , ˜ K k ) 0It follows upon passing to cohomology that(4.24) max { i Q j ,Q k +1 , Φ c ( x − c ) , i Q j , ˜ K k , Φ c ( x ) } ≥ i Q j ,Q k , Φ c ( x ) . By induction, we find that(4.25) i Q j , ˜ K µ , Φ c ( x − ( µ − k ) c ) ≥ µ − k ) i Q j ,Q k , Φ c ( x )for some k ≤ µ ≤ m . By a similar argument using the exact sequence(4.26) 0 → hom( ˜ K j , ˜ K µ ) x − c → hom( Q j , ˜ K µ ) x − c → hom( Q j +1 , ˜ K µ ) x → , one can show that(4.27) i K ν ,K µ , Φ ( y − ( j − ν ) c ) = i ˜ K ν , ˜ K µ , Φ c ( y − ( j − ν ) c ) ≥ j − ν ) i Q j , ˜ K µ , Φ c ( y ) . for any y ∈ R and some j ≤ ν ≤ m . Since ( j − ν ) + ( µ − k ) ≤ m −
1, the result follows. (cid:3)
Corollary 4.9.
Let ( A , Φ) be a filtered A ∞ category. If Tw A has exponential algebraic growth,then there exists a pair of objects ( K, L ) which has exponential filtered growth with respect to Φ .Proof. If Tw A has exponential algebraic growth, then so does Tw A Φ ,c for some c > (cid:3) The length filtration on the based loop of a manifold.
Before considering filtrationson wrapped Fukaya categories, we consider the simpler case of a filtration of the dg algebra ofbased Moore loops of a smooth manifold.Let ( N, ∗ ) be a pointed, connected closed manifold and let(4.28) Ω M ∗ N = { ( γ, T ) | γ : S → N, T ∈ [0 , ∞ ) } be the space of smooth Moore loops. This is well-known to be homotopy equivalent to theusual based loop space (see [11, Sec. 5.1]). There is a natural associative productΩ M ∗ N × Ω M ∗ N → Ω M ∗ N (4.29) (( γ , T ) , ( γ , T )) (cid:55)→ ( γ ∗ γ , T + T ) , where γ ∗ γ is the concatenation of loops. This product of spaces induces a product onsingular k -chains C −• (Ω M ∗ N ) (with grading reversed for future convenience), which coincideswith the Pontryagin product after passing to homology. We may therefore view C −• (Ω M ∗ N )as a Z -graded dg algebra over k .Let us now endow X with a Riemannian metric g . Given a loop γ : S → N , its length (cid:96) ( γ ) is defined by the formula (cid:96) ( γ ) = (cid:82) S | ˙ γ | g . Note that the length is independent of theparametrization and additive under concatenation of loops.For l ≥
0, let Ω
M,l ∗ N ⊂ Ω M ∗ N be the space of Moore loops of length at most l . We have afiltration of spaces { Ω M,l ∗ N } l ≥ , which is extended to an R -filtration by setting Ω M,l ∗ N = ∗ for l ≤
0. We can then consider the induced filtration on singular chains(4.30) C −• (Ω M ∗ N ) l := C −• (Ω M,l ∗ N ) . The following proposition is immediate.
Lemma 4.10.
The data (4.30) defines a filtration on the dg algebra C −• (Ω M ∗ N ) in the senseof Definition 4.1. (cid:3) Corollary 4.11.
Suppose that N is a simply-connected closed manifold. If Tw C −• (Ω M ∗ N ) has exponential algebraic growth, then N is rationally hyperbolic.Proof. Indeed, it follows from Corollary 4.9 that(4.31) limsup n log dim( H • ( C −• (Ω M ∗ N ) n ) → H −• (Ω M ∗ N )) n > . Now a theorem of Gromov (see [32, Thm. 1.4] or [33, Thm. 7.3]) states that(4.32) dim( H • (Ω M,mC ∗ N ) → H • (Ω ∗ N )) ≤ (cid:88) i ≤ m dim H i (Ω M ∗ N )for some C > H • ( C −• (Ω M ∗ N ) n ) → H −• (Ω M ∗ N )) = dim( H −• (Ω M,n ∗ N ) → H −• (Ω M ∗ N )) by defi-nition, it follows that(4.33) limsup n log( (cid:80) i ≤ n dim H i (Ω M ∗ N )) n > . According to Definition 2.19, this means that N is rationally hyperbolic. (cid:3) Remark . The assumption that N is a manifold in the statement of Corollary 4.11 wasused to construct the length-filtration on C −• (Ω ∗ N ), and to appeal to Gromov’s theorem. Wedo not know whether the statement Corollary 4.11 generalizes to arbitrary simply-connectedfinite CW complexes. Proof of Theorem 1.9.
Suppose that (
X, λ ) algebraically hyperbolic and is moreover Liouvilleequivalent to ( T ∗ N, λ can ), where N is a simply-connected spin manifold. This means in par-ticular that Tw W ( T ∗ N ) has exponential algebraic growth.It was shown by Abouzaid [2] (see also [31, Cor. 6.1]) that there is a quasi-equivalence of A ∞ algebras(4.34) C −• (Ω M ∗ N ; L k ) → CW • ( F p , F p ) T ∗ N which in fact induces an equivalence on twisted complexes(4.35) Tw C −• (Ω M ∗ N ; L k ) → Tw W ( T ∗ N ) . Since N is spin, the local system L k is trivial. Hence Corollary 4.11 implies that N is rationallyhyperbolic. (cid:3) Action filtrations and wrapped Fukaya categories
An action filtration on the wrapped Fukaya category of a Liouville manifold.
Let (
X, λ ) be a Liouville manifold. Let W lin ( X ; H ) be the wrapped Fukaya category of ( X, λ ),presented via Construction 2.15 for some choice of Hamiltonian H (and additional auxiliarydata). Following [5, Lem. 3.12], for any pair of objects L α , L β and any integer w ∈ Z , weconsider the chain sub-complexes C w ( L α , L β ) ⊂ CW • ( L α , L β ) defined as follows(5.1) C w ( L α , L β ; H ) = CF • ( L α , L β ; H )[ q ] ⊕ · · · ⊕ CF • ( L α , L β ; ( w − H )[ q ] ⊕ CF • ( L α , L β ; wH ) , (if w ≤
0, we let C w ( L α , L β ) = 0).Given an element c ∈ CW • ( K, L ; H ), it will be convenient to write | c | = w if c ∈ C w ( K, L ; H )but c / ∈ C w − ( K, L ; H ). LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 23
Definition 5.1.
Let Φ be the assignment, to any pair of objects L α , L β of W lin ( X ; H ) andany a ∈ R , of the (finite dimensional) subspace(5.2) CW • ( L α , L β ; H ) a := C (cid:98) a (cid:99) ( L α , L β ; H ) ⊂ CW • ( L α , L β ; H ) . The following lemma shows that Φ defines a filtration on the A ∞ category W lin ( X ). Lemma 5.2.
The data Φ defines a filtration on the A ∞ category W lin ( X ; H ) (see Defini-tion 4.1).Proof. This is just a matter of tracing through the definition of the operations µ d in [5, Sec.3h]. For any d ≥
1, choose a collection of d + 1 objects L , . . . , L d ∈ W lin ( X ). As explained in[5], the operation µ d is constructed by counting moduli spaces of stable popsicles. The relevantmoduli spaces (i.e. those that can have a nonzero count) are parametrized by triples ( d, F, w ),where F ⊂ { , . . . , d } and w = ( w , . . . , w d ) is a ( d + 1)-tuple of positive integers satisfying w = w + · · · + w d + | F | .For each such triple, there is a map(5.3) µ ( d,F, w ) : CF • ( L , L ; w H )[ q ] ⊗ · · · ⊗ CF • ( L d − , L d ; w d H )[ q ] → CF • ( L , L ; w H )[ q ]which we decompose as µ ( d,F, w ) = µ ( d,F, w )1 + qµ ( d,F, w )2 .For i = 1 , . . . , d , choose c i = a i + qb i ∈ CF • ( L i − , L i ; w i H )[ d ].We first claim that | µ ( d,F, w )1 ( c , . . . , c d ) | ≤ | c | + · · · + | c d | . By linearity, we may assumefor all i that either a i = 0 or b i = 0. In fact, we may assume that b i (cid:54) = 0 if and only if i ∈ F (indeed, one can check (see [5, Sec. 3h]) that µ ( d,F, w )1 vanishes otherwise). We thereforehave that | c i | = w i + 1 if i ∈ F and | c i | = w i otherwise. Hence | µ ( d,F, w )1 ( c , . . . , c d ) | ≤ w = w + · · · + w d + | F | ≤ | c | + · · · + | c d | .Next, we show that | qµ ( d,F, w )2 ( c , . . . , c d ) | ≤ | c | + · · · + | c d | . To see this, note (see [5, Sec.3h]) that(5.4) µ ( d,F, w )2 = d (cid:88) j =1 ± µ ( d,F, w )1 ( c , . . . , c j − , ∂ q c j , c j +1 , . . . , c d ) , where ∂ q ( a + qb ) := b . Hence | qµ ( d,F, w )2 ( c , . . . , c d ) | ≤ j | µ ( d,F, w )1 ( c , . . . , c j − , ∂ q c j , c j +1 , . . . , c d ) | (5.5) ≤ j {| c | + · · · + | c j − | + ( | c j | −
1) + | c j +1 | + · · · + | c d |} (5.6) = | c | + · · · + | c d | . (5.7)Finally, the operation µ d is obtained by summing the µ ( d,F, w ) over all triples ( d, F, w ) andextending by zero to all of CW • ( L , L ; H ) ⊗ · · · ⊗ CW • ( L d − , L d ; H ). We conclude that | µ d ( x , . . . , x d ) | ≤ | x | + · · · + | x d | , which proves that Φ defines a filtration. (cid:3) Algebraic varieties.
We carry over the notation from the previous section. For a ∈ R and any pair of objects K, L ∈ W lin ( X ; H ), we have inclusions of chain complexes(5.8) CF • ( K, L ; (cid:98) a (cid:99) H ) ⊂ CW • ( K, L ; H ) a ⊂ CW • ( K, L ; H ) . For b ≥ a , there is a natural continuation map (see [5, Sec. 3g])(5.9) κ : CF • ( K, L ; (cid:98) a (cid:99) H ) → CF • ( K, L ; (cid:98) b (cid:99) H ) . Lemma 5.3 (see Lem. 3.12 in [5]) . For any a ≤ b , there is a diagram of chain complexes (5.10) CF • ( K, L ; (cid:98) a (cid:99) H ) CF • ( K, L ; (cid:98) b (cid:99) H ) CW • ( K, L ; H ) a CW • ( K, L ; H ) bκ commuting up to homotopy. Moreover, the vertical arrows are quasi-isomorphisms. (cid:3) Corollary 5.4.
There is a canonical isomorphism (5.11) lim a ∈ R HF • ( K, L ; (cid:98) a (cid:99) H ) (cid:39) −→ lim a ∈ R H • ( CW • ( K, L ; H ) a ) = HW • ( K, L ; H ) . Moreover dim k im( HF • ( K, L ; (cid:98) a (cid:99) H ) → lim a ∈ R HF • ( K, L ; (cid:98) a (cid:99) H )) = dim k im( HF • ( K, L ; (cid:98) a (cid:99) H ) → HW • ( K, L ; H ))= dim k im( H • ( CW • ( K, L ; H ) a ) → HW • ( K, L ; H ))= i K,L, Φ ( a ) . (cid:3) We now state the following theorem of McLean.
Theorem 5.5 (McLean; see Thm. 1.2 and Def. 2.4 in [38]) . Suppose that ( X, dλ ) is symplec-tomorphic to an affine variety, or that ( ∂X, λ | ∂X ) is contactomorphic to the link of an isolatedcomplex singularity. Then (5.12) limsup n n log(dim k ( HF • ( K, L ; nH ) → HW • ( K, L ; H ))) ≤ dim M . Corollary 5.6.
Suppose that there exists a pair of objects
K, L ∈ W lin ( X ; H ) which has expo-nential filtered growth with respect to the filtration Φ . Then ( X, dλ ) is not symplectomorphicto an affine variety and ( ∂X, λ | ∂X ) is not contactomorphic to the link of an isolated complexsingularity.Proof. Suppose for contradiction that the conclusion is false. According to Theorem 5.5, itfollows that(5.13) limsup n n log(dim k ( HF • ( K, L ; nH ) → HW • ( K, L ; H ))) ≤ dim M . On the other hand, we have by hypothesis that(5.14) lim sup n ∈ N + n log i K,L, Φ ( n ) > . It follows that there exists a sequence of natural numbers n i → ∞ such that log i K,L, Φ ( n i ) >αn i for some α >
0. It then follows from Corollary 5.4 that we havelim sup i →∞ αn i log n i ≤ lim sup n log i K,L, Φ ( n )log n (5.15) = lim sup n log(dim k ( HF • ( K, L ; nH ) → HW • ( K, L ; H )))log n (5.16) ≤ dim M . (5.17)This gives the desired contradiction. (cid:3) LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 25
Proof of Theorem 1.10.
Let (
X, λ ) be an algebraically hyperbolic Liouville manifold and sup-pose that the conclusion is false. According to Fact 2.17, we may use W lin ( X ; H ) as a model forthe wrapped Fukaya category. By Lemma 5.2, W lin ( X ; H ) can be endowed with the filtrationΦ. Now combine Corollary 4.9 and Corollary 5.6. (cid:3) Topological entropy.
The purpose of this section is to prove Theorem 1.11 using thework of Alves and Meiwes [9].Let (
X, λ ) be a Liouville manifold. As in the previous sections, let us fix data for defin-ing the wrapped Fukaya category of (
X, λ ) according to Construction 2.15; that is, we fix adecomposition X = X ∪ [0 , ∞ ) × Y , a strictly positive Hamiltonian H , a collection of (spin)Lagrangians { L α } and additional auxiliary data. We assume that this data satisfies the vari-ous assumptions described in Construction 2.15. Note in particular that this implies that the( Y, λ | Y ) is non-degenerate as a contact manifold and that N ∩ S ( λ | Y ) = ∅ , where S ( λ | Y ) is theReeb action spectrum.Let us now choose a pair of (spin) Lagrangians K, L ∈ { L α } . Let us say that a (time-independent) Hamiltonian H (cid:48) : X → R is AM-admissible if H (cid:48) < X and H (cid:48) ( t, x ) = µe t + β on [0 , ∞ ) × Y for some µ > β ≤ − µ . We call µ the slope of the Hamiltonian H (cid:48) . Wedefine a partial order (cid:22) on the set of all AM-admissible Hamiltonians by setting H (cid:48) (cid:22) ˜ H (cid:48) if H (cid:48) ≤ ˜ H (cid:48) . We let H be the resulting poset of AM-admissible Hamiltonians, and we let H n (foreach n >
1) be the sub-poset of AM-admissible Hamiltonians of slope strictly less than n .For n >
1, we define following [9](5.18) HW • , ( n ) ( K, L ) := lim H (cid:48) ∈H n HF • ( K, L ; H (cid:48) ) . Remark . Our definition of HW • , ( n ) ( K, L ) does not a priori coincide with the definitiongiven by Alves and Meiwes in [9, Def. 2.14]. However, the definitions are shown to be equivalentin this particular case in [9, Sec. 2.2.4]. More generally, Alves and Meiwes adopt different Floertheoretic conventions than us for setting up Floer theory (for example, they work with Z / n ≥
1, we have a natural map HW • ;( n ) ( K, L ) → lim H (cid:48) ∈H HF • ( K, L ; H (cid:48) ) (cid:39) −→ lim m ∈ Z HF • ( K, L ; mH )(5.19) (cid:39) −→ HW • ( K, L ; H ) , (5.20)The isomorphism (5.20) is the one furnished by Corollary 5.4. The isomorphism in (5.19)can be seen as follows. First, note that we can restrict to the (cofinal) family of AM-admissibleHamiltonians H (cid:48) having integral slope. Next, recall that H is strictly positive and has slope1. It follows that there is a continuation map HF • ( K, L ; H (cid:48) ) → HF • ( K, L ; nH ) whenever n ≥ slope H (cid:48) , which is an isomorphism when slope H (cid:48) = n (see [38, Lem. 2.5]).We let i K,L ; AM ( n ) be the dimension of the image of (5.19).Observe that if H (cid:48) ∈ H n , we have H (cid:48) ≤ nH . We therefore a factorization of (5.19) as follows(5.21) HF • ;( n ) ( K, L ; H (cid:48) ) → HF • ( K, L ; nH ) → lim m ∈ N + HF • ( K, L ; mH ) = HW • ( K, L ; H ) . Moreover, since N ∩ S ( Y, λ | Y ) = ∅ , a standard continuation argument shows that the first mapin (5.21) is an isomorphism. Hence, by Corollary 5.4, we find that(5.22) i K,L ; AM ( n ) = dim( HF • ( K, L ; nH ) → lim m ∈ Z HF • ( K, L ; mH ) = HW • ( K, L ; H )) = i K,L, Φ ( n ) . Let us now consider a possibly different contact form α on ( Y, λ | Y ). We write α = f α λ | Y for some f α : Y → R . Alves and Meiwes prove the following theorem. Theorem 5.8 (Alves–Meiwes; see Thm. 1.7 in [9]) . Suppose that ∂K and ∂L are diffeomorphicto a sphere. Assuming without loss of generality that f α ≥ , we have: (5.23) h top ( φ α ) ≥ f α lim sup n ∈ N + n i K,L ; AM ( n ) . Remark . The proof of Theorem 5.8 in [9, Thm. 1.7] is written under the more restrictiveassumption that K = L . However, as noted by the authors in [9, Sec. 1.3], their argumentalso applies mutatis mutandis if ∂K ∩ ∂L = ∅ . Indeed, the basic idea is that HW • ( K, L ; H ) isessentially generated by Reeb chords ∂K → ∂L ; hence, a lower bound for the action growthof HW • ( K, L ; H ) gives a lower bound of the number of Reeb chords ∂K → ∂L as a functionof their length (see [9, Cor. 4.8]. One can then obtain a lower bound for topological entropyof the Reeb flow in terms of the growth of the number of Reeb chords ∂K → ∂L as in [9, Sec.4.2] (see also [8], where a similar argument is carried out). Corollary 5.10.
Suppose that there exists a pair of objects
K, L ∈ W lin ( X ; H ) which hasexponential filtered growth with respect to the filtration Φ (see Definition 4.6). Then anycontact form α on the contact manifold ( Y = ∂X, λ | Y ) has the property that its Reeb flow haspositive topological entropy.Proof. Combine (5.22) and Theorem 5.8. (cid:3)
Proof of Theorem 1.11.
Let
A ⊂ W ( X ; H ) be the full subcategory whose objects are the { L α } . By hypothesis, there is a quasi-equivalence Perf A →
Perf W ( X ; H ). Since ( X, λ ) isalgebraically hyperbolic, it follows by Lemma 3.8 that Tw A has exponential algebraic growth.By Lemma 5.2, W lin ( X ) can be endowed with a filtration, which thus induces a filtrationon A . Now combine Corollary 4.9 and Corollary 5.10. (cid:3) Proof of Corollary 1.12.
Let α be a contact form on ( ∂ ∞ X, ξ ∞ ) and let h top ( φ α ) be the topo-logical entropy of the associated Reeb flow. For T ≥
0, let π α ( T ) ∈ N be the number ofgeometrically distinct simple closed Reeb orbits of length at most T . Then [35, Thm. 1.1]states that(5.24) π α ( T ) ≥ Ce T · h top ( φ α ) /T, for some constant C > T large enough. The corollary now follows from Theorem 1.11. (cid:3) Computations with Liouville sectors
Sectorial decomposition.
In this section, we apply the theory of Ganatra–Pardon–Shende [30] to study the wrapped Fukaya category of a Liouville manifold (
X, λ ) which isobtained by gluing a pair of Liouville manifolds ( X , λ ) , ( X , λ ) along a Weinstein hypersur-face, as described in Section 2.3. We are particularly interested in describing gluings with theproperty that Tw W ( X ) or Tw W ( X ) embeds faithfully into Tw W ( X ).We assume in this section that the reader is familiar with the definition and basic propertiesof homotopy colimits of A ∞ categories. We refer to [30, A.4] for a self-contained exposition.The key tool which we will use in this subsection is the following theorem, which allows oneto express the wrapped Fukaya category of a Liouville manifold in terms of a decompositionof this manifold into Liouville sectors. Before stating this theorem, let us recall [29, Def. 2.10]that the boundary of a Liouville sector ( X, λ ) admits a canonical “symplectic reduction” map
LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 27 ∂X → F which simply quotients by the characteristic foliation. The pullback of λ | ∂X by asection of the quotient map endows F with a Liouville form which is canonical up to adding acompactly-supported exact form. Theorem 6.1 (Ganatra–Pardon–Shende; see Thm. 1.20 in [30]) . Let ( X, λ ) be a Liouvillemanifold. Suppose that X admits a decomposition as a union of two Liouville sectors X = X ∪ X meeting along a hypersurface H = X ∩ X . Suppose that F = ∂X/ R is Weinsteinup to deformation. Then there is a fully faithful embedding (6.1) hocolim( W ( X ) ← W ( F ) → W ( X )) → W ( X ) . If X and X are Weinstein, then this map is a pre-triangulated equivalence. The maps in (6.1) arise as follows. Up to deforming (
X, λ ), we can assume that F has aneighborhood of the form F × T ∗ [0 ,
1] (with the product Liouville structure). We can alsoup slightly thicken X , X so that they overlap in the set F × T ∗ [3 / , / W ( F ) → W ( X i ) is the composition of the K¨unneth embedding W ( F ) → W ( F × T ∗ [3 / , / W ( F × T ∗ [3 / , / → W ( X i ). (Recall [29, Sec. 3.6] thatdeformations of sectors induce quasi-equivalences on the wrapped Fukaya category.)We have the following corollary of Theorem 6.1 as a consequence of the discussion in Sec-tion 2.3. Corollary 6.2.
Let ( X , λ ) and ( X , λ ) be Liouville manifolds. Suppose that ( X, λ ) is aLiouville manifold obtained by gluing X , X along a Weinstein hypersurface F . Then there isa fully faithful embedding (6.2) hocolim( W ( ˇ X ) ← W ( F ) → W ( ˇ X )) → W ( X ) , where ˇ X i = X i \ ( R + × ( F × [ − (cid:15), (cid:15) ])) as explained in Section 2.3. Moreover, (6.2) is a pre-triangulated equivalence if the X i are Weinstein. Note that we have W ( ˇ X i ) = W ( X i , F ) = W ( X i , c F ) according to [30, Sec. 2.5] , where c F denotes the skeleton of F following the notationin [30] . We now state some algebraic lemmas which will be used in the next subsection.
Lemma 6.3 (Lem. A.5 in [30]) . Let {C σ } σ ∈ Σ → {C (cid:48) σ } σ ∈ Σ be a map of diagrams of A ∞ cat-egories indexed by a poset Σ . If C σ → C (cid:48) σ is a quasi-equivalence (resp. a pre-triangulatedequivalence, a Morita equivalence) for each σ ∈ Σ , then so is the induced functor (6.3) hocolim σ ∈ Σ C σ → hocolim σ ∈ Σ C (cid:48) σ . Lemma 6.4.
Consider the following maps of diagrams of A ∞ categories (6.4) {C σ } σ ∈ Σ → {C (cid:48) σ } σ ∈ Σ → {C (cid:48)(cid:48) σ } σ ∈ Σ indexed by a poset Σ . Suppose that the compositions C σ → C (cid:48) σ → C (cid:48)(cid:48) σ are Morita equivalences(i.e. fully faithful and the image split generates) for each σ ∈ Σ . Then the natural functors (6.5) hocolim σ ∈ Σ C σ → hocolim σ ∈ Σ C (cid:48) σ and (6.6) Tw(hocolim σ ∈ Σ C σ ) → Tw(hocolim σ ∈ Σ C (cid:48) σ ) are faithful (i.e. the induced cohomological functor is faithful). Proof.
According to Lemma 6.3, the induced map hocolim σ ∈ Σ C σ → hocolim σ ∈ Σ C (cid:48)(cid:48) σ is a Moritaequivalence, so in particular it is fully faithful. According to [45, Lem. 3.23], this map remainsfully faithful upon passing to twisted complexes. The desired claims now follow immediatelyfrom the functoriality of the homotopy colimit and of Tw( − ). (cid:3) Lemma 6.5.
Consider a diagram of A ∞ categories (6.7) A B C . i j If the image of i is acyclic (i.e. every object in the image is acyclic), then there is a fullyfaithful embedding A → hocolim(
A ← B → C ) .Proof. Let G = Groth( A ← B → C ) be the Grothendieck construction as described in [30, A.4].Recall that the set of objects of G is the union of the set of objects of A , B and C .Let Q be a subset of objects in G which is constructed by choosing a quasi-representative ofcone( K e −→ i ( K )) and cone( K e −→ j ( K )) for all K ∈ B . Here the morphisms e are the “adjacentmorphisms” which correspond to the identity in H (hom G ( K, i ( K ))) = H (hom A ( i ( K ) , i ( K )))and H (hom G ( K, j ( K ))) = H (hom C ( j ( K ) , j ( K ))). Since H (hom A ( i ( K ) , i ( K ))) is acyclic byhypothesis, we may (and do) in fact assume that the chosen quasi-representative for cone( K e −→ i ( K )) is a shift of K .By definition, hocolim( A ← B → C ) = G / Q, where the quotient is defined as in [29, Def.3.10]. Recall that for any objects X ∈ A and Y ∈ B ∪ C , we have G ( X, Y ) = 0. It also followsthat if
Y, Z are objects in
B ∪ C and Y → Z is a morphism, then G ( X, cone( K → Z )) = 0.Inspecting now the definition of the morphisms in the quotient category, we see that given X, X (cid:48) ∈ A , we tautologically have G / Q( X, X (cid:48) ) = A ( X, X (cid:48) ), and similarly for the higheroperations. (cid:3)
The subcritical case.
Given a Liouville sector (
X, λ ) of dimension 2 n , recall that aLiouville hypersurface F ⊂ ∂X is said to be subcritical if its its skeleton is contained in thesmooth image of a second countable manifold of dimension < n − Proposition 6.6.
Let ( X , λ ) and ( X , λ ) be Liouville manifolds. Suppose that ( X, λ ) isa Liouville manifold obtained by gluing X , X along a subcritical Weinstein hypersurface F .Then there are fully faithful embeddings W ( X ) → W ( X ) and W ( X ) → W ( X ) .Proof. Since F is subcritical we have W ( F ) = 0. We also have W ( X i , c F ) = W ( X i ) by stopremoval (see [30, Thm. 1.16]). Now combine Corollary 6.2 and Lemma 6.5. (cid:3) Lemma 6.7.
Let ( X, λ ) be a Weinstein manifold and let F ⊂ ∂ ∞ X be a Liouville hypesurface.Suppose that Skel( F ) ⊂ ∂ ∞ X is a (smooth) sphere and let D ⊂ W ( X, Λ) be the linking disk.If Λ is a loose Legendrian sphere, then CW • ( D, D ) X, Λ is acyclic.Proof. Let ˜ X be obtained by attaching a Weinstein handle along Λ and let C ⊂ ˜ X be thecorresponding cocore. According to the main result of [17], CW • ( C, C ) ˜ X is isomorphic as an A ∞ algebra to the Chekanov–Eliashberg dg algebra of the attaching sphere Λ. It follows that CW • ( C, C ) ˜ X is acyclic since Λ is loose. However, [30, Cor. 1.21] implies that(6.8) CW • ( C, C ) ˜ X = CW • ( D, D ) X, Λ ⊗ C −• (Ω S n − ) k. It follows by Lemma 6.5 that CW • ( D, D ) X, Λ is also acyclic (indeed, put k = C = C (cid:48) = C (cid:48)(cid:48) = B = B (cid:48)(cid:48) , A = A (cid:48) = A (cid:48)(cid:48) = CW • ( D, D ) X, Λ and B (cid:48) = C −• (Ω S n − )). (cid:3) Proposition 6.8.
Let ( X , λ ) and ( X , λ ) be Liouville manifolds. Suppose that ( X, λ ) isobtained by gluing X , X along the a Weinstein hypersurface F . If Skel( F ) ⊂ ∂X is a loose LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 29
Legendrian sphere Λ and ( X , λ ) is Weinstein up to deformation, then there exists a fullyfaithful embedding W ( X ) → W ( X ) .Proof. Since F is Weinstein, it follows from Proposition 2.11 that F = T ∗ Λ up to deformation.The map W ( F ) → W ( ˇ X ) = W ( X , F ) = W ( X , Λ) sends the cotangent fiber to the linkingdisk of Λ. In particular W ( F ) → W ( ˇ X ) has acyclic image according to Lemma 6.7. Theproposition now follows from Lemma 6.5. (cid:3) The critical case.
The input for this subsection consists in the following: • A Liouville manifold (
X, λ ) which is the union of two Liouville sectors X = V ∪ T ∗ D n ,where H = V ∩ T ∗ D n is a hypersurface; • A Liouville sector ( X (cid:48) , λ ) which is the union of two Liouville sectors X (cid:48) = V ∪ V ,where H (cid:48) = V ∩ V is a hypersurface disjoint from ∂X (cid:48) .As mentioned in Section 6.1, we may deform λ near H (resp. λ (cid:48) near H (cid:48) ) so that we havethe following diagram of Liouville sectors(6.9) T ∗ D n T ∗ S n − × T ∗ [3 / , / V X V X (cid:48) By precomposing with the K¨unneth embedding W ( T ∗ S n − ) → W ( T ∗ S n − × T ∗ [3 / , / A ∞ categories(6.10) W ( T ∗ D n ) W ( T ∗ S n − ) W ( V ) W ( X ) W ( V ) W ( X (cid:48) ) i i i Proposition 6.9.
Suppose that the following two conditions are satisfied: (i)
The Liouville sector V is Weinstein; (ii) There exists a functor (cid:15) : Tw W ( V ) → Tw k with the property that the image of acotangent fiber L (cid:48) ⊂ T ∗ S n − under the composition (cid:15) ◦ i is isomorphic to k as anobject of H (Tw k ) .Then there is a faithful functor (6.11) Tw W ( X ) → Tw W ( X (cid:48) ) . Proof.
Let L = L (cid:48) × T ∗ / be the image of L (cid:48) under the above K¨unneth embedding. First of all,it follows from (i) and Theorem 6.1 that the natural map(6.12) Tw(hocolim( W ( T ∗ D n ) i ←− W ( T ∗ S n − ) i −→ W ( V ))) → Tw W ( X )induced by (6.10) is a quasi-equivalence. It also follows from Theorem 6.1 (and the fact [45,Lem. 3.23] that fully faithful functors remain fully faithful under passing to twisted complexes)that(6.13) Tw(hocolim( W ( V ) i ←− W ( T ∗ S n − ) i −→ W ( V ))) → Tw W ( X (cid:48) )is fully-faithful.According to [31, Cor. 6.1], there is a quasi-equivalence CW • ( L, L ) T ∗ D n (cid:39) C −• ( ∗ ) = k . Wethus obtain from (6.10) a diagram of A ∞ categories (6.14) kCW • ( L, L ) T ∗ D n CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − W ( T ∗ D n ) W ( T ∗ S n − ) W ( V )It follows from Lemma 6.3 that the induced map(6.15)hocolim CW • ( L, L ) T ∗ D n CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − W ( V ) → hocolim kCW • ( L (cid:48) , L (cid:48) ) T ∗ S n − W ( V ) is a quasi-equivalence.According to [31, Cor. 6.1], the wrapped Fukaya category of the cotangent bundle of a con-nected, compact manifold (with possibly non-empty boundary) is generated by any cotangentfiber. By Lemma 6.3, it follows that the induced map(6.16)hocolim CW • ( L, L ) T ∗ D n CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − W ( V ) → hocolim W ( T ∗ D n ) W ( T ∗ S n − ) W ( V ) is a pre-triangulated equivalence.By strict unitality, the composition CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − → CW • ( L, L ) T ∗ D n → k admits asection σ : k → CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − . We thus have a diagram of A ∞ categories(6.17) k CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − Tw k W ( V ) W ( T ∗ S n − ) W ( V ) σ(cid:15) i i By (ii), the composition k → Tw k sends the single object in the source to an object whichis isomorphic to k in H (Tw k ). It is therefore a pre-triangulated equivalence (see [30, Lem.A.3]). Similarly, CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − → W ( T ∗ S n − ) is also a pre-triangulated equivalence sincethe cotangent fiber generates. It follows according to Lemma 6.4 that(6.18)Tw hocolim( k ← CW • ( L (cid:48) , L (cid:48) ) T ∗ S n − → W ( V )) → Tw hocolim( W ( V ) ← W ( T ∗ S n − ) → W ( V ))is faithful.The desired conclusion now follows from combining (6.12), (6.16), (6.15), (6.18) and (6.13)(in that order). (cid:3) LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 31
Lemma 6.10.
Let M be a closed manifold and let F ⊂ T ∗ M be a cotangent fiber. Then thereis a natural functor (well-defined up to homotopy equivalence) (6.19) Tw W ( T ∗ M ) → Tw k which takes F to the object k ∈ Tw k .Proof. Observe that the zero section 0 M ⊂ T ∗ M defines an A ∞ module over CW • ( F p , F p ) T ∗ M ,which is the same thing as a map of A ∞ algebras(6.20) CW • ( F p , F p ) T ∗ M → hom Ch k ( CW • ( F p , M ) T ∗ M , CW • ( F p , M ) T ∗ M ) = k. Since the cotangent fiber generates [31, Cor. 6.1], we may invert the quasi-equivalence(6.21) Tw CW • ( F p , F p ) T ∗ M → Tw W ( T ∗ M )uniquely up to homotopy, and thus composition with (6.20) induces the desired functor. (cid:3) Lemma 6.11.
Let ( X, λ ) be a Liouville manifold. Suppose that there is a Liouville inclusion i : ( T ∗≤ (cid:15) M, λ std ) (cid:44) → ( X, λ ) and an inclusion of Liouville sectors j : ( X , λ ) (cid:44) → ( X, λ ) . Let L ⊂ X be a Lagrangian (cylindrical at infinity and equipped with a spin structure) and supposethat the restriction of L to the image of i agrees with a cotangent fiber F . Then the Viterbotransfer map described in [30, Sec. 8.3](6.22) Tw W ( X ) → Tw W ( T ∗ M ) sends L to an object isomorphic to F in H (Tw W ( T ∗ M )) .Proof. Inspecting the construction of the Viterbo transfer map in [30, Sec. 8.3], we see that itsuffices to verify that the image L (cid:55)→ L ˜ × i R under the composition(6.23) W ( X ) (cid:44) → W ( X × C Re ≥ , c X ) → W ( X × C Re ≥ , c T ∗ M )is isomorphic to the image F (cid:55)→ F × i R under the stabilization embedding(6.24) W ( T ∗ M ) (cid:44) → W ( X × C Re ≥ , c T ∗ M ) . Note first that we may assume that the “cylindrization” procedure described in [30, Sec.6.2] which replaces L × i R with L ˜ × i R acts trivially on ( U ∩ L ) × i R , where U is small opencontaining 0 M ∩ L . Indeed, this follows from the fact that the Liouville form vanishes identicallyon U ∩ L .We now give a similar argument to the proof of Thm. 1.11 in [30]. As explained in [30, Sec.7.2], F × i R is a linking disk for the stop c T ∗ M = 0 M . On the other hand, consider an isotopy i R (cid:32) ( i R ) (cid:48) of the line i R ⊂ C through exact cylindrical Lagrangians which fixes the end+ i ∞ and moves the end − i ∞ past the stop { + ∞} ⊂ C in the positive direction. We may inaddition assume that the Lagrangian arcs in this isotopy have compactly supported primitive(see [30, Lem. 6.1]). This induces an isotopy L ˜ × i R (cid:32) L ˜ × ( i R ) (cid:48) ⊂ X × C which passes throughthe stop c T ∗ M positively in a single point. But L ˜ × ( i R ) (cid:48) is clearly the zero object (e.g. because( i R ) (cid:48) is displaceable in the complement of the stop). Hence it follows from the wrapping exacttriangle [30, Thm. 1.9] that L ˜ × i R is isomorphic to a linking disk. (cid:3) Corollary 6.12.
Let ( W, λ W ) be a Liouville manifold. With the notation of Proposition 6.9,suppose that ( V , λ ) is constructed by gluing ( W, λ W ) to the Liouville sector ( T ∗ D n , λ std ) along some Weinstein hypersurface F ⊂ ∂ ∞ T ∗ D n , as in Section 2.3. Then there is a functor (cid:15) : Tw W ( V ) → Tw k satisfying condition (ii) in the statement of Proposition 6.9. Proof.
We may assume that the closure of F is disjoint from the image of the embedding T ∗ S n − × T ∗ [3 / , / → T ∗ D n in (6.9).Note that ( T ∗ S n , λ std ) is the union of two copies of ( T ∗ D n , λ std ) which intersect along thehypersurface H consisting of the union of fibers over the equators. Since ∂V = ∂T ∗ D n = H by construction, we may form the Liouville manifold(6.25) X (cid:48)(cid:48) = V ∪ H T ∗ D n by gluing.We now have an obvious sectorial embedding j : V (cid:44) → X . Observe also that there is atautological embedding i : T ∗≤ (cid:15) S n → X (for some metric and suitable (cid:15) > (cid:15) : Tw W ( V ) → Tw k as the composition of (6.22) and (6.19). (cid:3) Proofs of theorems from Sections 1.3 and 1.5
Cotangent bundles.
This section is an elaboration on the discussion in Section 1.3. Let X be a connected finite CW-complex. Let L be the Z -local system on Ω ∗ X whose holonomyrepresentation along the loop ( γ ts ) t ∈ S is defined by evaluating w ( X ) along the torus ( s, t ) (cid:55)→ γ ts . It will be convenient to set L k := L ⊗ Z k . The Pontryagin product defines a ring structureon H • (Ω ∗ X ; L ), and more generally on H • (Ω ∗ X ; L k ), where the identity element is given bythe class of the constant map (see [6, Sec. 1.3]).Let us now consider a closed (possibly non-orientable) manifold M . By the celebratedresult of Abbondandolo–Schwarz [1] and Abouzaid [2] (see also [31, Cor. 6.1]), there is aquasi-equivalence of A ∞ algebras(7.1) CW • ( F p , F p ) T ∗ M (cid:39) C −• (Ω M ; L k )for any p ∈ M , where L k is defined as above and the Z -grading on the right-hand side iscollapsed to a Z / Z -graded A ∞ algebras with Z -coefficients (and with L instead of L k on the right hand side), but we only consider theweaker version for consistency with the rest of the paper.Passing to cohomology in (7.1), we find that H • ( F p , F p ) (cid:39) H −• (Ω p M ; L k ) as associativealgebras. It follows by Lemma 3.4 that a sufficient condition for W ( X ) and hence Tw W ( X )to have exponential algebraic growth is that the associative algebra H −• (Ω p M ; L k ) has expo-nential algebraic growth. Recall from Definition 1.2 that a manifold M is said to be in theclass E if H −• (Ω p M ; L k ) has exponential algebraic growth. Lemma 7.1.
The following manifolds are in the class E . (i) Any compact manifold admitting a metric with all sectional curvatures strictly lessthan zero. This of course includes all orientable surface of genus at least , as well asall non-orientable surfaces of non-orientable genus at least (i.e. any non-orientablesurface obtained by connect-summing at least copies of RP ). (ii) Any simply-connected, spin manifold M which has the rational homotopy type of awedge of m ≥ spheres. (iii) Any simply-connected, spin manifold M which does not have the rational homotopytype of a sphere, and such that H i ( M ; k ) = 0 for all i > . (iv) Any connected sum of spin manifolds, each of whose rational cohomology algebra re-quires at least two generators. (v)
Any simply-connected rationally hyperbolic spin manifold such that H • (Ω X ; k ) is finitelygenerated as an algebra. LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 33
The list of examples in Lemma 7.1 is far from exhaustive. Note that if the Avramov–F´elixconjecture mentioned in Section 1.2 is true, then it follows from Lemma 2.18 that any rationallyhyperbolic spin manifold is in the class E . Proof.
First of all, let us argue that we have H −• (Ω p M ; L k ) = H −• (Ω p M ; k ) for all the exam-ples under consideration. For (i), the Cartan–Hadamard theorem implies that any manifold M in (i) is aspherical, which means L k is trivial. For (ii-v), the assumption that M is spinimmediately implies that L k is trivial.For (i), we have k [ π ( M ; ∗ )] = H (Ω ∗ M ; k ), and a well-known theorem of Milnor [40, Thm.2] implies that the left-hand side has exponential algebraic growth. For (ii), [20, Sec. 33(c);Ex. 1] implies that U L M is a free associative algebra on m generators, which has exponentialalgebraic growth since m ≥
2. (iii) follows from (ii) and a result of Baues [20, Sec. 24(c)], whichstates that any simply-connected finite CW complex with rational homology concentrated inodd degrees has the rational homotopy type of a wedge of spheres. For (iv), a standard Mayer–Vietoris argument shows that the connected sum of spin manifolds is spin. The claim nowfollows from [27, Cor. 3.4 and Rmk. 3.5]. Finally, (v) is immediate from the definition ofrational hyperbolicity (Definition 2.19). (cid:3)
Remark . In relation to Lemma 7.1(v), we remark that there exists a simply-connectedsmooth manifold whose Pontryagin algebra is not finitely generated. Lemaire [34] constructeda simply-connected finite CW complex X whose Pontryagin algebra is not finitely-generated.It can be shown using the Barge–Sullivan theorem [27, Sec. 3.1.1] that H • (Ω ∗ X ; k ) is a retractof the Pontryagin algebra of a smooth manifold M , implying in particular that H • (Ω ∗ M ; k ) isnot finitely generated. The author is grateful to Y. F´elix for communicating this argument tohim.7.2. Products.
We now give the proof of Proposition 1.4, which was stated in the introduc-tion.
Proof of Proposition 1.4.
According to [30, Thm. 1.5], there is a fully faithful embedding(7.2) W ( X ) ⊗ W ( X (cid:48) ) (cid:44) → W ( X × X (cid:48) ) , which thus remains fully faithful upon passing to Tw according to [45, Lem. 3.23].There is also a fully faithful embedding(7.3) Tw W ( X ) ⊗ Tw W ( X (cid:48) ) (cid:44) → Tw(Tw W ( X ) ⊗ Tw W ( X (cid:48) )) (cid:39) −→ Tw( W ( X ) ⊗ W ( X (cid:48) )) . Combining these, we obtain a fully faithful embedding(7.4) Tw W ( X ) ⊗ Tw W ( X (cid:48) ) → Tw W ( X × X (cid:48) )which induces a fully faithful functor(7.5) H • Tw W ( X ) ⊗ H • Tw W ( X (cid:48) ) = H • (Tw W ( X ) ⊗ Tw W ( X (cid:48) )) → H • (Tw W ( X × X (cid:48) )) . The result now follows from Lemma 3.7 and Lemma 3.4. (cid:3)
Gluing algebraically hyperbolic Liouville manifolds.
Proof of Theorem 1.6.
In the subcritical case, it follows by Proposition 6.6 that there is a fullyfaithful functor W ( X ) → W ( X ). In the flexible case, it follows similarly by Proposition 6.8that there is a fully faithful functor W ( X ) → W ( X ). The induced functor Tw W ( X ) → Tw W ( X ) is also fully faithful (see e.g. [45, Lem. 3.23]). The theorem now follows fromLemma 3.4. (cid:3) Figure 3.
This a priori complicated arboreal skeleton admits an obvious em-bedding of the genus 2 surface with the shaded region removed. Its Weinsteinthickening (which exists in light of [7]) can thus be recognized as algebraicallyhyperbolic by Theorem 7.3.
Proof of Theorem 1.7.
Fix an embedding ( T ∗ D n , λ std ) → ( X, λ ) which covers F ⊂ ∂ ∞ X . Let H = ∂T ∗ D n and observe that X is the union of two Liouville sectors V ∪ V meeting along H , where V = X − int( T ∗ D n ) and V is obtained by gluing X to T ∗ D n along F . This isprecisely the input considered at the start of Section 6.3.Corollary 6.12 furnishes a map (cid:15) : Tw W ( V ) → Tw k which satisfies (ii) in Lemma 3.4.Moreover, V is Weinstein since X is Weinstein by assumption. We may therefore applyProposition 6.9, which implies that there is a faithful morphism Tw W ( X ) → Tw W ( X ). Thetheorem now follows from Lemma 3.4. (cid:3) Skeleta and Lagrangian immersions.
The following theorem gives a criterion to checkthat a Weinstein manifolds is hyperbolic in terms of the stratified topology of its skeleton.
Theorem 7.3.
Let ( X (cid:48) , λ (cid:48) ) be a Weinstein manifold of dimension n . Let M be a closed,connected manifold of dimension n and let B ⊂ M be a ball. Suppose that there exists asmooth embedding f : ˜ M := M − B → X (cid:48) which satisfies the following two conditions: (i) f is a smooth embedding into the Lagrangian stratum of Skel( X (cid:48) , λ ) ; (ii) the image of f is contained in a closed, exact Lagrangian K ⊂ ( X (cid:48) , λ ) .If M ∈ E , then then Tw W ( X (cid:48) ) has exponential algebraic growth.Proof. According to (i) and Proposition 2.11, we have (after possibly deforming λ ) an embed-ding of Liouville sectors i : ( T ∗ ˜ M , λ std ) → ( X (cid:48) , λ (cid:48) ). Let V = i ( T ∗ ˜ M ), let V = X (cid:48) − int V and let H (cid:48) = V ∩ V ⊂ X (cid:48) . Let us also set ( X = T ∗ M, λ = λ std ). This is precisely the datawhich is considered at the start of Section 6.3; in order to apply Proposition 6.9 to this data,we must verify that the required hypotheses are satisfied.We now appeal to (ii) and Lemma 6.11, which furnishes a functor(7.6) Tw W ( X (cid:48) ) → Tw W ( T ∗ K ) . Recall that the functor (7.6) takes a cotangent fiber in F p ⊂ T ∗ M ⊂ X (cid:48) , for any p ∈ ∂M , toan object which is isomorphic in H ( W ( T ∗ K )) to a cotangent fiber of T ∗ K . By composing(7.6) with the functor Tw W ( T ∗ K ) → Tw k in (6.19), we obtain a functor (cid:15) : Tw W ( X (cid:48) ) → Tw k which satisfies (ii) in the statement of Proposition 6.9. Since (i) in the statement of LGEBRAICALLY HYPERBOLIC LIOUVILLE MANIFOLDS 35
Proposition 6.9 is automatically satisfied, it follows from Proposition 6.9 that there exists afaithful morphism(7.7) Tw W ( T ∗ M ) → Tw W ( X (cid:48) ) . The claim follows from Proposition 1.3 and Lemma 3.4. (cid:3)
Remark . Both assumptions in the statement of Theorem 7.3 are needed. Indeed, Propo-sition 2.14 implies that any manifold can be embedded in the skeleton of an algebraicallyhyperbolic Weinstein manifold: this explains why one must impose that the image of f iscontained in the Lagrangian stratum of Skel( X, λ ).To see that (ii) is also needed, let N be a closed hyperbolic manifold and let g be a Morsefunction with a single local maximum. Consider the induced handle decomposition on X = T ∗ N . Now let X (cid:48) be obtained by “loosifying” the top handle attachment, that is, by attachingthe top handle along a stabilization of the original Legendrian knot. Then the skeleton of X (cid:48) admits an embedding of N − B into the Lagrangian stratum. However, X (cid:48) is a flexibleWeinstein manifold, so its wrapped Fukaya category is zero. In particular, it is not algebraicallyhyperbolic. Remark . Using the operations described in Section 1.3, it is easy to construct examplesof algebraically hyperbolic Weinstein manifolds which cannot be detected using Theorem 7.3.One obvious source of examples is furnished by Example 1.5. A simple and concrete example isas follows: let X be obtained by plumbing T ∗ S with T ∗ Σ where Σ is an orientable surfaceof genus 2, and let X = T ∗ S . Then X = X × X is algebraically hyperbolic. However,Skel( X, λ ) consists in two Lagrangians L = S × S and L = S × Σ which intersect alongan embedded S . This Weinstein manifold does not satisfy the criterion of Theorem 7.3. Proof of Corollary 1.14.
Let B ⊂ M be a ball which contains the preimages of all the doublepoints. Let K be obtained by performing Polterovich surgery on the double points. Now applyTheorem 7.3. (cid:3) Proof of Corollary 1.16.
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