aa r X i v : . [ m a t h . S G ] S e p HAMILTONIAN NO-TORSION
MARCELO S. ATALLAH AND EGOR SHELUKHIN
Abstract.
In 2002 Polterovich has notably established that on closed aspheri-cal symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which wecall Hamiltonian torsion, must in fact be trivial. In this paper we prove the firsthigher-dimensional Hamiltonian no-torsion theorems beyond the symplecticallyaspherical case. We start by showing that closed symplectic Calabi-Yau and neg-ative monotone symplectic manifolds do not admit Hamiltonian torsion. Goingstill beyond topological constraints, we prove that every closed positive mono-tone symplectic manifold (
M, ω ) admitting Hamiltonian torsion is geometricallyuniruled by holomorphic spheres for every ω -compatible almost complex struc-ture, partially answering a question of McDuff-Salamon. This provides manyadditional no-torsion results, and as a corollary yields the geometric uniruled-ness of monotone Hamiltonian S -manifolds, a fact closely related to a celebratedresult of McDuff from 2009. Moreover, the non-existence of Hamiltonian torsionimplies the triviality of Hamiltonian actions of lattices like SL ( k, Z ) for k ≥ , aswell as those of compact Lie groups. Finally, for monotone symplectic manifoldsadmitting Hamiltonian torsion, we prove an analogue of Newman’s theorem onfinite transformation groups for several natural norms on the Hamiltonian group:such subgroups cannot be contained in arbitrarily small neighborhoods of theidentity. Our arguments rely on generalized Morse-Bott methods, as well as onquantum Steenrod powers and Smith theory in filtered Floer homology. Contents
1. Introduction and main results 21.1. Introduction 21.2. Main results 52. Preliminary material 142.1. Basic setup 142.2. Floer theory 162.3. Quantum homology and PSS isomorphism 192.4. Spectral invariants in Floer theory 203. Isolated connected sets of periodic points 233.1. Generalized perfect Hamiltonians 233.2. Floer cohomology 314. Cluster structure of the essential spectrum 325. Hamiltonian torsion 365.1. Proof of Theorem C 36
Introduction and main results
Introduction.
The question of the existence of finite group actions on man-ifolds has been of interest in topology for a long time. In particular, it is in orderto study this question that P. A. Smith [81] has developed in the 1930s what isnow called Smith theory for cohomology with F p coefficients in the context of con-tinuous actions of finite p -groups. We refer the reader to [4, 5, 23, 35] for furtherreferences on Smith theory.Quite a lot of progress regarding this question has been obtained in low-dimensionaltopology [9, 46] and in smooth topology in arbitrary dimension - see for example[48] and references therein. As a first easy example, we remark that it is not hardto classify finite group actions on closed surfaces.In symplectic topology, it was shown by Polterovich [57] that non-trivial Hamilton-ian finite group actions, that we refer to as
Hamiltonian torsion , on symplecticallyaspherical manifolds do not exist. Furthermore, Chen and Kwasik [9] rule outsymplectic finite group actions acting trivially on homology on certain symplecticCalabi-Yau 4-manifolds (see [91] for a further development). Moreover, general con-straints were obtained by Mundet i Riera [47], showing roughly speaking that finitegroups acting effectively in a Hamiltonian way must be approximately abelian.Furthermore, Hamiltonian actions of cyclic groups on rational ruled symplectic 4-manifolds, that is symplectic S -bundles over S , were recently shown to be inducedby S -actions [10] (see [7, 8, 11] for related and for contrasting results). Compatibly,the strongest restriction to date on manifolds admitting non-trivial Hamiltonian S -actions was obtained by McDuff [42] who showed that all such manifolds mustbe uniruled , in the sense that at least one genus zero k -point Gromov-Witteninvariant, for k ≥ , involving the point class must not vanish. Of course, rationalruled symplectic 4-manifolds satisfy this condition, in fact with k = 3: they are strongly uniruled. Either condition implies that these manifolds are geometrically
AMILTONIAN NO-TORSION 3 uniruled: for each ω -compatible almost complex structure J and each point p ∈ M, there is a J -holomorphic sphere passing through p. Finally, in [77] a newnotion of uniruledness, F p -Steenrod uniruledness, was introduced for p = 2 , andwas generalized to odd primes p > p -th powerof the cohomology class Poincar´e dual to the point is defined and deformed in thesense of not coinciding with the classical Steenrod p -th power. This notion similarlyimplies geometric uniruledness. It is currently not known whether or not it impliesuniruledness in the sense of McDuff, but it is expected that it does (see [69, 75] forfirst steps in this direction).In this paper we prove the first higher-dimensional Hamiltonian no-torsion re-sults since that of Polterovich, that hold beyond the symplectically aspherical case.Firstly, we prove that, in addition to symplectically aspherical manifolds, symplec-tically Calabi-Yau manifolds and negative monotone symplectic manifolds do notadmit Hamiltonian torsion. An elementary argument then shows, in summary, thatif a closed symplectic manifold M admits Hamiltonian torsion, then it admits aspherical homology class A such that h c ( T M ) , A i > , h [ ω ] , A i > S -actions: indeed,negative monotone and Calabi-Yau manifolds are not geometrically uniruled, andneither are the symplectically aspherical ones.Going far beyond topological restrictions, we further study restrictions on Hamil-tonian torsion in the positive monotone case. Using recently discovered techniques,we show that in this case the existence of non-trivial Hamiltonian torsion implies F p Steenrod-uniruledness for certain primes p, and hence geometric uniruledness. Thisagain fits well with the result of McDuff and in fact provides a partial solution toProblem 24 from the monograph [44] of McDuff-Salamon. Studying the propertiesof the quantum Steenrod operations and their relation to Gromov-Witten invari-ants further (see [75, 88, 89] for first inroads in this direction) might show that oursolution is in fact quite complete. Furthermore, we are tempted to conjecture thefollowing analogue of the result of McDuff. Conjecture 1.
Each closed symplectic manifold with non-trivial Hamiltonian tor-sion must be uniruled.For the special case of symplectically Calabi-Yau 4-manifolds, our results com-pare to those of Chen-Kwasik [9] as follows: the paper [9] rules out more generalsymplectic finite group actions than Hamiltonian ones, but in a more restrictivecontext, as they require these 4-manifolds to have non-zero signature, and b +2 atleast 2 , while we make no such assumptions. Furthermore, the results of Wu-Liu[91], which hold only in dimension 4 , cover a different class of manifolds than ourresults. This is a smooth map u : C P → M satisfying Du ◦ j = J ◦ Du for the standard complexstructure j on C P . Such spheres and their significance in symplectic topology were discovered byGromov [31]. We refer to McDuff-Salamon [43] for a detailed modern description of this notion.
MARCELO S. ATALLAH AND EGOR SHELUKHIN
Before addressing further results on the metric properties of Hamiltonian torsion,when it exists in the positive monotone case, we comment on our methods of proof.Curiously enough, our arguments involve a recently discovered analogue of Smiththeory in filtered Hamiltonian Floer homology [73, 76, 80], and related notionsof quantum Steenrod powers [74, 88, 89]. Previously these methods were appliedto questions of existence of infinitely many periodic points [74, 76] and, more re-strictively, of obstructions on manifolds to admit Hamiltonian pseudo-rotations[12, 77, 78]. In fact, a general theme of this paper is that a Hamiltonian diffeomor-phism of finite order behaves in many senses like a counterexample to the Conleyconjecture. For example, the statement of Corollary 2 is analogous to that of [30,Theorem 1.1] that provides the most general setting wherein the Conley conjectureis known to hold.Additional results in this paper include the following. First, in the special case when(
M, ω ) has minimal Chern number N = n + 1 , we deduce from the work of Seideland Wilkins [75], as in [77], that non-trivial Hamiltonian torsion implies that thequantum product [ pt ] ∗ [ pt ] does not vanish. This means that the manifold is stronglyrationally connected: it implies strong uniruledness, and also shows that for eachpair of distinct points p , p in M, and each ω -compatible almost complex structure J, there exists a J -holomorphic sphere in M passing through p , p . Second, weprove that the spectral norm [51, 68, 86] of a Hamiltonian torsion element φ oforder k on a closed rational symplectic manifold, that is h [ ω ] , π ( M ) i = ρ · Z ,ρ > , satisfies γ ( φ ) ≥ ρ/k, and as an immediate consequence, the same estimateapplies for the Hofer norm [33, 38].More importantly, in our final main result, we prove that in the monotone case,given φ ∈ Ham(
M, ω ) \ { id } of order k, that is φ k = id , there exists m ∈ Z /k Z suchthat(1) γ ( φ m ) ≥ ρ/ . This last result should be considered a Hamiltonian analogue of the celebratedresult of Newman [49] (see also [15, 82]), the C -distance having been replaced bythe spectral distance. Moreover we prove the stronger statement that if k is prime,then γ ( φ m ) ≥ ρ ⌊ k/ ⌋ /k for a certain m ∈ Z /k Z , and provide a similar statementin the context of Hamiltonian pseudo-rotations.The bound (1) can further be seen to imply Newman’s result in a special caseas follows. By [79, Theorem C] (see also [36]), when M = C P n is the complexprojective space with the standard symplectic form normalized so that C P hasarea 1 , there is a constant c n depending only on the dimension, such that for all φ ∈ Ham(
M, ω ) , the usual C -distance of φ to the identity satisfies d C ( φ, id) ≥ c n γ ( φ ) . Hence if φ is of finite order, then by (1) there exists m ∈ Z such that d C ( φ m , id) ≥ c n / . AMILTONIAN NO-TORSION 5
It would be very interesting to see if the results of this paper can be generalizedto the case of Hamiltonian homeomorphisms, as defined in [6]. This generalizationdoes not seem to be straightforward because we use the properties of the lineariza-tion of the Hamiltonian diffeomorphism at its fixed points, as well as Smith theoryin filtered Floer homology, which is not in general stable in the C -topology.We close the introduction by noting that we expect that our results in the mono-tone case should generalize to the semi-positive case, once the relevant results of[76],[74] have been generalized to the requisite setting. Since these generalizationswould not considerably differ, in a conceptual way, from the arguments presentedin this paper, but would necessitate more lengthy technical proofs, we defer theirinvestigation to further publications.1.2. Main results.
We start with the following theorem of Polterovich [57], orig-inally stated in the case where π = 0 . For the reader’s convenience we include itsproof in Section 5.4.
Theorem A (Polterovich) . Let ( M, ω ) be a closed symplectically aspherical sym-plectic manifold. Then each homomorphism G → Ham(
M, ω ) , where G is a finitegroup, is trivial. In this paper we prove a number of additional “no-torsion” theorems of this kind,going beyond the symplectically aspherical case, and study the metric properties ofHamiltonian diffeomorphisms of finite order when such obstructions do not hold.Our conditions on the manifold that imply the absence of Hamiltonian torsionare of two kinds: the first is purely topological, and the second, perhaps moresurprisingly, is in terms of pseudo-holomorphic curves.1.2.1.
Topological conditions.
The first set of results of this paper is as follows.
Theorem B.
Let ( M, ω ) be a closed negative monotone or closed symplecticallyCalabi-Yau symplectic manifold. Then each homomorphism G → Ham(
M, ω ) , where G is a finite group, is trivial. A simple exercise in linear algebra shows that the class of manifolds, which wecall symplectically non-positive, covered by Theorems A and B can be describedconcisely as those closed symplectic manifolds (
M, ω ) for which h [ ω ] , A i · h c ( T M ) , A i ≤ A ∈ π ( M ) . In other words, the following holds.
Corollary 2.
If a closed symplectic manifold ( M, ω ) admits a non-trivial homo-morphism G → Ham(
M, ω ) from a finite group, then there exists A ∈ π ( M ) , suchthat h [ ω ] , A i > , h c ( T M ) , A i > . MARCELO S. ATALLAH AND EGOR SHELUKHIN
For details of this implication see [30, Proof of Theorem 4.1].Both these results are given by the following two steps that essentially generalizethe notion of a perfect Hamiltonian diffeomorphism, namely one that has a finitenumber of contractible periodic points of all periods, to the case of compact path-connected isolated sets of fixed points. We call such an isolated set of fixed pointsof φ ∈ Ham(
M, ω ) a generalized fixed point of φ. Recall that a fixed point x of aHamiltonian diffeomorphism φ = φ H is called contractible whenever the homotopyclass α ( x, φ ) of the path α ( x, H ) = { φ tH ( x ) } for a Hamiltonian H generating φ ,is trivial. This class does not depend on the choice of Hamiltonian by a classicalargument in Floer theory. We call a generalized fixed point F of φ contractible, ifall fixed points x ∈ F are contractible.We call φ ∈ Ham(
M, ω ) generalized perfect if there exists a sequence k j → ∞ ofiterations, such that for all j ∈ Z > the diffeomorphism φ k j has a finite numberof contractible generalized fixed points, this set of contractible generalized fixedpoints does not depend on j, and moreover the following condition regarding indicesholds: for each capping F of the generalized periodic orbit F corresponding toa generalized fixed point F ∈ π (Fix( φ k j )) , and Hamiltonian H generating φ, themean-index ∆( H ( k j ) , x ) , where x is a capped 1-periodic orbit of H ( k j ) correspondingto x ∈ F and capping F is constant as a function of x ∈ F . We refer to Section2.1.3 for the definition of the mean-index.Finally we call a diffeomorphism φ with a finite number of (contractible) generalizedfixed points weakly non-degenerate if for each (contractible) fixed point x of φ, the spectrum of the differential D ( φ ) x at x contains points different from 1 ∈ C . Using the existence of ω -compatible almost complex structures invariant undera Hamiltonian diffeomorphism of finite order, we prove the following structuralresult. Theorem C.
Let ( M, ω ) be a closed symplectic manifold. Then a p -torsion Hamil-tonian diffeomorphism φ ∈ Ham(
M, ω ) for a prime p, is a weakly non-degenerategeneralized perfect Hamiltonian diffeomorphism. In fact, it is Floer-Morse-Bottand its generalized fixed points are symplectic submanifolds. Following the index arguments of Salamon-Zehnder [66], and their generalizationdue to Ginzburg-G¨urel [29], we prove the following obstruction to the existence ofweakly non-degenerate generalized perfect Hamiltonian diffeomorphisms.
Theorem D.
Let a closed symplectic manifold ( M, ω ) be negative monotone orsymplectically Calabi-Yau. Then ( M, ω ) does not admit weakly non-degenerate gen-eralized perfect Hamiltonian diffeomorphisms. Together with Theorem C, Theorem D immediately implies Theorem B, after theelementary observation that in view of Cauchy’s theorem for finite groups, to ruleout all Hamiltonian finite group actions it is sufficient to rule out all Hamiltoniantorsion of prime order. One can say that, almost paradoxically, we use the large
AMILTONIAN NO-TORSION 7 time asymptotic behavior of our Hamiltonian system to study its periodic dynamics!This can be considered to be the main general idea of this paper.We remark that, as easy examples show, generalized perfect Hamiltonian diffeo-morphisms do indeed exist on the manifolds of Theorem D, if one drops the weaknon-degeneracy assumption. For example, one can take T = S × S , S = R / Z to be the standard torus with ( x, y ) denoting a general point, and ω st = dx ∧ dy the standard symplectic form, and pick φ ∈ Ham( T , ω st ) , φ = φ tH , t > , for H ∈ C ∞ ( T , R ) given by H ( x, y ) = cos(2 πy ) . It is easy to see that the set ofcontractible periodic points of φ consists precisely of the two isolated sets { y = 0 } , and { y = } . Conditions in terms of pseudo-holomorphic curves.
Our second set of re-sults deals with the class of monotone symplectic manifolds. It is evident that farmore than topological conditions are necessary to rule out Hamiltonian torsion inthis case, since each Hamiltonian S -manifold, such as C P n for example, admitsHamiltonian torsion. We formulate our restriction on the existence of Hamiltoniantorsion geometrically as follows. For an ω -compatible almost complex structure J on a closed symplectic manifold ( M, ω ) we say that the manifold is geometricallyuniruled if for each point p ∈ M, there exists a J -holomorphic sphere u : C P → M, such that p ∈ im( u ) . Theorem E.
Let ( M, ω ) be a closed monotone symplectic manifold, that is notgeometrically uniruled for a certain ω -compatible almost complex structure J. Theneach homomorphism G → Ham(
M, ω ) , where G is a finite group, is trivial. This is a corollary of the following more precise result involving the quantum Steen-rod power operations.
Theorem F.
Let ( M, ω ) be a closed monotone symplectic manifold that admitsa Hamiltonian diffeomorphism of order d > . Then the p -th quantum Steenrodpower of the cohomology class µ ∈ H n ( M ; F p ) Poincar´e dual to the point class isdeformed for all primes p coprime to d. Theorems E and F provide an obstruction to the existence of Hamiltonian diffeo-morphisms of finite order in terms of pseudo-holomorphic curves. The existence ofan obstruction of this type was conjectured by McDuff and Salamon, and publi-cized as Problem 24 in their introductory monograph [44]. Therefore we providea solution to a reasonable variant of this problem. Indeed, further investigationsinto the enumerative nature of quantum Steenrod operations might prove that oursolution is in fact complete, in the framework of monotone symplectic manifolds.Such investigations were initiated in [75, 88, 89].The proof of Theorem F relies on two steps. These steps are aimed at showingthat torsion Hamiltonian diffeomorphisms of closed monotone symplectic mani-folds, that by Theorem C are generalized perfect and weakly non-degenerate, are
MARCELO S. ATALLAH AND EGOR SHELUKHIN moreover homologically minimal in the following sense. To formulate it precisely,we first discuss a useful technical notion.Let K be a coefficient field. For a generalized fixed point F of a Hamiltonian diffeo-morphism ψ we define a generalized version HF loc ( ψ, F ) of local Floer homology.Such notions date back to the original work of Floer [21, 22] and have been revisteda number of times: for example by Pozniak in [61]. It is naturally Z / Z -graded .We call a Hamiltonian diffeomorphism a generalized K pseudo-rotation with thesequence k j if it is generalized perfect with the sequence k j , and furthermore HF loc ( ψ, F ) = 0 for all F ∈ π (Fix( ψ )) , and the homological count N ( ψ, K ) := X F∈ π (Fix( ψ )) dim K HF loc ( ψ, F )of generalized fixed points of ψ = φ k j satisfies N ( ψ, K ) = dim K H ∗ ( M ; K )for all j ∈ Z > . We recall that usually an F p pseudo-rotation is defined anal-ogously, with the sequence k j = p j − with the additional hypothesis that each F ∈ π (Fix( ψ )) for ψ = φ k j consists of a single point. We note that k j = p j − shalloften be a convenient sequence for us to work with.In view of the discussion in [76, 79] this homological minimality for a Hamiltoniandiffeomorphism ψ with a finite number of generalized fixed points is equivalent tothe absence of finite bars in the barcode B ( ψ ) of ψ, a notion of recent interestin symplectic topology (see [37, 59, 60, 76, 79] for example). It also implies theequality Spec ess ( F ; K ) = Spec vis ( F ; K )between two homologically defined subsets of the spectrum associated to a Hamil-tonian F ∈ H generating ψ. Recall that the spectrum Spec( F ) of F is the set ofcritical values of the action functional of F. For a coefficient field K , there is anested sequence of subsetsSpec ess ( F ; K ) ⊂ Spec vis ( F ; K ) ⊂ Spec( F ) . Here the essential spectrum
Spec ess ( F ; K ) is the set of values of all spectral invari-ants associated to F, in other words the set of starting points of infinite bars in thebarcode of F, and the visible spectrum Spec vis ( F ; K ) is the set of action values ofcapped (generalized) periodic orbits of F that have non-zero local Floer homology,in other words the set of endpoints of all bars in the barcode. It is not hard tomodify the definitions of the two homological spectra to include multiplicities, inwhich case their equality would be equivalent to homological minimality.The first step in the proof of Theorem F, which is non-trivial and uses the mainmethods of [76], is the following reduction. We also define a Z -graded version for a capped generalized 1-periodic point F lifting F . AMILTONIAN NO-TORSION 9
Theorem G.
Let ( M, ω ) be a closed monotone symplectic manifold. Suppose that φ ∈ Ham(
M, ω ) is a Hamiltonian diffeomorphism of prime order q ≥ . Then:(i) For each prime p different from q, the q -torsion diffeomorphism φ is aweakly non-degenerate generalized pseudo-rotation over F p , with the se-quence k j given by the monotone-increasing ordering of any infinite subsetof the set { k ∈ Z | k = 0 (mod q ) } . (ii) Moreover Spec ess ( H ; K ) = Spec vis ( H ; K ) for each Hamiltonian H generating φ, and each coefficient field K , of char-acteristic p = q and Spec ess ( H ( k ) ; Q ) = k · Spec ess ( H ; Q ) + ρ · Z for all k coprime to q. (iii) Finally, part (i) holds also for p = q, and in part (ii) Spec ess ( H ; K ) = Spec vis ( H ; K ) , Spec ess ( H ( k ) ; K ) = k · Spec ess ( H ; K ) + ρ · Z hold with arbitary coefficient field K , and moreover Spec vis ( H ; K ) = Spec( H ) . We briefly explain the approach used to prove part (i) of Theorem G. Following themain theme of the proof of Theorem C, we use information about large iterationsof H to study the periodic Hamiltonian diffeomorphism φ = φ H that it generates.More precisely, let ψ = φ k , for k coprime to q. Combining the theory of barcodesof Hamiltonian diffeomorphisms (see Proposition 22), and Smith-type inqualitiesin filtered Floer homology (see Theorem M), we observe that for the bar-lengths β ( ψ, F p ) ≤ . . . ≤ β K ( ψ, F p ) ( ψ, F p )of ψ, we have the following inequality. Set β tot ( ψ, F p ) = β ( ψ, F p ) + . . . + β K ( ψ, F p ) ( ψ, F p )to be the total bar-length of ψ. Then β tot ( ψ p m , F p ) ≥ p m · β tot ( ψ, F p ) . However, β tot ( ψ p m , F p ) is bounded, since it can take at most q − β tot ( ψ, F p ) = 0which in turn, implies part (i), by the theory of barcodes (see Proposition 22). Remark . We separate part (iii) of Theorem G because it requires a differentproof, relying on Proposition 4 below. The first statement of part (iii) is obtainedvia Proposition 4 by classical Smith theory combined with classical homologicalArnol’d conjecture (outlined in [10, Remark 7.1] with details for p = 2). One couldalso obtain this statement by a suitable generalization of Theorem M on Smiththeory in filtered Floer homology, which is however out of the scope of this paper.The following statement is a key component of part (iii) of the proof of TheoremG. It relies on the generalization of the Morse-Bott theory of Pozniak [61, Theorem3.4.11] to the situation with signs and orientations, as in for example [67, Chapter9], [26, Chapter 8], or [87]. However it is not entirely straightforward, because asclassical examples show, it is false in the general Floer-Morse-Bott situation, whilein our case it holds because of the existence of special ω -compatible almost complexstructures adapted to the situation. Proposition 4.
Let ( M, ω ) be a closed symplectic manifold, and φ ∈ Ham(
M, ω ) a Hamiltonian diffeomorphism of finite order d ≥ . Let F be a path-connectedcomponent of the fixed-point set of φ. Finally, let R be a commutative unital ring.Then the local Floer homology of φ at F with coefficients in R satisfies: HF loc ( φ, F ) ∼ = H ( F ; R ) . We also note that the proof of Theorem G has the following by-product, whichis a new analogue, for Hamiltonian torsion, of the classical consequence of Floertheory, whereby the map π (Ham( M, ω )) → π ( M ) is trivial. Proposition 5.
Let ( M, ω ) be a closed monotone symplectic manifold, and φ in Ham(
M, ω ) be a Hamiltonian diffeomorphism of prime order. Then all the fixedpoints of φ are contractible. The second step in the argument proving Theorem F is the following statement.It essentially follows the arguments of [78] and [74].
Theorem H.
Let ( M, ω ) be a closed monotone symplectic manifold that admits aweakly non-degenerate generalized F p pseudo-rotation for a prime p ≥ . Then the p -th quantum Steenrod power of the cohomology class µ ∈ H n ( M ; F p ) Poincar´edual to the point class is deformed.
These two results immediately imply Theorem F and therefore, by a Gromov com-pactness argument, Theorem E.1.2.3.
Applications to actions of Lie groups and lattices.
To conclude the discussionof our first two sets of results, we discuss their implications to the question ofexistence of Hamiltonian actions of possibly disconnected Lie groups and latticesin Lie groups, on closed symplectic manifolds.
AMILTONIAN NO-TORSION 11
A well-known result of Delzant [14] (see [58] for an alternative argument) impliesthat a connected semi-simple group can only act on a closed symplectic manifoldif it is compact. A compact zero-dimensional Lie group is finite, whence TheoremsB and E provide topological and geometrical obstructions to their action. Theidentity component K of a compact Lie group K of positive dimension is a compactconnected Lie group of positive dimension, and as such admits a maximal torus T ∼ =( S ) k of positive dimension whose conjugates cover the whole group. Therefore,the absence of Hamiltonian torsion, as in Theorems A, B, E, and F implies that anon-trivial K -action yields a non-trivial K -action, since otherwise it would factorthrough K/K which is finite. This in turn yields a non-trivial T -action and afortiori a non-trivial S -action. A celebrated result of McDuff [42] then shows thatnon-trivial S -actions imply uniruledness in the sense of k -point genus 0 Gromov-Witten invariants and hence geometric uniruledness. Corollary 6 (McDuff) . Let ( M, ω ) be a closed symplectic manifold that is notgeometrically uniruled. Then each homomorphism K → Ham(
M, ω ) for a compactconnected Lie group K must be trivial. Moreover, by a simple continuity argument, a non-trivial continuous S -actionimplies a non-trivial Z /p Z -action for each prime p. Therefore Theorems B and Eimply the above corollary of the result of McDuff for symplectically aspherical,symplectically Calabi-Yau, negative monotone, or monotone symplectic manifolds.However, Theorem F also implies that in this case if the manifold is monotone,it must be F p Steenrod-uniruled for all primes p. It is seen from examples due toSeidel and Wilkins [75] that there exist closed monotone symplectic manifolds thatare uniruled in the sense of Gromov-Witten invariants, and yet not F p Steenrod-uniruled for certain primes p. We note that McDuff’s theorem was proven by showing that certain loops of Hamil-tonian diffeomorphisms in a blow-up of the manifold are non-trivial, and detectableby Seidel’s representation [70]. It would be interesting to investigate the existence ofnon-trivial Hamiltonian loops associated to Hamiltonian diffeomorphisms of finiteorder. We note that for a Hamiltonian H generating φ ∈ Ham(
M, ω ) of order d, theHamiltonian H ( d ) generates a loop homotopic to { ( φ tH ) d } . The non-contractibilityof this loop is not obvious since for a rotation φ π/ of S by angle 2 π/ z -axis, the loop { φ t · π/ } is not contractible in Ham( S , ω st ) , while the loop { φ − t · π/ } is contractible therein, yet φ − π/ = φ π/ . Finally we can argue, following the work of Polterovich [57] on the HamiltonianZimmer conjecture, that SL ( k, Z ) for k ≥ SL ( k, Z ) for k ≥ SL ( k, Z ) , k ≥ , ismuch more difficult and seems to be currently out of reach of our methods. Metric properties.
Our third and final set of results studies the metric prop-erties of Hamiltonian torsion diffeomorphisms, in cases that are not ruled out byour previous arguments, for example on C P n . Recall that the spectral pseudo-norm of a Hamiltonian H ∈ C ∞ ( S × M, R ) ona closed symplectic manifold ( M, ω ) is defined in terms of Hamiltonian spectralinvariants as γ ( H ) = c ([ M ] , H ) + c ([ M ] , H ) , and the spectral norm of φ ∈ Ham(
M, ω ) is set as γ ( φ ) = inf φ H = φ γ ( H ) . We refer to Section 2 for a more in-depth discussion of this interesting notion,remarking for now that this is a conjugation-invariant and non-degenerate normon Ham(
M, ω ) , yielding a bi-invariant metric d γ ( f, g ) = γ ( gf − ) . This was shown in large generality in [51, 68, 86].Furthermore, whenever defined, γ ( φ ) provides a lower bound on the celebratedHofer distance [33, 38] d Hofer ( φ, id) , defined as d Hofer ( φ, id) = inf φ H = φ Z max M H ( t, − ) − min M H ( t, − ) dt. Finally in [6, 36, 79] it was shown in various degrees of generality that γ ( φ ) isbounded by the C -distance d C ( φ, id) of φ to the identity, at least in a small d C -neighborhood of the identity.These and numerous other recent results show that the spectral norm γ is an im-portant measure of a Hamiltonian diffeomorphism. Here, we provide lower boundson γ ( φ ) , under the assumption that φ is of finite order. Our first result is relativelygeneral and quite straightforward, and follows essentially from the homogeneityof the action under iteration, however it underlines the fact that the finite ordercondition implies certain metric rigidity. Theorem I.
Let ( M, ω ) be a closed rational symplectic manifold, with rationalityconstant ρ > , that is h [ ω ] , π ( M ) i = ρ · Z . Let φ ∈ Ham(
M, ω ) be a non-trivialHamiltonian diffeomorphism of order d, that is φ d = id . Then γ ( φ ) ≥ ρ/d. As a further consequence of Theorem G, which requires considerably more complexmethods, we obtain the following analogue of Newman’s theorem for the spectralnorm of Hamiltonian torsion elements.
Theorem J.
Let ( M, ω ) be a closed monotone symplectic manifold. Consider aHamiltonian diffeomorphism φ ∈ Ham(
M, ω ) of order d > . Then if the rationalityconstant of ω is ρ > , there exists m ∈ Z /d Z such that γ ( φ m ) ≥ ρ/ . AMILTONIAN NO-TORSION 13
Here the coefficients are in an arbitrary field K . In fact, if d = p is prime, we provethe stronger statement, that there exists m ∈ Z /p Z , such that γ ( φ m ) ≥ ρ · ⌊ p/ ⌋ /p. The key notion in the proof of this result is a new invariant of a Hamiltoniandiffeomorphism φ ∈ Ham(
M, ω ) , which we call the spectral length l ( φ, K ) of φ withcoefficients in a field K . It is defined as the minimal diameter of Spec ess ( H ; K ) ∩ I, for an interval I = ( a − ρ, a ] ⊂ R of length ρ (with fixed H as I varies). In particular,we show that this minimum does not depend on the choice of Hamiltonian H with φ H = φ. We show the key property that l ( φ, K ) ≤ γ ( φ, K ) and that in our casethe spectral length behaves in a controlled way with respect to iterations. By acombinatorial analysis of our situation we consequently deduce Theorem J. Weexpect l ( φ, K ) to have additional applications in quantitative symplectic topologythat we plan to investigate.Theorem J is generally speaking sharp, as can be seen from the rotation φ of S by 2 π/ z -axis. In this case φ = id and γ ( φ ) = γ ( φ ) = γ ( φ − ) = ρ/ , where ρ is the area of the sphere. Observe moreover that the lower bound inTheorem J does not depend on the order of φ. In particular if d = 2 then TheoremI gives the stronger lower bound γ ( φ ) ≥ ρ/ , which is again sharp for the π -rotationof S about the z -axis. We recall that Newman’s theorem is the same assertion, butfor the C -distance to the identity, in the setting of homeomorphisms of smoothmanifolds. In contrast to our result, the constant in Newman’s theorem is notexplicit.Finally, we remark that analogous statements hold for generalized F p pseudo-rotations φ with sufficiently large admissible sequences. For example, for the se-quence k j = p j − , we get the lower bound γ ( φ k j ) ≥ ρ/ ( p + 1) for some j ∈ Z > , which is saturated by the rotation of S by 2 π/ ( p + 1) about the z -axis. For thesequence k j = j, we obtain the following lower bound, which is saturated by any2 πθ -rotation on S about the z -axis, where θ / ∈ Q . Theorem K.
Let φ ∈ Ham(
M, ω ) be a generalized K pseudo-rotation with sequence k j = j on a closed monotone symplectic manifold ( M, ω ) with rationality constant ρ. Then sup j ∈ Z > γ ( φ k j ) ≥ ρ/ , the coefficients being taken in K . This result is new in this generality even for strongly non-degenerate pseudo-rotations. In the special case where (
M, ω ) is a complex projective space, thisresult can also be obtained in a different way by following the methods of [28]. Preliminary material
Basic setup.
In this section, we recall established aspects of the theory ofHamiltonian diffeomorphisms on symplectic manifolds. Throughout the article,(
M, ω ) denotes a 2 n -dimensional closed symplectic manifold. Definition 7 (Monotone, negative monotone and symplectically Calabi-Yau) . Suppose that the cohomology class of the symplectic form ω is proportional tothe first Chern class [ ω ] = κ · c ( T M ) , for κ = 0, on the image H S ( M ; Z ) of the Hurewicz map π ( M ) → H ( M ; Z ) . If κ < M, ω ) negative monotone , and if κ > monotone . Ifthe first Chern class c ( T M ) vanishes on the image of the Hurewicz map, we say(
M, ω ) is symplectically Calabi-Yau .The symplectic manifold (
M, ω ) is called rational whenever P ω = (cid:10) [ ω ] , H S ( M ; Z ) (cid:11) is a discrete subgroup of R . If P ω = { } , then P ω = ρ · Z for ρ >
0, which wecall the rationality constant of (
M, ω ). If P ω = 0 we call ( M, ω ) symplecticallyaspherical .Finally we recall that the minimal Chern number of ( M, ω ) is the index N = N M = [ Z : im( c ( T M ) : π ( M ) → Z )] . Hamiltonian isotopies and diffeomorphisms.
We consider normalized 1-periodicHamiltonian functions H ∈ H ⊂ C ∞ (S × M, R ), where H is the space of Hamilto-nians normalized so that H ( t, − ) has zero ω n mean for all t ∈ [0 , . For each H ∈ H we have the corresponding time-dependent vector field X tH defined by the relation ω ( X tH , · ) = − dH t . In particular, to each Hamiltonian function we can associatea Hamiltonian isotopy { φ tH } induced by X tH and its time-one map φ H = φ H . Weomit the H from this notation whenever it is clear from context. Such maps φ H arecalled Hamiltonian diffeomorphisms and they form a group denoted by Ham( M, ω ).For a Hamiltonian diffeomorphism φ ∈ Ham(
M, ω ), we denote the set of its con-tractible fixed points by Fix( φ ), and by x ( k ) , for x ∈ Fix( φ ), its image under theinclusion Fix( φ ) ⊂ Fix( φ k ). Contractible means the homotopy class α ( x, φ ) of thepath α ( x, H ) = { φ tH ( x ) } for a Hamiltonian H ∈ H generating φ , is trivial. Thisclass does not depend on the choice of Hamiltonian by a classical argument in Floertheory.We denote by H ( k ) the 1-periodic k -th iteration of a Hamiltonian function H ,where H ( k ) ( t, x ) = kH ( kt, x ) and φ H ( k ) = φ kH . There is a bijective correspondencebetween Fix( φ H ) and contractible 1-periodic orbits of the isotopy { φ tH } , thus for x ∈ Fix( φ H ), we denote by x ( t ) the 1-periodic orbit given by x ( t ) = φ tH ( x ) and,similarly, by x ( k ) ( t ) the 1-periodic orbit given by x ( k ) ( t ) = φ tH ( k ) ( x ( k ) ). AMILTONIAN NO-TORSION 15
The Hamiltonian action functional.
Let x : S → M be a contractible loop.It is then possible to extend this map to a capping of x , namely, a map x : D → M such that x | S = x . Let L pt M denote the space of contractible loops in M andconsider the equivalence relation on capped orbits given by( x, x ) ∼ ( y, y ) ⇐⇒ x = y and x − y ) ∈ ker[ ω ] , where x − y ) stands for gluing disks along their boundaries with the orientationof y reversed. The quotient space e L pt M of capped orbits by the above equivalencerelation is a covering of L pt M with the group of deck transformations isomorphicto Γ = π ( M ) / ker[ ω ]. We write ( x, x ) or simply x for the equivalence class ofthe capped orbit. With this notation, to each A ∈ Γ we associate the deck trans-formation sending a capped orbit x to x A . We define the Hamiltonian actionfunctional A H : e L pt M → R of a 1-periodic Hamiltonian H by A H ( x ) = Z H ( t, x ( t )) dt − Z x ω. Observe that the critical points of the Hamiltonian action functional are exactly( x, x ) for x a contractible 1-periodic orbit O ( H ) satisfying the equation x ′ ( t ) = X tH ( x ( t )). We denote by e O ( H ) the set of critical points of A H . The action spectrum of H is defined as Spec( H ) = A H ( e O ( H )). We remark following [68] that in themonotone and negative monotone cases the action spectrum is a closed nowheredense subset of R . In addition, if A ∈ Γ then A H ( x A ) = A H ( x ) − Z A ω, and for x ( k ) , the k − th iteration of x with the naturally inherited capping, we have A H ( k ) ( x ( k ) ) = k A H ( x ) . Definition 8 (Non-degenerate and weakly non-degenerate orbits) . A 1-periodicorbit x of H is called non-degenerate if 1 is not an eigenvalue of the linearized time-one map D ( φ H ) x (0) . We call x weakly non-degenerate if there exists at least oneeigenvalue of D ( φ H ) x (0) different from 1. A Hamiltonian H is said non-degenerate if all its 1-periodic orbits are non-degenerate.The non-degeneracy of an orbit x of H is equivalent to graph ( φ H ) = { ( x, φ H ( x )) | x ∈ M } intersecting the diagonal ∆ M ⊂ M × M transversely at ( x (0) , x (0)). Following[66] for any Hamiltonian H and ǫ > H ′ satisfying k H − H ′ k C < ǫ . This fact is key in the definition of filtered FloerHomology of degenerate Hamiltonians and for local Floer homology. Mean-index and the Conley-Zehnder index.
Following [29, 66], the mean-index ∆( H, x ) of a capped orbit x of a possibly degenerate Hamiltonian H is a realnumber measuring the sum of the angles swept by the eigenvalues of { D ( φ tH ) x ( t ) } lying on the unit circle. Here a trivialization induced by the capping is used inorder to view { D ( φ tH ) x ( t ) } as a path in Sp (2 n, R ). One can show that for the time-one map φ = φ H generated by the Hamiltonian H the mean-index depends onlyon the class e φ of { φ tH } t ∈ [0 , in the universal cover ] Ham(
M, ω ) making the notation∆( e φ, x ) suitable. In addition, the mean-index depends continuously on both φ andthe capped orbit and it behaves well with iterations,∆( e φ k , x ( k ) ) = k · ∆( e φ, x ) . Meanwhile, the
Conley-Zehnder index
CZ(
H, x ) of a non-degenerate Hamiltonianis integer valued and roughly measures the winding number of the above men-tioned eigenvalues. Once again, the index only depends on e φ so we can also writeCZ( H, x ) = CZ( e φ, x ). We shall use the same normalization as in [29], namely,CZ( H, x ) = n if x is a non-degenerate maximum of an autonomous Hamiltonian H with small Hessian and x is the constant capping. We shall omit the H or e φ in thenotation when it is clear from the context. We remark that for an element A ∈ Γ∆( x A ) = ∆( x ) − h c ( T M ) , A i and CZ( x A ) = CZ( x ) − h c ( T M ) , A i . Also, in the case that x is non-degenerate we have(2) | ∆( x ) − CZ( x ) | < n. Following [17, 58, 62] we observe that a version of the Conley-Zehnder index canbe defined even in the case where the capped orbit is degenerate. It is called theRobbin-Salamon index and it coincides with the usual Conley-Zehnder index in thenon-degenerate case. Furthermore we note that the mean-index can be equivalentlydefined by(3) ∆( e φ H , x ) = lim k →∞ k CZ( e φ kH , x ( k ) ) , where we are slightly abusing notation in the sense that CZ here means the Robbin-Salamon index so as to include the degenerate case. The limit in (3) exists, as theRobbin-Salamon index is a quasi-morphism CZ : f Sp (2 n, R ) → R (see [17, Section3.3.4], [13]). In particular, as can also be seen directly from its definition in [66],the mean-index is induced by a homogeneous quasi-morphism ∆ : f Sp (2 n, R ) → R . Morevoer, this map is continuous, and satisfies the additivity property∆(ΦΨ) = ∆(Φ) + ∆(Ψ)for all Φ ∈ π ( Sp (2 n, R )) ⊂ f Sp (2 n, R ) and all Ψ ∈ f Sp (2 n, R ) . Floer theory.
Floer thoery was first introduced by A.Floer [19, 20, 21] asa generalization of the Morse-Novikov homology for the Hamiltonian action func-tional defined above. We refer to [53] and references therein for details on the
AMILTONIAN NO-TORSION 17 constructions described in this subsection, and to [1, 71, 92], as well as to refer-ences therein, for a discussion of canonical orientations.2.2.1.
Filtered and total Floer Homology.
In this subsection we review the con-struction of filtered Floer homology in order to recall some basic properties and setnotation.Let H be a non-degenerate 1-periodic Hamiltonian on a rational symplectic mani-fold ( M, ω ) and K a fixed base field. For a ∈ R \ Spec( H ) and { J t ∈ J ( M, ω ) } t ∈ S a generic loop of ω -compatible almost complex structures, denote CF k ( H ; J ) c } < ∞ for every c ∈ R , the vector space over K generated by critical points of the Hamiltonian action functional of filtration level < a . The graded K -vector space CF ∗ ( H, J )
M, ω ) of the Hamiltonian group Ham( M, ω ). Also, when M is rational the above construction extends by a standard continuity argument todegenerate Hamiltonians. Remark . Since we deal with negative monotone symplectic manifolds, it is im-portant to emphasize that for our arguments to apply to this case in general, wemust make use of the machinery of virtual cycles (see [26, 27, 41] and referencestherein) to guarantee that the Floer differential is defined. In this case, the groundfield K should be of characteristic zero. However, if the minimal Chern number of( M, ω ) is N ≥ n − , then ( M, ω ) fits into the framework of semi-positive symplecticmanifolds, and hence classical transversality techniques suffice (see [34]).2.2.2. The irrational case. In this paper we also consider the case in which themanifold M is symplectically Calabi-Yau, which includes the possibility of it beingirrational. In this case we have to work a little harder if H is degenerate since thecontinuation argument above does not work as before, since non-spectral a, b for H do not necessarily remain non-spectral even for arbitrarily small perturbations H of H . Moreover, the resulting homology groups depend on the choice of non-degenerate perturbation H . We shall follow [32] to work around this issue.For a fixed Hamiltonian H and action window I = ( a, b ) with a, b ∈ R \ Spec( H ),consider the set of non-degenerate perturbations e H whose action spectrums do notinclude a and b and H ≤ e H (i.e. H ( t, x ) ≤ e H ( t, x ) for all x ∈ M and t ∈ S ).Observe that ≤ induces a partial order in the set of perturbations. In addition,by considering a monotone decreasing homotopy e H s from e H to e H , one obtainsan induced homomorphism between the Floer homology groups. These give riseto transition maps HF ∗ ( H ′ ) I → HF ∗ ( H ′′ ) I whenever H ′′ ≤ H ′ . Therefore, we candefine the filtered Floer homology of H by taking the direct limit HF ∗ ( H ) I = lim −→ HF ∗ ( H ′ ) I over the homology groups of the perturbations satisfying the aforementioned con-ditions. We remark that in the case where H is non-degenerate or M is rational,this definition coincides with the usual filtered Floer homology groups.2.2.3. Local Floer homology. In this section we shall follow [29] in order to brieflyreview the construction of the local Floer homology of a Hamiltonian H at a capping x of an isolated 1-periodic orbit x .Since x is isolated we can find an isolating neighbourhood U of x in the extendedphase-space S × M whose closure does not intersect the image { ( t, y ( t )) } t ∈ [0 , ofany other orbit y of H . For a C small enough non-degenerate perturbation H ′ the orbit x splits into finitely many 1-periodic orbits O ( H ′ , x ) of H ′ which arecontained in U and whose cappings are inherited from x . We denote by O ( H ′ , x )the capped 1-periodic orbits x splits into. Moreover, we can also guarantee that anyFloer trajectory and any broken trajectory, between capped orbits in O ( H ′ , x ) arecontained in U . For a base field K we consider the vector space CF ∗ ( H, x ) generated AMILTONIAN NO-TORSION 19 by O ( H ′ , x ), which by the above observation naturally inherits a Floer differentialand a grading by the Conley-Zehnder index. The homology of this chain complexis independent of the choice of the perturbation H ′ once it is close enough and itis called the local Floer homology of H at x ; it is denoted by HF loc ∗ ( H, x ). Thisgroup depends only on the class e φ of { φ tH } in the universal cover ] Ham( M, ω ) andthe capped orbit x, in the sense that homotopic paths have choices of cappings oforbits corresponding to a fixed point x ∈ Fix( φ ) in bijection, and the correspondinggroups are canonically isomorphic. Hence we write HF loc ∗ ( H, x ) = HF loc ∗ ( e φ, x ) . Ifwe ignore the Z -grading, then the group depends only on φ = φ H and x ∈ Fix( φ ) . Inthis case, we write HF loc ( φ, x ) for the corresponding local homology group whichis naturally only Z / (2)-graded.Let x be a capped 1-periodic orbit of a Hamiltonian H . We define the support of HF loc ∗ ( H, x ) to be the collection of integers k such that HF loc k ( H, x ) = 0. By thecontinuity of the mean-index and by equation (2) it follows that HF loc ∗ ( H, x ) issupported in the interval [∆( x ) − n, ∆( x ) + n ]. One can show that if x is weaklynon-degenerate then it is actually supported in (∆( x ) − n, ∆( x ) + n ). We shallexplore the idea behind the proof of this second fact later as we use the sameargument to prove a similar claim in slightly greater generality, namely, that of anisolated compact path-connected family of contractible fixed points.2.3. Quantum homology and PSS isomorphism. In the present section wedescribe the quantum homology of a symplectic manifold. It might be helpful tothink of it as the Hamiltoniana Floer homology in the case the Hamiltonian is givenby a C -small time-independent Morse function. Alternatively, one may considerit as the cascade approach [24] to Morse homology for the unperturbed symplecticarea functional on the space e L pt M . For a more detailed inspection of these subjectswe refer to [39, 53, 72].2.3.1. Quantum homology. For a fixed ground field K and the Novikov field Λ K = K [[ q − , q ] of ( M, ω ), where deg( q ) = 2 N , we set QH ( M ) = QH ( M, K ) = H ∗ ( M ; Λ K )as a Λ K -module. This module has the structure of a graded-commutative unitalalgebra over Λ K whose product, denoted by ∗ , is defined in terms of 3-point genus0 Gromov-Witten invatiants [40, 43, 63, 64, 90]. It can be thought of as a deforma-tion of the usual intersection product on homology. As in the classical homologyalgebra, the unit for this quantum product is the fundamental class [ M ] of M .2.3.2. Piunikhin-Salamon-Schwarz isomorphism. Under our conventions for theConley-Zehnder index, one obtains a map P SS : QH ∗ ( M ) → HF ∗− n ( H ) , by counting (for generic auxiliary data) isolated configurations of negative gradienttrajectories γ : ( −∞ , → M incident at γ (0) with the asymptotic lim s →−∞ u ( s, · ) of a map u : R × S → M of finite energy, satisfying the Floer equation ∂u∂s + J t ( u ) (cid:18) ∂u∂t − X tk ( u ) (cid:19) = 0 , for ( s, t ) ∈ R × S and K ( s, t ) ∈ C ∞ ( M, R ) a small perturbation β ( s ) H t , such thatit coincides with H for s ≪ − s ≫ +1. Also, we ask that β : R → [0 , β ( s ) = 0 for s ≪ − s ≫ +1. This mapproduces an isomorphism of Λ K -modules, which intertwines the quantum producton QH ( M ) with the pair of pants product on Hamiltonian Floer homology. Thismap is called the Piunikhin-Salamon-Schwarz isomorphism .2.4. Spectral invariants in Floer theory. We review the theory of spectralinvariants following the works of [29, 53, 58] which contain a more exhaustive listof properties and finer details of the construction.Let (