aa r X i v : . [ m a t h . S G ] A ug ON THE RANK OF π ( Ham ) ANDR´ES PEDROZA
Abstract.
We show that for any positive integer k there exists a closedsymplectic 4-manifold, such that the rank of the fundamental group of thegroup of Hamiltonian diffeomorphisms is at least k. Introduction
The problem of determining the homotopy type of the group of Hamiltoniandiffeomorphisms for a closed symplectic manifold is nowadays a far reachingproblem in symplectic topology. In order to have an idea of the limited knowl-edge in the subject, absolutely nothing is known about the homotopy typeof the group Ham( T , ω ) where ( T , ω ) is the 4-dimensional torus with thestandard symplectic form. Note that we stated an example of a 4-dimensionalmanifold, since for 2-dimensional symplectic manifolds the problem is partiallyunderstood due to the fact that in two dimensions symplectic geometry agreeswith area and orientation preserving geometry. See for instance [11, Sec. 7.2].In this direction, Ham( S , ω ) = Symp ( S , ω ) has the same homotopy typeas SO (3) [13]; and for the surface of genus g ≥
1, Ham( M g , ω ) is simplyconnected.In higher dimensions there are some cases where the homotopy type ofHam( M, ω ) is completely understood, due the techniques of holomorphic curves.For instance, Ham( C P , ω F S ) has the homotopy type of
P U (3); Ham( C P × C P , ω F S ⊕ ω F S ) has the homotopy type of SO (3) × SO (3). These results aredue to M. Gromov [4]. The rational homotopy type for the case of one-pointblow up of ( C P , ω FS ) was settled by M. Abreu and D. McDuff in [1]. For moreexamples, see the work of F. Lalonde and M. Pinsonnault [6]; and J. Evans[3].Leave behind the problem of determining the homotopy type and focus onfirst stage of the problem: the fundamental group of Ham( M, ω ). Recall thatthe fundamental group of a topological group is an abelian group. Hence isnatural to ask: Given any positive integer k , does there exists a symplecticmanifold such that the free part of π (Ham( M, ω )) is isomorphic to Z k . Usingcartesian products of symplectic manifolds together with Seidel’s representa-tion [12] or Weinstein’s morphism [14] if possible to provide a weak answer tothe problem. Namely, is possible to construct a symplectic manifold ( M, ω ) such that the rank of π (Ham( M, ω )) is at least k . See for instance [9], whereSeidel’s morphism on cartesian products is studied. In this note we arrive tothe same conclusion but on 4-dimensional symplectic manifolds. That is, inthe smallest possible dimension. Theorem 1.1.
Given a positive integer k , there exists a closed, connected andsimply connected symplectic -manifold ( M, ω ) such that rank π (Ham( M, ω )) ≥ k. The proof that we provide is a hands-on proof. The symplectic 4-manifoldof the theorem turns out to be the blow up of ( C P , ω FS ) at k points of dis-tinct weights. The techniques used throughout this note are soft techniques ofsymplectic topology, where Weinstein’s morphism plays a key role.If Symp ( M, ω ) stands for the connected component of Symp(
M, ω ) thatcontains the identity map, then the inclusion Ham(
M, ω ) ⊂ Symp ( M, ω )induces an injective map π (Ham( M, ω )) → π (Symp ( M, ω ))due to the Flux morphism, [8, Ch. 10]. Therefore, Theorem 1.1 also holds ifthe group Ham(
M, ω ) is replaced by Symp ( M, ω ).Unlike the group of Hamiltonian diffeomorphisms of a closed symplecticmanifold, the group of symplectic diffeomorphisms is not necessarily connected.Therefore, following the same line of ideas is natural to ask if given a positivenumber k does there exists a closed connected symplectic manifold ( M, ω )such that the number of connected components of Symp(
M, ω ) is equal to k . Recently, D. Aroux and I. Smith solved this problem in [2, Thm 1.3] viaFloer-theoretic arguments.As a byproduct of the arguments used to prove the main result, we are alsoable to show that Calabi’s morphism on the one-point blow up of ( R , ω ) isnon trivial. The first examples of open manifolds whose Calabi’s morphism isnon trivial are due to A. Kislev [5]. Alongside we prove that the rank of thefundamental group of Ham( f M , e ω κ ) is positive where ( f M , e ω κ ) is the one-pointblow of weight κ of ( M, ω ). Hence our result improves the one obtained byD. McDuff [7], since information about π ( M ) and π ( M ) is irrelevant in ourarguments. Theorem 1.2.
Let ( f M , e ω κ ) be the one-point blow up of weight κ of the closedmanifold ( M, ω ) . Then for infinitely many values of κ , the rank of π (Ham( f M , e ω κ )) is positive. It is worth mentioning the case M = T with the standard symplectic form.Therefore, π (Ham( e T , e ω κ )) has positive rank. However, nothing is knownabout the group π (Ham( T , ω )).We are grateful to L. Polterovich from bringing [5] to our attention. N THE RANK OF π (Ham) 3 Preliminary computations
A compactly supported path of Hamiltonian diffeomorphismson ( R , ω ) . Consider 1-periodic smooth functions a , a , a , a : R → R suchthat a (0) = a (0) = 1 and a (0) = a (0) = 0. Let α : R → R be also asmooth function such that α (0) ∈ Z . Furthermore, the functions a j are alsosubject to the condition that A t := (cid:18) a ( t ) e πi α ( t ) a ( t ) e − πit a ( t ) e πi α ( t ) a ( t ) e − πit (cid:19) t ∈ [0 , × α is α (0) ∈ Z , itdoes not have to be periodic unlike the functions a j . Thus { A t } ≤ t ≤ is a pathin U (2) that stars at the identity. Let ψ a t : ( C , ω ) → ( C , ω ) be the pathof Hamiltonian diffeomorphism induced by { A t } . A direct computation yieldsthe Hamiltonian H a t and time-dependent vector field X a t induced by { ψ a t } . Lemma 2.3.
The path { ψ a t } ≤ t ≤ induced by the path of unitary matrices { A t } induces in ( R , ω ) the time-dependent vector field X a t = (cid:8) πy ( a − a α ′ ) + x ( a ′ a + a ′ a ) + 2 πy ( a a − a a α ′ ) (cid:9) ∂∂x + (cid:8) πx ( a α ′ − a ) + 2 πx ( a a α ′ − a a ) + y ( a ′ a + a ′ a ) (cid:9) ∂∂y + (cid:8) x ( a a ′ + a a ′ ) + 2 πy ( a a − a a α ′ ) + 2 πy ( a − a α ′ ) (cid:9) ∂∂x + (cid:8) πx ( a a α ′ − a a ) − y ( a a ′ + a a ′ ) + 2 πx ( a α ′ − a ) (cid:9) ∂∂y , and Hamiltonian function H a t ( x , y , x , y ) = π ( − a α ′ + a )( x + y ) + π ( − a α ′ + a )( x + y )+2 π ( a a − a a α ′ )( x x + y y )+( a a ′ + a a ′ )( x y − x y ) . For r > B r the open ball of radius r in R centered at the origin.Using the fact that the functions a j define the unitary matrix (1), it followsthat the integral of H a t over the ball B r only depends on α and r . Lemma 2.4. If H a t is the Hamiltonian function of Lemma 2.3, then Z Z B r H a t ω dt = (cid:18) π r (cid:19) (1 + α (0) − α (1)) . ANDR´ES PEDROZA
Proof.
The integral over B r of the last two terms of H a t are zero. Now sincethe functions a j are the entries of a unitary matrix, it follows that Z Z B r H a t ω dt = π Z B r ( x + y ) ω · Z − a α ′ + a − a α ′ + a dt = π (cid:18) π r (cid:19) Z − α ′ dt = π (cid:18) π r (cid:19) (1 + α (0) − α (1)) . (cid:3) We are interested in the case when the above integral is non zero. Thereby,by the last result we neglect the functions a j and focus our attention on α .Next we used a bump function to obtain a compactly supported Hamiltonianpath. This is the reason that we considered a Hamiltonian path instead of aHamiltonian loop. If a Hamiltonian function induces a loop, then multiplyingit by a bump function will no longer induce a loop. To that end fix r , R ∈ R such that 0 < r < R and also fix a bump function ρ : R → R such that ρ ≡ B r and ρ ≡ R \ B R . Consider the compactly supportedsmooth function H a ,ρt : ( R , ω ) → R defined as H a ,ρt := ρ · H a t and let { ψ a ,ρt } ≤ t ≤ be the induced Hamiltonian path. Observe that ψ a ,ρt ∈ Ham c ( R , ω ) and that on the ball B r we have that ψ a ,ρt = ψ a t and H a ,ρt = H a t for all t ∈ [0 , A compactly supported loop of Hamiltonian diffeomorphismson ( R , ω ) . Next we define a loop in Ham c ( R , ω ) based at the identity byconcatenating two paths of the kind defined above. Fix a , . . . a , b , . . . b smooth 1-periodic functions and α and β also smooth functions, such that allfunctions are subject to the conditions previously imposed. Further, we alsoimpose the condition that the two path of matrices { A t } and { B t } agree in aneighborhood of t = 1. Therefore, the exponent functions satisfy α (1) − β (1) ∈ Z .Define the Hamiltonian loop ψ = { ψ t } ≤ t ≤ in Ham c ( R , ω ) as ψ t := (cid:26) ψ a ,ρt t ∈ [0 , ψ b ,ρ − t t ∈ [1 , . (2)Thus, its compactly supported Hamiltonian function is given by H t := (cid:26) H a ,ρt t ∈ [0 , H b ,ρ − t t ∈ [1 , . (3) N THE RANK OF π (Ham) 5 The above observations on the paths { ψ a ,ρt } and { ψ b ,ρt } and the computationof Lemma 2.4 imply the following facts about the Hamiltonian loop ψ. Proposition 2.5.
Given r , R ∈ R such that R > r > there is a Hamil-tonian loop ψ = { ψ t } ≤ t ≤ defined as in (2) such that it is supported in B R andon B r it agrees with a loop of unitary matrices. Moreover, its Hamiltonianfunction H t satisfies Z Z B r H t ω dt = (cid:18) π r (cid:19) ( α (0) − α (1) − β (0) + β (1)) . (4)The condition that the paths { ψ a ,ρt } and { ψ b ,ρt } agree in a neighborhood of1, imply that α (0) − α (1) − β (0) + β (1) is a integer. Thus Eq. (4) is a kind of winding number for the loop of unitary matrices. Furthermore, it is possibleto choose the functions α and β so that the paths agree in a neighborhood of 1and α (0) − α (1) − β (0) + β (1) is non zero. From now on we fix the Hamiltonianloop ψ in Ham c ( R , ω ) defined in (2) so that α (0) − α (1) − β (0) + β (1) is equalto 1. The actual value is not important, what is relevant at this point is thatit is a non zero integer.On an open manifold ( M, ω ) the Calabi morphism Cal : π (Ham c ( M, ω )) → R is defined as Cal( φ ) := Z Z M F t ω n dt, where the loop φ is generated by the compactly supported Hamiltonian F t .Moreover, if ( M, ω ) is exact then Calabi’s morphism is identically zero. Con-sequently, for the the loop ψ defined in (2) we have that0 = Z Z R H t ω dt, (5)in contrast with the integral over B r that is non zero.Equation (5) will be useful when we consider the loop ψ in a Darboux charton a closed manifold. For, in this case the normalization condition correspondsto zero mean.3. A loop of Hamiltonian diffeomorphisms in the one-point blowup of infinite order
In [10] it is proved that if a closed symplectic manifold admits a Hamilton-ian circle action, then after blowing up one point the fundamental group ofthe group of Hamiltonian diffeomorphisms has positive rank. In this sectionwe prove that the above result always holds, namely we prove Theorem 1.2.Henceforth, the hypothesis about the Hamiltonian circle action is no longerrequired. We restrict to the four-dimensional case in view of the main result of
ANDR´ES PEDROZA this paper; however the result presented in this section holds on any symplecticmanifold of dimension greater than or equal than four.This section can be considered has the first step of the proof of the maintheorem. Recall that for any R >
0, the loop ψ = { ψ t } ≤ t ≤ defined in (2)is supported in the open ball B R ⊂ ( R , ω ). Let ( M, ω ) be a closed rationalsymplectic 4-manifold. For R > ψ ∈ Ham c ( R , ω ) can be regarded in Ham( M, ω ).Since we are in a closed symplectic manifold (
M, ω ), we must normalizedthe Hamiltonian function H t : ( M, ω ) → R . Thus, for t ∈ [0 ,
1] define c t := 1Vol( M, ω ) Z M H t ω . Afterwards define H N t : M → R as H N t := H t − c t . Thus, H N t is the normalizedHamiltonian function that induces the same loop ψ in Ham( M, ω ).Call ιB r ⊂ M the image of B r ⊂ R under the Darboux embedding. Thenthe loop ψ = { ψ t } satisfies the following: • ψ t ( ι (0))) = ι (0) for all t ∈ [0 , • ψ t behaves like a unitary matrix on ιB r for all t ∈ [0 , ιB r ⊂ M , define ( f M , e ω r ) to be the one-pointblow up at ι (0) of ( M, ω ) of weight r . From the above remarks on the loop ψ ,it follows from [10, Sec. 3] that ψ induces a Hamiltonian loop e ψ = { e ψ t } ≤ t ≤ in Ham( f M , e ω r ). For appropriate values of r , we claim that [ e ψ ] has infiniteorder in π (Ham( f M , e ω r )) . We prove this using Weinstein’s morphism A : π (Ham( f M , e ω r )) → R / P ( f M , e ω r ) , where P ( f M , e ω r ) is the period group.Next we prove Theorem 1.2 that was stated at the Introduction. We givea more precise statement of the theorem in terms of the loop of Hamiltoniandiffeomorphisms ψ defined above and the weight of the blow up. Keep in mindthat we state this result for four-dimensional symplectic manifolds, but thesame argument works in higher dimensions. Theorem 3.6.
Let r > such that πr is a transcendental number. If ( M, ω ) is a rational symplectic 4-manifold, then the induced loop [ e ψ ] has infinite orderin π (Ham( f M , e ω r )) . Proof.
Since (
M, ω ) is rational, there are q , . . . , q s ∈ Q such that P ( M, ω ) = Z h q , . . . , q s i . In fact, P ( f M , e ω r ) = Z h q , . . . , q s , πr i since the area of the linein the exceptional divisor is πr . N THE RANK OF π (Ham) 7 From [10, Thm. 1.1], it is possible to compute A ( e ψ ) in terms solely of theloop ψ . Namely, A ( e ψ ) = " A ( ψ ) + 1Vol( f M , e ω r ) Z Z ιB r H N t ω dt . Since { ψ t } is null homotopic, it follows that A ( ψ ) = 0 in R / P ( M, ω ). Observethat the integral of H N t over ιB r ⊂ M is the same as the integral over B r ⊂ R . Therefore, from Proposition 2.5 we have that Z Z ιB r H N t ω dt = Z Z ιB r H t − c t ω dt = (cid:18) π r (cid:19) · − Vol( B r , ω ) Z c t dt. The functions H t are supported B R ⊂ R , thus they can also be consideredas a functions on M . Hence using Eq. (5) it follows that Z c t dt = Z M, ω ) Z M H t ω dt = 1Vol( M, ω ) Z Z M H t ω dt = 0 . If V stands for Vol( M, ω ), then Vol( f M , e ω r ) = V − π r /
2. Substitutingthe above computations, we have A ( e ψ ) = (cid:20) V − π r / (cid:18) π r (cid:19)(cid:21) ∈ R / Z h q , . . . , q s , πr i . Then the equation A ( e ψ m ) = 0, for m ∈ N , is equivalent to a polynomialequation on πr with rational coefficients. Recall that V ∈ Q . Since πr istranscendental, the Hamiltonian loop e ψ = [ { e ψ t } ≤ t ≤ ] has infinite order in π (Ham( f M , e ω r )) . (cid:3) Proof of the main theorem
The proof of Theorem 3.6 gives the blueprint that we follow in order to provethe main result. The proof of the main theorem deals with k distinct Hamil-tonian loops supported in k mutually disjoint balls. Thereby, the hypothesisin Theorem 3.6, about πr being transcendental, is replaced by the followinglemma. Lemma 4.7.
Given k ∈ N there exist k distinct real numbers y , . . . , y k suchthat for any j, s ∈ { , . . . , k } the equation ( a + q y + · · · + q k y k )( b − ( y + · · · + y k )) + cy j + y s = 0 ANDR´ES PEDROZA has no solution for any q , . . . , q k , a, b, c ∈ Q . The proof of the lemma is a consequence of the fact that R is an infinitedimensional Q -vector space. Next we provide the proof of the main theorembuilded on the ideas of the proof of Theorem 3.6.For the proof of the main theorem is important to consider the complex2-dimensional projective space ( C P , ω F S ) endowed with its standard Fubini-Study symplectic form. If ω F S is normalized so that ( C P \ C P , ω F S ) issymplectomorphic to the open ball of radius R in ( R , ω ), then the area of acomplex line is πR . Therefore, the period group of ( C P , ω F S ) is Z h πR i andVol( C P , ω F S ) = ( πR ) / Proof of Theorem 1.1.
Given k >
0, it follows from Lemma 4.7 that there are k distinct numbers r , . . . , r k ∈ R > where y j := πr j for j ∈ { , . . . , n } .Let R , . . . , R k ∈ R be any numbers such that R j > r j for all j ∈ { , . . . , k } .On R , fix k mutually disjoint balls B , . . . , B k centered at p , . . . , p k and ofradii R , . . . , R k respectively. Inside each ball B j fix a a smaller ball B r j ofradius r j centered at p j . Next, consider the complex 2-dimensional projectivespace ( C P , ω F S ) such that the symplectic form is normalized so that ( C P \ C P , ω F S ) is symplectomorphic to an open ball of radius R in ( R , ω ) thatcontains all the balls B , . . . , B k and πR is a rational number. Therefore theperiod group of ( C P , ω F S ) is Z h πR i .Call ι j B r j ⊂ C P the image of the fixed ball B r j ⊂ R . Hence in ( C P , ω F S )there are k mutually disjoint embedded balls ι B r , . . . , ι k B r k . Denote by( C P k C P , e ω r ) the symplectic manifold that is obtained by blowing up the k points ι ( p ) , . . . , ι k ( p k ) in ( C P , ω F S ) where the weight at ι j ( p j ) is r j . Thatis, the embedded ball ι j B r j is removed from the torus for j ∈ { , . . . , k } . ( C P k C P , e ω r ) is the desired closed symplectic 4-manifold of the main the-orem. Notice that it is simply connected.Next we define k Hamiltonian loops. From Proposition 2.5 for each j ∈{ , . . . , k } there is a loop ψ ( j ) = { ψ ( j ) t } ≤ t ≤ in Ham c ( R , ω ) supported in theball of radius R j such that inside the ball of radius r j it agrees with a loopof unitary matrices. Therefore, the k loops in Ham c ( R , ω ) induced k loopsHam( C P , ω FS ) that we also denoted by ψ (1) , . . . , ψ ( k ) .For every j ∈ { , . . . , k } , the loop ψ ( j ) on ( C P , ω FS ) behaves as a loopof unitary matrices on each of the embedded ball ι s B r s . Moreover, if s = j it is the constant loop on ι s B r s . In any case, the loop ψ ( j ) fixes the points ι ( p ) , . . . , ι k ( p k ). Therefore, from [10, Sec. 3] it follows that ψ ( j ) inducesa loop e ψ ( j ) in Ham( C P k C P , e ω r ). We claim that the loops e ψ (1) , . . . , e ψ ( k ) generate a subgroup isomorphic to Z k in π (Ham( C P k C P , e ω r )). N THE RANK OF π (Ham) 9 Fix j ∈ { , . . . k } . The corresponding Hamiltonian H ( j ) t of ψ ( j ) is compactlysupported, as before let H ( j ) , N t : ( C P , ω FS ) → R be its normalization. As inthe proof of Theorem 3.6 we have that Z Z ι j B rj H ( j ) , N t ω dt = (cid:18) π r j (cid:19) · A ( e ψ ( j ) ) = (cid:20) V − π ( r + · · · + r k ) / (cid:18) π r j (cid:19)(cid:21) ∈ R / Z h πR , πr , . . . , πr k i . where V := Vol( C P , ω ) ∈ Q .Since πR ∈ Q and the numbers πr , . . . , πr k were chosen according toLemma 4.7, it follows that for any m ∈ Z > the equation A (( e ψ ( j ) ) m ) = 0does not hold. Therefore [ e ψ ( j ) ] has infinite order in π (Ham( C P k C P , e ω r )).The same reasoning implies that A (( e ψ ( j ) ) m ) = A (( e ψ ( s ) ) n ) for distinct j, s ∈{ , . . . , k } and any m, n ∈ Z > . Henceforth the rank of π (Ham( C P k C P , e ω r ))is at least k . (cid:3) Calabi’s morphism
As noted before, Calabi’s morphism on π (Ham c ( R , ω )) is trivial. However,the arguments used before show that Calabi’s morphism on the one-point blowup of ( R , ω ) is non trivialFix r >
0, let ( f R , e ω r ) be the blowup of the origin in ( R , ω ) of weight r . Aswe have seen the loop ψ in Ham c ( R , ω ) induces the loop e ψ in Ham c ( f R , e ω r ). Proposition 5.8.
For any r > , the Calabi morphism on Ham c ( f R , e ω r ) doesnot vanish.Proof. Let ψ be the Hamiltonian loop in Ham c ( R , ω ) defined in Eq. 2 andlet e ψ be the induced loop in Ham c ( f R , e ω r ). According to [10, Sec. 1], Cal( e ψ )and Cal( ψ ) are related asCal( e ψ ) = Cal( ψ ) − Z Z B r H t ω dt (6)where H t : ( R , ω ) → R is the compactly supported Hamiltonian function ofthe loop ψ defined in Eq. (3). Thus from Proposition 2.5,Cal( e ψ ) = − π r · . (cid:3) Remark.
The above result is true for any dimension; Calabi’s morphism onHam c ( g R n , e ω r ) does not vanish for n ≥ r > π (Ham c ( f R , e ω r ))it does not descend to Ham c ( f R , e ω r ). Furthermore, for the Hamiltonian loop e ψ in Ham c ( f R , e ω r ) that appears in the previous proof we have that ℓ ( e ψ ) ≥ π r . Here ℓ ( · ) stands for the Hofer length of the class in π (Ham( · )). For the precisedefinition of ℓ ( · ), see [11, Sec. 7.3]. References [1]
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Facultad de Ciencias, Universidad de Colima, Bernal D´ıaz del Castillo No.340, Colima, Col., Mexico 28045
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