##### Document Text Contents

Page 1

Geometry & Topology 10 (2006) 1635–1747 1635

Geometry of contact transformations and domains:

orderability versus squeezing

YAKOV ELIASHBERG

SANG SEON KIM

LEONID POLTEROVICH

Gromov’s famous non-squeezing theorem (1985) states that the standard symplectic

ball cannot be symplectically squeezed into any cylinder of smaller radius. Does

there exist an analogue of this result in contact geometry? Our main finding is that

the answer depends on the sizes of the domains in question: We establish contact non-

squeezing on large scales, and show that it disappears on small scales. The algebraic

counterpart of the (non)-squeezing problem for contact domains is the question of

existence of a natural partial order on the universal cover of the contactomorphisms

group of a contact manifold. In contrast to our earlier beliefs, we show that the

answer to this question is very sensitive to the topology of the manifold. For instance,

we prove that the standard contact sphere is non-orderable while the real projective

space is known to be orderable. Our methods include a new embedding technique in

contact geometry as well as a generalized Floer homology theory which contains both

cylindrical contact homology and Hamiltonian Floer homology. We discuss links to

a number of miscellaneous topics such as topology of free loops spaces, quantum

mechanics and semigroups.

53D10, 53D40; 53D35, 53D50

Dedicated to Dusa McDuff on the occasion of her 60th birthday

1 Introduction and main results

1.1 Contact (non)-squeezing

Consider the standard symplectic vector space R2n endowed with the symplectic form

! D dp^ dq D

nP

1

dpi ^ dqi . We often identify R2n with Cn and write z D pC iq

for the complex coordinate. Symplectic embeddings preserve the volume, and hence

the Euclidean ball

B2n.R1/ WD f�jzj2 <R1g

Published: 28 October 2006 DOI: 10.2140/gt.2006.10.1635

Page 2

1636 Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich

cannot be symplectically embedded into B2n.R2/ if R2 <R1 . Gromov’s famous non-

squeezing theorem states that there are much more subtle obstructions for symplectic

embeddings and, in particular, B2n.R1/ cannot be symplectically embedded into the

cylinder

C 2n.R2/ WD B2.R2/�R2n�2

when R2 <R1 , see [28]. This result led to the first non-trivial invariants of symplectic

domains in dimension 2n� 4.

In the present paper we address the question whether there are any analogues of

non-squeezing results in contact geometry. Consider the prequantization space of

R2n , that is the contact manifold V D R2n �S1; S1 D R=Z, with contact structure

� D Ker.dt � ˛/ where ˛ is the Liouville form 1

2

.pdq � qdp/. Given a subset

D � R2n , write yD D D � S1 for its prequantization. The naive attempt to extend

the non-squeezing from D to yD fails. It is is easy to show (see Proposition 1.24 and

Section 2.2) that for any R1;R2 > 0 there exists a contact embedding of yB.R1/ into

yB.R2/ which, for n> 1, is isotopic to the inclusion through smooth embeddings into

V . Furthermore, due to the conformal character of the contact structure, the domain

yB.R/ can be contactly embedded into an arbitrarily small neighborhood of a point in

V (see Corollary 1.25 below).

However, the situation becomes more sophisticated if one considers only those contact

embeddings which come from globally defined compactly supported contactomor-

phisms of .V; �/. We write G D Cont .V; �/ for the group of all such contactomor-

phisms.

Given two open subsets U1 and U2 of a contact manifold V , we say that U1 can be

squeezed into U2 if there exists a contact isotopy ‰t W Closure.U1/! V; t 2 Œ0; 1�;

such that ‰0 D 1 and

‰1.Closure.U1//� U2:

The isotopy f‰tg is called a contact squeezing of U1 into U2 . If, in addition, W � V

is an open subset such that Closure.U2/�W and ‰t .Closure.U1//�W for all t , we

say that U1 can be squeezed into U2 inside W . If the closure of U1 is compact, the

ambient isotopy theorem (see, for instance, Geiges [24]) guarantees that any squeezing

of U1 into U2 inside W extends to a contactomorphism from G whose support lies in

W .1

1If the group G is not connected than the possibility to squeeze by an isotopy is stronger than by a

global contactomorphism. All squeezing and non-squeezing results in this paper are proven in the strongest

sense, ie, squeezing is always done by a contact isotopy while in our non-squeezing results we prove

non-existence of the corresponding global contactomorphism.

Geometry & Topology, Volume 10 (2006)

Page 57

Geometry of contact transformations and domains 1691

This implies, in particular, that for every pair of orbits

C;

� 2 P˙ with CZ.

C/�

CZ.

�/D 0 we have

dimM.Con; I;

C;

�/D 0:

This enables us to define a monotonicity morphism

monW C .a;b/.�!HC;JC/! C .a;b/.

�!

H�;J�/

by the formula

(57) mon.

C/D

X

2P.a;b/.RC/

CZ.

C/�CZ.

�/D0

�.

C;

�/

� ;

where �.

C;

�/ is the mod 2 number of elements of the finite set M.Con; I;

C;

�/.

This morphism respects the gradings of the complexes. The next standard fact from

the Floer theory is as follows.

Theorem 4.29 Under the above assumptions, for any a< b<min.C�; CC/ and for all

.e; i/ 2 I.HC/ formula (57) defines a homomorphism of generalized Floer complexes

monW

�

C

.a;b/

e;i

.

�!

HC;JC/; dC

�

!

�

C

.a;b/

�.e;i/

.

�!

H�;J�/; d�

�

;

so that we have

d� ımonDmon ı dC:

In particular, mon defines a homology homomorphism

monW GFH.a;b/

e;i

.

�!

H;JC/! GFH.a;b/�.e;i/.

�!

H;J�/ :

The homomorphism mon constructed above a priori depends on the following data:

� a concordance ConD Con.H�;HC/D .W; .…//;

� an almost complex structure I adjusted to Con which respects almost complex

structures J˙ adjusted to

�!

H˙ .

Standard arguments of Floer theory show that the homomorphism mon does not change

under the following operations:

� replacing the pair .Con; I/ by a pair .Con0; I 0/ where Con0 is a concordance

equivalent to Con and I 0 is the corresponding (see Lemma 4.26) almost complex

structure adjusted to Con0 ;

� a homotopy of the concordance structure .…/ on W through the concordance

structures;

Geometry & Topology, Volume 10 (2006)

Page 58

1692 Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich

� a homotopy of the adjusted almost complex structure I on W through almost

complex structures adjusted to Con and respecting J˙ .

Consider now three framed Hamiltonian structures

�!

H1;

�!

H2 and

�!

H3 equipped with

adjusted almost complex structures J1;J2 and J3 respectively. Assume that we are

given directed concordances

ConiC1;i D Con.Hi ;HiC1/ ; i D 1; 2;

which induce monotonicity morphisms

moniC1;i W GFH.�!H iC1;JiC1/! GFH.

�!

H i ;Ji/ :

A standard argument of Floer theory shows that the gluing Con32 ˘Con21 of concor-

dances corresponds to the composition mon32 ımon21 of the monotonicity morphisms.

The next result is crucial for our proof of non-squeezing results in contact geometry. It

will enable us to reduce calculations in contact homology to more traditional calculations

in Floer homology.

Proposition 4.30 For a regular Hamiltonian structure the generalized Floer homology

does not depend on the choice of a framing and of an adjusted almost complex structure.

We start with two auxiliary lemmas whose proof is elementary and is left to the reader.

Lemma 4.31 Let Aı

i

, where ı 2 f�; 0;Cg and i 2 f0; 1g be a collection of six

linear spaces. Suppose that we are given eight morphisms between them such that the

following diagram commutes:

(58) AC

0

//

��@

@@

@@

@@

A0

1

//

��?

??

??

??

?

A�

0

A

C

1

//

??~~~~~~~

A0

0

//

??��������

A�

1

Assume that the diagonal arrows are isomorphisms. Then all the arrows are isomor-

phisms.

Lemma 4.32 Let E be a topological linear space and let ƒ � E be a convex cone.

Given any �0; �1 2 E with �1 � �0 2 Interior.ƒ/ and any �0; �1 2ƒ, there exists a

smooth path �s , s 2 Œ0; 1�, connecting �0 with �1 such that its derivative P�s satisfies

the following conditions:

(i) P�s D �0 for s near 0 and P�s D �1 for s near 1;

Geometry & Topology, Volume 10 (2006)

Page 113

Geometry of contact transformations and domains 1747

[39] G I Olshanskii, Invariant orderings in simple Lie groups. Solution of a problem of E.

B. Vinberg, Funktsional. Anal. i Prilozhen. 16 (1982) 80–81 (Russian) MR684139

English translation: Functional Anal. Appl. 16 (1982) 311–313

[40] J Palis, Jr, W de Melo, Geometric theory of dynamical systems, Springer, New York

(1982) MR669541An introduction, Translated from the Portuguese by A K Manning

[41] J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827–844

MR1241874

[42] W Rudin, Function theory in the unit ball of Cn , Grundlehren series 241, Springer,

New York (1980) MR601594

[43] D Salamon, J Weber, Floer homology and the heat flow, Geom. Funct. Anal. (GAFA).

to appear arXiv:math.SG/0304383

[44] I Ustilovsky, Infinitely many contact structures on S4mC1 , Internat. Math. Res. Notices

(1999) 781–791 MR1704176

[45] C Viterbo, Functors and computations in Floer homology with applications, Part II,

preprint (1996)

[46] C Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math.

Soc. 13 (2000) 411–431 MR1750956

[47] J Weber, Noncontractible periodic orbits in cotangent bundles and Floer homology,

Duke Math. J. 133 (2006) 527–568 MR2228462

[48] M-L Yau, Cylindrical contact homology of subcritical Stein-fillable contact manifolds,

Geom. Topol. 8 (2004) 1243–1280 MR2087083

Department of Mathematics, Stanford University

Stanford, CA 94305-2125, USA

Departamento de Matemática, Instituto Superior Técnico

Av Roviso Pais, 1049-001 Lisboa, Portugal

School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact

Sciences, Tel Aviv University, 69978 Tel Aviv, Israel

[email protected], [email protected],

[email protected]

Proposed: Eleny Ionel Received: 12 February 2006

Seconded: Tomasz Mrowka, Peter Ozsváth Revised: 30 September 2006

Geometry & Topology, Volume 10 (2006)

Geometry & Topology 10 (2006) 1635–1747 1635

Geometry of contact transformations and domains:

orderability versus squeezing

YAKOV ELIASHBERG

SANG SEON KIM

LEONID POLTEROVICH

Gromov’s famous non-squeezing theorem (1985) states that the standard symplectic

ball cannot be symplectically squeezed into any cylinder of smaller radius. Does

there exist an analogue of this result in contact geometry? Our main finding is that

the answer depends on the sizes of the domains in question: We establish contact non-

squeezing on large scales, and show that it disappears on small scales. The algebraic

counterpart of the (non)-squeezing problem for contact domains is the question of

existence of a natural partial order on the universal cover of the contactomorphisms

group of a contact manifold. In contrast to our earlier beliefs, we show that the

answer to this question is very sensitive to the topology of the manifold. For instance,

we prove that the standard contact sphere is non-orderable while the real projective

space is known to be orderable. Our methods include a new embedding technique in

contact geometry as well as a generalized Floer homology theory which contains both

cylindrical contact homology and Hamiltonian Floer homology. We discuss links to

a number of miscellaneous topics such as topology of free loops spaces, quantum

mechanics and semigroups.

53D10, 53D40; 53D35, 53D50

Dedicated to Dusa McDuff on the occasion of her 60th birthday

1 Introduction and main results

1.1 Contact (non)-squeezing

Consider the standard symplectic vector space R2n endowed with the symplectic form

! D dp^ dq D

nP

1

dpi ^ dqi . We often identify R2n with Cn and write z D pC iq

for the complex coordinate. Symplectic embeddings preserve the volume, and hence

the Euclidean ball

B2n.R1/ WD f�jzj2 <R1g

Published: 28 October 2006 DOI: 10.2140/gt.2006.10.1635

Page 2

1636 Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich

cannot be symplectically embedded into B2n.R2/ if R2 <R1 . Gromov’s famous non-

squeezing theorem states that there are much more subtle obstructions for symplectic

embeddings and, in particular, B2n.R1/ cannot be symplectically embedded into the

cylinder

C 2n.R2/ WD B2.R2/�R2n�2

when R2 <R1 , see [28]. This result led to the first non-trivial invariants of symplectic

domains in dimension 2n� 4.

In the present paper we address the question whether there are any analogues of

non-squeezing results in contact geometry. Consider the prequantization space of

R2n , that is the contact manifold V D R2n �S1; S1 D R=Z, with contact structure

� D Ker.dt � ˛/ where ˛ is the Liouville form 1

2

.pdq � qdp/. Given a subset

D � R2n , write yD D D � S1 for its prequantization. The naive attempt to extend

the non-squeezing from D to yD fails. It is is easy to show (see Proposition 1.24 and

Section 2.2) that for any R1;R2 > 0 there exists a contact embedding of yB.R1/ into

yB.R2/ which, for n> 1, is isotopic to the inclusion through smooth embeddings into

V . Furthermore, due to the conformal character of the contact structure, the domain

yB.R/ can be contactly embedded into an arbitrarily small neighborhood of a point in

V (see Corollary 1.25 below).

However, the situation becomes more sophisticated if one considers only those contact

embeddings which come from globally defined compactly supported contactomor-

phisms of .V; �/. We write G D Cont .V; �/ for the group of all such contactomor-

phisms.

Given two open subsets U1 and U2 of a contact manifold V , we say that U1 can be

squeezed into U2 if there exists a contact isotopy ‰t W Closure.U1/! V; t 2 Œ0; 1�;

such that ‰0 D 1 and

‰1.Closure.U1//� U2:

The isotopy f‰tg is called a contact squeezing of U1 into U2 . If, in addition, W � V

is an open subset such that Closure.U2/�W and ‰t .Closure.U1//�W for all t , we

say that U1 can be squeezed into U2 inside W . If the closure of U1 is compact, the

ambient isotopy theorem (see, for instance, Geiges [24]) guarantees that any squeezing

of U1 into U2 inside W extends to a contactomorphism from G whose support lies in

W .1

1If the group G is not connected than the possibility to squeeze by an isotopy is stronger than by a

global contactomorphism. All squeezing and non-squeezing results in this paper are proven in the strongest

sense, ie, squeezing is always done by a contact isotopy while in our non-squeezing results we prove

non-existence of the corresponding global contactomorphism.

Geometry & Topology, Volume 10 (2006)

Page 57

Geometry of contact transformations and domains 1691

This implies, in particular, that for every pair of orbits

C;

� 2 P˙ with CZ.

C/�

CZ.

�/D 0 we have

dimM.Con; I;

C;

�/D 0:

This enables us to define a monotonicity morphism

monW C .a;b/.�!HC;JC/! C .a;b/.

�!

H�;J�/

by the formula

(57) mon.

C/D

X

2P.a;b/.RC/

CZ.

C/�CZ.

�/D0

�.

C;

�/

� ;

where �.

C;

�/ is the mod 2 number of elements of the finite set M.Con; I;

C;

�/.

This morphism respects the gradings of the complexes. The next standard fact from

the Floer theory is as follows.

Theorem 4.29 Under the above assumptions, for any a< b<min.C�; CC/ and for all

.e; i/ 2 I.HC/ formula (57) defines a homomorphism of generalized Floer complexes

monW

�

C

.a;b/

e;i

.

�!

HC;JC/; dC

�

!

�

C

.a;b/

�.e;i/

.

�!

H�;J�/; d�

�

;

so that we have

d� ımonDmon ı dC:

In particular, mon defines a homology homomorphism

monW GFH.a;b/

e;i

.

�!

H;JC/! GFH.a;b/�.e;i/.

�!

H;J�/ :

The homomorphism mon constructed above a priori depends on the following data:

� a concordance ConD Con.H�;HC/D .W; .…//;

� an almost complex structure I adjusted to Con which respects almost complex

structures J˙ adjusted to

�!

H˙ .

Standard arguments of Floer theory show that the homomorphism mon does not change

under the following operations:

� replacing the pair .Con; I/ by a pair .Con0; I 0/ where Con0 is a concordance

equivalent to Con and I 0 is the corresponding (see Lemma 4.26) almost complex

structure adjusted to Con0 ;

� a homotopy of the concordance structure .…/ on W through the concordance

structures;

Geometry & Topology, Volume 10 (2006)

Page 58

1692 Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich

� a homotopy of the adjusted almost complex structure I on W through almost

complex structures adjusted to Con and respecting J˙ .

Consider now three framed Hamiltonian structures

�!

H1;

�!

H2 and

�!

H3 equipped with

adjusted almost complex structures J1;J2 and J3 respectively. Assume that we are

given directed concordances

ConiC1;i D Con.Hi ;HiC1/ ; i D 1; 2;

which induce monotonicity morphisms

moniC1;i W GFH.�!H iC1;JiC1/! GFH.

�!

H i ;Ji/ :

A standard argument of Floer theory shows that the gluing Con32 ˘Con21 of concor-

dances corresponds to the composition mon32 ımon21 of the monotonicity morphisms.

The next result is crucial for our proof of non-squeezing results in contact geometry. It

will enable us to reduce calculations in contact homology to more traditional calculations

in Floer homology.

Proposition 4.30 For a regular Hamiltonian structure the generalized Floer homology

does not depend on the choice of a framing and of an adjusted almost complex structure.

We start with two auxiliary lemmas whose proof is elementary and is left to the reader.

Lemma 4.31 Let Aı

i

, where ı 2 f�; 0;Cg and i 2 f0; 1g be a collection of six

linear spaces. Suppose that we are given eight morphisms between them such that the

following diagram commutes:

(58) AC

0

//

��@

@@

@@

@@

A0

1

//

��?

??

??

??

?

A�

0

A

C

1

//

??~~~~~~~

A0

0

//

??��������

A�

1

Assume that the diagonal arrows are isomorphisms. Then all the arrows are isomor-

phisms.

Lemma 4.32 Let E be a topological linear space and let ƒ � E be a convex cone.

Given any �0; �1 2 E with �1 � �0 2 Interior.ƒ/ and any �0; �1 2ƒ, there exists a

smooth path �s , s 2 Œ0; 1�, connecting �0 with �1 such that its derivative P�s satisfies

the following conditions:

(i) P�s D �0 for s near 0 and P�s D �1 for s near 1;

Geometry & Topology, Volume 10 (2006)

Page 113

Geometry of contact transformations and domains 1747

[39] G I Olshanskii, Invariant orderings in simple Lie groups. Solution of a problem of E.

B. Vinberg, Funktsional. Anal. i Prilozhen. 16 (1982) 80–81 (Russian) MR684139

English translation: Functional Anal. Appl. 16 (1982) 311–313

[40] J Palis, Jr, W de Melo, Geometric theory of dynamical systems, Springer, New York

(1982) MR669541An introduction, Translated from the Portuguese by A K Manning

[41] J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827–844

MR1241874

[42] W Rudin, Function theory in the unit ball of Cn , Grundlehren series 241, Springer,

New York (1980) MR601594

[43] D Salamon, J Weber, Floer homology and the heat flow, Geom. Funct. Anal. (GAFA).

to appear arXiv:math.SG/0304383

[44] I Ustilovsky, Infinitely many contact structures on S4mC1 , Internat. Math. Res. Notices

(1999) 781–791 MR1704176

[45] C Viterbo, Functors and computations in Floer homology with applications, Part II,

preprint (1996)

[46] C Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math.

Soc. 13 (2000) 411–431 MR1750956

[47] J Weber, Noncontractible periodic orbits in cotangent bundles and Floer homology,

Duke Math. J. 133 (2006) 527–568 MR2228462

[48] M-L Yau, Cylindrical contact homology of subcritical Stein-fillable contact manifolds,

Geom. Topol. 8 (2004) 1243–1280 MR2087083

Department of Mathematics, Stanford University

Stanford, CA 94305-2125, USA

Departamento de Matemática, Instituto Superior Técnico

Av Roviso Pais, 1049-001 Lisboa, Portugal

School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact

Sciences, Tel Aviv University, 69978 Tel Aviv, Israel

[email protected], [email protected],

[email protected]

Proposed: Eleny Ionel Received: 12 February 2006

Seconded: Tomasz Mrowka, Peter Ozsváth Revised: 30 September 2006

Geometry & Topology, Volume 10 (2006)