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Essential Manifolds with Extra Structures

Permanent Link: http://ufdc.ufl.edu/UFE0045303/00001

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Title: Essential Manifolds with Extra Structures
Physical Description: 1 online resource (67 p.)
Language: english
Creator: Kutsak, Sergii M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

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Subjects / Keywords: algebra -- algebraic -- contact -- essential -- fundamental -- group -- hard -- invariant -- lefschetz -- lie -- manifold -- nilmanifold -- property -- structure -- symplectic
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
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Electronic Thesis or Dissertation

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Abstract: We consider classes of algebraic manifoldsA, of symplectic manifolds S, of symplectic manifolds with the hard Lefschetzproperty HS and the class of cohomologically symplectic manifolds CS. For everyclass of manifolds C we denote by EC(pi,n) a subclass of n-dimensional rationally essential manifolds with fundamental group pi. We prove that for allthe above classes with symplectically aspherical form the condition EC(pi,2n)?Ø implies that EC(pi,2n-2)?Ø for every n>2. Also weprove that all the inclusions EA subset EHS subset ES subset ECS are proper. We give the list of all 7-dimensional nilpotent real Lie algebra that admits acontact structure. Based on this list, we describe all 7-dimensional nilmanifolds that admit an invariant contact structure.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sergii M Kutsak.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Dranishnikov, Alexander Nikolae.
Local: Co-adviser: Rudyak, Yuli B.

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045303:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045303/00001

Material Information

Title: Essential Manifolds with Extra Structures
Physical Description: 1 online resource (67 p.)
Language: english
Creator: Kutsak, Sergii M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: algebra -- algebraic -- contact -- essential -- fundamental -- group -- hard -- invariant -- lefschetz -- lie -- manifold -- nilmanifold -- property -- structure -- symplectic
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We consider classes of algebraic manifoldsA, of symplectic manifolds S, of symplectic manifolds with the hard Lefschetzproperty HS and the class of cohomologically symplectic manifolds CS. For everyclass of manifolds C we denote by EC(pi,n) a subclass of n-dimensional rationally essential manifolds with fundamental group pi. We prove that for allthe above classes with symplectically aspherical form the condition EC(pi,2n)?Ø implies that EC(pi,2n-2)?Ø for every n>2. Also weprove that all the inclusions EA subset EHS subset ES subset ECS are proper. We give the list of all 7-dimensional nilpotent real Lie algebra that admits acontact structure. Based on this list, we describe all 7-dimensional nilmanifolds that admit an invariant contact structure.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sergii M Kutsak.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Dranishnikov, Alexander Nikolae.
Local: Co-adviser: Rudyak, Yuli B.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045303:00001


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ESSENTIALMANIFOLDSWITHEXTRASTRUCTURES By SERGIIKUTSAK ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c r 2013SergiiKutsak 2

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Idedicatethisworktomyfamilyandmyteachers 3

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ACKNOWLEDGMENTS FirstIwanttothankmyadvisorsDr.AlexanderDranishnikov andDr.YuliRudyak fortheirsupportandguidanceduringmystudiesingraduate school.Itwaspleasureto workunderthesupervisionofsuchknowledgeablemathemati cians.Theiradvisesand encouragementmademyresearchfruitfulandenjoyable.Iam gratefultoDr.Alexander DranishnikovfororganizingsuchagreatTopologyandDynam icalSystemsseminarand givingmeanopportunitytoparticipateinit. AlsoIthanktotheDepartmentofMathematicsattheUniversi tyofFloridaforgiving meanopportunitytocarryoutresearch. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 ABSTRACT .........................................6 CHAPTER 1INTRODUCTION ...................................7 2CLASSESOFESSENTIALMANIFOLDS .....................12 2.1Preliminaries ..................................12 2.2SymplecticManifolds ..............................15 2.3SymplecticEssentialManifolds ........................23 3INVARIANTCONTACTSTRUCTURES ......................34 3.1Nilmanifolds ...................................34 3.2ClassicationofNilpotentRealLieAlgebras .................39 3.3Preliminaries:MinimalModels ........................46 3.4ContactStructuresonDecomposableNilpotentLieAlgeb ras .......47 3.5ContactStructuresonIndecomposableNilpotentLieAlg ebras ......56 3.6TheCaseofDim < 7 ..............................63 REFERENCES .......................................64 BIOGRAPHICALSKETCH ................................67 5

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ESSENTIALMANIFOLDSWITHEXTRASTRUCTURES By SergiiKutsak May2013 Chair:AlexanderDranishnikovCochair:YuliRudyakMajor:Mathematics Weconsiderclassesofalgebraicmanifolds A ,ofsymplecticmanifolds S ,of symplecticmanifoldswiththehardLefschetzproperty HS andtheclassofcohomologically symplecticmanifolds CS .Foreveryclassofmanifolds C wedenoteby EC ( n ) a subclassof n -dimensionalrationallyessentialmanifoldswithfundame ntalgroup .We provethatforalltheaboveclasseswithsymplecticallyasp hericalformthecondition EC ( ,2 n ) 6 = ; impliesthat EC ( ,2 n 2) 6 = ; forevery n > 2 .Alsoweprovethatallthe inclusions EAEHSESECS areproper. Wegivethelistofall 7 -dimensionalnilpotentrealLiealgebrasthatadmitaconta ct structure.Basedonthislist,wedescribeall 7 -dimensionalnilmanifoldsthatadmitan invariantcontactstructure. 6

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CHAPTER1 INTRODUCTION RecallthatanEilenberg-Maclanespace X = K ( G n ) isapathconnected CW-complexallofwhosehomotopygroupsvanish,exceptposs iblyfordimension n inwhich n ( X ) G .Forthecase n =1 thecondition k ( X ) 0 forall k > 1 can bereplacedbytheconditionthat X havecontractibleuniversalcover.Wecanbuild aCW-complex X = K ( G n ) forany n and G ,assuming G isabelianif n > 1 ,inthe followingway.First,wecanconstructan ( n 1) connectedCW-complexofdimension n +1 suchthat n ( X ) G .Thenwecanattachhigherdimensionalcellsto X tomake k ( X ) trivialfor k > n withoutaffecting n ( X ) orthelowerhomotopygroups. Let M beaclosed,connected,orientablemanifoldofdimension n andlet bethe fundamentalgroupof M .Recallthatamap f : M K ( ,1) iscalledaclassifyingmap for M if f inducesanisomorphism f : 1 ( M x 0 ) 1 ( K ( ,1), f ( x 0 )) forall x 0 2 M .Itis well-knownthataclassifyingmapexistsandisuniqueuptoh omotopy. Notethatthediscretegroup actstransitivelyontheuniversalcoverof K ( ,1) Sincetheuniversalcoverof K ( ,1) iscontractiblethenitfollowsthattheEilenberg-MacLane spaceistheclassifyingspace B .Hencetheuniversalcover e M M of M is isomorphictothepullbackoftheuniversalcover E B undersomemap g : M K ( ,1) .ItcanbeprovedbyusingtheBorelconstructionthat g induces anisomorphismonthefundamentalgroups.Hence, g isaclassifyingmap. Gromovcalledaclosedmanifold M inessentialifthereexistsaclassifyingmap f : M K ( ,1) tothe ( n 1) -skeletonof K ( ,1) .Otherwise M iscalledessential, [ 20 ].Gromovnoticedthatmanifoldswithpositivescalarcurva turetendtobeinessential. Heintroducedseveralclassesofessentialmanifolds(hype rspherical,hypereuclidean, enlargeable,[ 21 ])forwhichhejointlywithLawsonprovedthatmanifoldsoft hose classescannotcarryametricwithpositivescalarcurvatur e,[ 24 ]. 7

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Fororientablemanifoldsitisconvenienttouseanotherde nitionofessential manifoldwhichisequivalentthedenitionstatedabove.Th efollowingProposition 1.1 is foundinreference[ 13 ]. Denition1. Let f : M n K n beamapofaclosedorientablemanifold M toan n -dimensionalCWspace K ,andlet e beanopen n -cellof K .Considerthemap h = h ( f e ): M f K K = ( Ke ) = S n .Assumethat M and e areoriented.Then weput deg e f=deg(h) and deg e f iscalledthedegreeof f at e Itisclearthatif f f 0 : M K arehomotopicthen deg e f=deg e f 0 Denition2. Amap f : X Y ofpathconnectedspacesiscalled -surjectiveif f : 1 ( X ) 1 ( Y ) isanepimorphism. Lemma1. Let f : M n K n beamapofaclosedorientablemanifold M toan n -dimensionalCWspace K with n > 2 .Let e 1 ,..., e q beopen n -cellsin K .If f is surjectiveand deg e j f =0 for j =1,..., q ,then f ishomotopictoamap g : M K such that g ( M ) T e j = ; for j =1,..., q Lemma2. Let f : M n K n > 2 bea -surjectivemapofaclosedorientablemanifold M toanarbitraryCWspace.If f : H n ( M ) H n ( K ) isthezerohomomorphismthen f canbedeformedintothe ( n 1) -skeletonof K Proposition1.1. Anorientable n -manifold M isessentialifandonlyifthehomomorphism f : H n ( M ) H n ( K ( ,1)) inducedbytheclassifyingmap f isnontrivial. Equivalently,iftheimageofthefundamentalclass [ M ] 2 H n ( M ) under f isnontrivialin the n thintegralhomologygroup H n ( K ( ,1)) oftheEilenberg-MacLanespace K ( ,1) Example1.1. Amanifold M iscalledasphericalifallhigherhomotopygroups k k 2 aretrivial.Forinstance,torus T n ,acompactorientablesurface M g ofgenus g are asphericalmanifolds.Itisclearthateveryorientableasp hericalmanifoldisessential, becausethecorrespondingEilenberg-Maclanespaceisthem anifolditselfanda classifyingmapishomotopictotheidentitymapandtherefo reinducesnon-trivial 8

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homomorphismonthe n thhomologygroups,where n istherealdimensionofthe manifold M .Furthermore,theproductofessentialmanifoldsisanesse ntialmanifold. Example1.2. Ifamanifold M ofdimension n > 0 issimplyconnectedthenitis inessential,becausethecorrespondingEilenberg-MacLan espaceisjustasinglepoint andaclassifyingmap f isconstant.Hence f inducestrivialhomomorphismonthe n th homologygroups.Example1.3. Therealprojectivespace R P 2 n +1 isanessentialmanifold,because 1 ( R P 2 n +1 ) = 1 ( R P 1 ) = Z 2 andtherefore R P 1 isanEilenberg-Maclanespace, becauseitsuniversalcover S 1 iscontractible.Now,theinclusion i : R P n R P 1 isa classifyingmap.Considerthehomologyexactsequence H 2 n +1 ( R P 2 n +1 ) i H 2 n +1 ( R P 1 ) @ H 2 n +1 ( R P 1 R P 2 n +1 ) ... (1–1) Since H 2 n +1 ( R P 1 R P 2 n +1 ) = H 2 n +1 ( R P 1 = R P 2 n +1 ) = 0 then i issurjectiveandtherefore non-trivial,because H 2 n +1 ( R P 1 ; Z ) 6 =0 Denition3. Let M beaclosed,connected,orientablemanifoldofdimension n and let bethefundamentalgroupof M .Wesaythatmanifold M isrationallyessentialifa classifyingmap f : M K ( ,1) inducesnontrivialhomomorphism f : H n ( M ; Q ) H n ( K ( ,1); Q ) Clearly,everyrationallyessentialmanifoldisessential butnotviseversa: R P 2 n +1 is notrationallyessential. Clearly,if H n ( K ( ,1))=0 thentherearenoessential(andhencerationally essential) n -manifoldswiththefundamentalgroup .Theconversealsoholds:if is anitelypresentedgroupand H n ( ; Q ) 6 =0 thenthereexistsarationallyessential n -manifoldwiththefundamentalgroup ,seeTheorem 2.2 below. BrunnbauerandHankegaveacharacterizationofGromovtype classesofrationally essentialmanifoldswithgivenfundamentalgroupintermso fgrouphomology,[ 4 ].We considersimilarproblemforcertainsymplectictypeclass es.Givenaclassofmanifolds 9

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C wedenoteby EC thesubclassthatconsistsofrationallyessentialmanifol ds.Herewe considerthefollowingclasses: AHSSCS (1–2) where A istheclassofalgebraicmanifolds, S istheclassofsymplecticmanifolds, HS istheclassofsymplecticmanifoldswiththehardLefschetz property,and CS istheclass ofcohomologicallysymplecticmanifolds(seesections 2.1 and 2.3 below).Itisknown thatalltheaboveinclusionsofclassesareproper,[ 6 11 16 43 ].Wewillshowthatthe inclusionsoftheessentialcounterpartsarealsoproper. Foreveryclassofmanifolds C wedenoteby C ( n ) asubclassof n -dimensional manifoldswithfundamentalgroup .Westudythefollowingquestion. MAINPROBLEM.Forwhichvalues and n ,is EC ( n ) non-empty? Inparticular,weaddressthefollowingconjecturepropose dbyDranishnikovand Rudyak: CONJECTURE.Fortherstthreeaboveclassesfor n > 2 thecondition EC ( ,2 n ) 6 = ; impliesthat EC ( ,2 n 2) 6 = ; Weproveforalltheaboveclassesaweakerversionoftheconj ecturethatdeals withsymplecticallyasphericalmanifolds,seeSection3fo rthedenition.Notethatevery complexprojectivealgebraicmanifold V issymplectic:thecorrespondingsymplectic formisgivenbytheK ¨ ahlerform,[ 23 ,p.109].Inparticular,weareabletospeakabout symplecticallyasphericalalgebraicmanifolds. Oneimportantclassofessentialmanifoldsistheclassofni lmanifolds.Every nilmanifoldisanEilenberg-MacLanespace K ( ,1) forsomenitelypresented nilpotenttorsionfreegroup ,[ 29 ].Let M = N = beanilmanifoldofdimension n withfundamentalgroup .Thenaclassifyingmapishomotopictotheidentity map i : M K ( ,1) .Thereforethemap i inducesnontrivialhomomorphism 10

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i : H n ( M ) H n ( K ( ,1)) .InChapter 3 wedescribeall 7 -dimensionalnilmanifolds whichadmitaninvariantcontactstructure. 11

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CHAPTER2 CLASSESOFESSENTIALMANIFOLDS 2.1Preliminaries Theorem2.1 (Whitehead) Let X and Y bepath-connectedpointedspacesandlet f :( X x 0 ) ( Y y 0 ) beamap.Ifthereis n 1 suchthat f ] : q ( X x 0 ) q ( Y y 0 ) (2–1) isanisomorphismfor q < n andanepimorphismfor q = n ,then f : H q ( X x 0 ) H q ( Y y 0 ) (2–2) isanisomorphismfor q < n andanepimorphismfor q = n .Conversely,if X and Y are simplyconnectedand f isanisomorphismfor q < n andanepimorphismfor q = n then f ] isanisomorphismfor q < n andanepimorphismfor q = n Nowwedescribesurgeryonmanifolds.Considertwosmoothma nifoldswith boundary X and Y .Thentheboundaryoftheproductmanifold X Y is @ ( X Y )=( @ X Y ) [ ( X @ Y ). (2–3) Notethatthespace S p S q 1 canbeunderstoodeitherastheboundaryof D p +1 S q 1 orastheboundaryof S p D q @ ( S p D q )= S p S q 1 = @ ( D p +1 S q 1 ). (2–4) Let M beasmoothclosedmanifoldofdimension n = p + q If : S p D q M isan embeddingthenwemaydeneanother n -dimensionalmanifold M 0 tobe M 0 =( M int(im )) [ j S p S q 1 (D p+1 S q 1 ). (2–5) Onesaysthatthemanifold M 0 isproducedbyasurgerycuttingout S p D q andgluingin D p +1 S q 1 orbya p -surgery. 12

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Surgeryiscloselyrelatedtohandleattaching.Ifwearegiv enan ( n +1) -manifold withboundary ( N @ N ) andanembedding : S p D q @ N ,where n = p + q ,wemay deneanother ( n +1) -manifoldwithboundary N 0 by N 0 = N [ ( D p +1 D q ). (2–6) Themanifold N 0 isobtainedbyattachinga p +1 -handle,with @ N 0 obtainedfrom @ N bya p -surgery @ N 0 =( @ N imint(im )) [ j S p S q 1 (D p+1 S q 1 ). (2–7) Noteifweperformasurgeryonasmoothmanifold M wenotonlyobtainanewmanifold M 0 butalsoabordism W between M and M 0 .Thetraceofthesurgeryisthebordism ( W ; M M 0 ) with W =( M I ) [ S p D q f 1 g ( D p +1 D q ) (2–8) the ( n +1) -dimensionalmanifoldwithboundary @ W = M [ M 0 obtainedfromtheproduct M I byattachinga ( p +1) -handle D p +1 D q Notethatifamanifold M 0 isobtainedfromamanifold M ofdimension n = p + q by a p -surgerythenthemanifold M canbeobtainedfrom M 0 bya ( q 1) -surgery,thetrace ofwhichcoincideswiththetraceoftheoriginalsurgery,up toorientation. Example2.1. Considerasphere S 2 .Ifwecutouttwodisks S 0 D 2 andgluebackthe cylinder S 1 D 1 thenweobtainthetorus S 1 S 1 ,providingthatthegluingmapshave thesameorientationontheboundarycircles,otherwiseweo btaintheKleinbottle. Thefollowingfactisknown(seeforexample[ 4 ],[ 12 ]).Sincethereisnodetailed proofofitinprint,wegiveacompleteproofhere.Theorem2.2. Foreverynitelypresentedgroup andeveryinteger n if H n ( ; Q ) 6 =0 thenforeverynontrivialelement z 2 H n ( ; Q ) thereexistsaclosed,connected, orientable n -manifold M ,aninteger k 6 =0 andamap f : M K ( ,1) suchthat f ([ M ])= kz and f : 1 ( M ) 1 ( K ( ,1)) isagroupisomorphism. 13

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Proof. Let beanitelypresentedgroupandlet n beanintegersuchthat H n ( ; Q ) 6 =0 Takeanynontrivialelement z in H n ( ; Q ) .BecauseofatheoremofThom,thereexist aclosed n -manifold N ,aninteger k 6 =0 andamap g : N K ( ,1) suchthat g ([ N ])= kz ,seee.g.[ 35 ,TheoremIV.7.36].Supposethat g : 1 ( N ) 1 ( K ( ,1)) is notsurjective.Let :[0,1] K ( ,1) bealoopsuchthat [ ] 2 1 ( K ( ,1)) n im ( g ) and (0)= (1)= y 0 .Withoutlossofgeneralitywecanassumethat y 0 2 im(g)sincethe fundamentalgroupsof K ( ,1) basedatdifferentpointsareisomorphicbecause K ( ,1) ispathconnected.Take x 0 2 N suchthat f ( x 0 )= y 0 .Considerchart ( U ) on N such that ( U )= R n and ( x 0 )=0 .Nowdenefunction h : R n R n ingeneralizedspherical coordinatesasfollows h ( r 1 ,..., n 1 )= 8><>: 0 if 0 r 1, ( r 1, 1 ,..., n 1 ) if r > 1. (2–9) Toperformasurgeryonamanifold N weshalldeneanewfunction eg by: e g ( x )= g ( x ) if x = 2 U e g ( x )= g ( 1 h ( x )) if x 2 U .Then eg ishomotopicto g because h is homotopictotheidentitymapon R n .Let D bethepreimageunder oftheunitball in R n centeredat 0 .Nowweperformasurgeryonthemanifold N .Thereexistsan embedding i : S 0 D n N suchthat i ( S 0 D n ) D and x 0 = 2 i ( S 0 D n ) .Formanew manifoldfromtheunionof N I and D 1 D n byattaching S 0 D n toitsimageunder i 1 .Wecanextendmap e g 1 bydening eg on D 1 D n asfollows e g ( t x )= ( t ) forall ( t x ) 2 D 1 D n 1 (2–10) Connectpoint x 0 withpoints (0, c ),(1, c ) in D 1 D n 1 forsome c 2 D n bypaths r 1 ( t ), r 2 ( t ) respectively.Let ( t )=( t c ) 2 D 1 D n forall t 2 [0,1] .Then ( e g 1) ( r 1 r 1 2 )= .Sowecanconstructamanifold e N andamap eg : e N K ( ,1) suchthat e g ([ e N ])= kz and e g inducesanepimorphismonfundamentalgroups.Now wewanttoperformsurgeriesthatannihilatetheelementsth atgeneratethekernelof e g .Notethatsince e N isorientabletheneveryloop r in e N canbehomotopedtoaloop 14

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e r thathastrivialnormalbundlein e N .Clearly,ifaloop e r istrivialthentheloop e r 1 isalsotrivialforeverypath :[0,1] e N suchthat (1)= e r (0) .SinceKer ( e g ) is normallynitelygenerated[ 45 ]thenwecanperformsurgeryon e N nitelymanytimes toconstructamanifold M andamap f : M K ( ,1) thatinducesisomorphism f : 1 ( M ) 1 ( K ( ,1)) andsuchthat f ([ M ])= kz Notethateveryorientedmanifoldofdimension 2 isessential,exceptforthe sphere S 2 .Anoriented3-manifold M isessentialiffthegroup 1 ( M ) isnotfree,[ 22 38 ]. Denition4. Wedeneacohomologyclass v 2 H m ( X ; G ) tobeasphericalif v = f ( a ) foraclassifyingmap f : X K ( 1 ( X ),1) andsome a 2 H m ( K ( 1 ( X ),1); G ) Notethatifaclass v isasphericaland v k 6 =0 then v k isaspherical. Proposition2.1. Let M beclosed,orientablemanifoldofdimension km ,andlet u 2 H m ( M ; Q ) beanasphericalclass.If u k 6 =0 ,then M isrationallyessential. Proof. Let u = f ( a ) foraclassifyingmap f : X K ( 1 ( X ),1) andsome a 2 H m ( K ( ,1); Q ) .Then f ( a k )= u k 6 =0 .Hence f : H km ( K ( ,1); Q ) H km ( M ; Q ) isnontrivial.Sincethecohomologyandhomologywithcoef cientsin Q aredualthen f : H km ( M ; Q ) H km ( K ( ,1); Q ) isalsonontrivial. 2.2SymplecticManifolds Denition5. Asymplecticstructureonasmoothmanifold M isanon-degenerateskewsymmetricclosed2-form 2 n 2 ( M ) .Asymplecticmanifoldisapair ( M ) where M is asmoothmanifoldand isasymplecticstructureon M Thenon-degeneracyconditionmeansthatforall p 2 M wehavethepropertythatif ( v w )=0 forall w 2 T p M then v =0 .Theskew-symmetryconditionmeansthatfor all p 2 M wehave ( v w )= ( w v ) forall v w 2 T p M .Theclosedconditionmeans thattheexteriorderivative d of isidenticallyzero.Sinceeachodd-dimensional skew-symmetricmatrixissingular,weseethat M hasevendimension. 15

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Notethatnoteverysmoothmanifoldadmitsasymplecticstru cture,therearesome topologicalrestrictions.Everysymplectic 2 n -dimensionalmanifold ( M ) isorientable sincethe n -foldwedgeproduct ^ ... ^ nevervanishes.Ifaclosedmanifold M admits asymplecticform thenthesecondcohomologygroup H 2 ( M R ) isnon-trivial,because symplecticformcannotbeexact.Example2.2. Considerthe2-sphere S 2 .Ifwethinkof S 2 astheunitsphere S 2 = f ( x 1 x 2 x 3 ) 2 R 3 j X j x 2 j =1 g (2–11) thentheareaformgivenby x ( )= h x i for 2 T x S 2 isasymplecticformon S 2 .Itisclearthatthespheres S 2 n n 2 cannotadmitasymplecticstructurebecause H 2 ( S 2 n )=0 forall n 2 Example2.3. TheEuclideanspace R 2 n n 1 isasymplecticmanifoldwiththestandard symplecticform 0 = n X j =1 dx j ^ dy j (2–12) Example2.4. [ 33 ]Cotangentbundlesformanimportantclassofsymplecticma nifolds. Thesearethephasespacesofclassicalmechanics,withcoor dinates q p corresponding topositionandmomentum.Acotangentbundle T M ofasmoothmanifold M isthe vectorbundlewhosesectionsare 1 -formson M .Wedescribethesymplecticstructure oncotangentbundles.Let x : U R n bealocalcoordinatecharton M .Wemaythinkof thecoordinates x 1 ,..., x n of x asthereal-valuedfunctionson U .Thentheirdifferentials dx j ( q ): T q M R at q 2 U formabasisofthedualspace T q M .Henceeveryvector v 2 T q M canbewrittenintheform v = n X j =1 y j dx j (2–13) Thecoordinates y j inthisformulaareuniquelydeterninedby q and v anddetermine coordinatefunctions T U R n :( q v ) 7! ( x ( q ), y ( q ), v )). Inthesecoordinatesthe 16

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canonical1-formisgivenby can = n X j =1 y j dx j (2–14) Itcanbeprovedthatthisformulafor can doesnotdependonthechoiceoflocal coordinate x .Then can = d can (2–15) isasymplecticformonthecotangentbundle T M ThesymplecticmanifoldsincontrasttotheRiemannianmani foldshavenolocal invariants,becauseaccordingtotheDarboux'stheoremeve rysymplecticform on M is locallydiffeomorphictothestandardform 0 on R 2 n Theorem2.3. Let ( M ) beasymplecticmanifoldofdimension 2 n .Thenthereexistan opencover f U g of M andcharts : U ( U ) R 2 n suchthat 0 = (2–16) Thisatlashassymplectictransitionmatrices d ( 1 )( x ) 2 Sp (2 n ) (2–17) for x 2 ( U \ U ) .ChartswiththesepropertiesarecalledDarbouxcharts. Denition6. Acomplexstructureonthetangentbundle TM ofa 2 n -dimensionalreal manifold M isatangentbundleautomorphism J : TM TM suchthat J 2 = 1 .An almostcomplexstructureon M isacomplexstructure J onthetangentbundle TM Withsuchastructure J everytangentspace T x M becomesacomplexvectorspacewith multiplicationby i = p 1 correspondingto J C T x M T x M :( s + it v ) 7! sv + tJv (2–18) Anondegenerate 2 -form 2 n 2 ( M ) iscalledcompatiblewith J if x ( Jv Jw )= x ( v w ) (2–19) 17

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forall v w 2 T x M x 2 M and x ( v Jv ) > 0 (2–20) forallnonzero v 2 T x M andall x 2 M .If J isacompatiblealmostcomplexstructureon M then g ( v w )= ( v Jw ) (2–21) denesaRiemannianmetricon M andthetriple ( J g ) iscalledcompatible. Theorem2.4. Everysymplecticmanifold ( M 2 n ) admitsacompatiblealmostcomplex structure,[ 33 ]. Notethatnoteveryeven-dimensionalmanifoldthatadmitsa nalmostcomplex structureissymplectic.Forexample,a 6 -dimensionalsphere S 6 admitsanalmost complexstructurebutisnotsymplectic,because H 2 ( S 6 R ) = 0 Wedenethe 2 n 2 n matrix J 0 asfollows: J 0 = 0B@ 0 I I 0 1CA (2–22) Denition7. Analmostcomplexstructure J ona 2 n -dimensionalmanifold M iscalled integrableifthereexistsanatlas f U g on M suchthatthealmostcomplexstructure isrepresentedbythematrix J 0 inlocalcoordinates,inotherwordsifforallpoints q 2 M wehave d ( q ) J q = J 0 d ( q ): T q ( M ) R 2 n (2–23) Thisimpliesthatthetransitionmatricescommutewith J 0 d ( 1 ( z )) 2 GL ( n ; C ), (2–24) inotherwordsthetransitionmaps 1 areholomorphic.Conversely,ifthetransition mapsareholomorphicthentheendomorphism J q : T q M T q M J q [ ]=[ J 0 ] (2–25) 18

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isawell-denedalmostcomplexstructureon T q M .Herewethinkofthetangentspace T q M asthesetofequivalenceclasses [ ] with q 2 U and 2 R 2 n underthe equivalencerelation [ ] [ ] iff = d ( 1 )( x ) for x = ( q ) .Acomplex structureisanintegrablealmostcomplexstructure. Let X Y : M TM bethevectoreldsonasmoothmanifold M ,[ 33 ].The Nijenhuistensor N J isdenedby N J [ X Y ]=[ JX JY ] J [ JX Y ] J [ X JY ] [ X Y ]. (2–26) Theorem2.5. Analmostcomplexstructure J isintegrableifandonlyif N J =0. Denition8. AK ¨ ahlermanifoldisasymplecticmanifold ( M ) withanintegrable almostcomplexstructure J Example2.5. EveryRiemannsurfaceisaK ¨ ahlermanifold. Example2.6. TheK ¨ ahlermanifold ( R 2 n J 0 0 ) canbeidentiedwith C n insuchaway that J 0 correspondstomultiplicationby i where i = p 1. Incomplexgeometryon C n it isconvenienttodealwiththecomplex-valuedfunctions z 1 ,..., z n z 1 ,..., z n (2–27) asiftheywereindependentvariables.Thusweintroducethe differentialforms dz j = dx j + idy j d z j = dx j idy j (2–28) Notethatthesearecomplex-valued 1 -formson R 2 n = C n Wedenetwovectoreldson C n asfollows: @ @ z j = 1 2 @ @ x j + i @ @ y j @ @ z j = 1 2 @ @ x j i @ @ y j (2–29) Forcomplex-valueddifferentialformson C n thedifferential d :n k n k +1 canbe convenientlyexpressedintheform d = @ + @ (2–30) 19

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where @ = n X j =1 @ @ z j dz j @ = n X j =1 @ @ z j d z j (2–31) Thecondition d 2 =0 implies @ 2 =0, @ 2 =0, @ @ + @@ =0. (2–32) Thestandardsymplecticformcannowbewrittenas 0 = i 2 @ @ f = i 2 n X j =1 dz j ^ d z j f ( z )= n X j =1 z j z j (2–33) Example2.7. Thecomplexprojectivespace C P n isthespaceofcomplexlinesin C n +1 Thusapointin C P n istheequivalenceclassofanonzerocomplex ( n +1) -vectors [ z ]=[ z 0 : : z n ] undertheequivalencerelation [ z 0 : : z n ] [ z 0 : : z n ] for 6 =0 Thetangentspaceof C P n at [ z ] isthequotient T [ z ] C P n = C n +1 = C z (2–34) andthestandardcomplexstructureismultiplicationby i : J : T [ z ] C P n T [ z ] C P n : 7! i (2–35) Considerthecoordinatepatch U j C P n where z j 6 =0 anddene j : U j C n :[ z 0 : : z n ] 7! z 0 z j ,..., z j 1 z j z j +1 z j ,..., z n z j (2–36) thealmostcomplexstructure J isintegrablebecausethetransitionmaps k 1 j are holomorphic.Considerthe 2 -form 0 = i 2 P n =0 z z 2 n X k =0 X j 6 = k z j z j dz k ^ d z k z j z k dz j ^ d z k (2–37) Ontheopensetwhere z j 6 =0 theform 0 canberepresentedas 0 = i 2 @ @ f j f j ( z )=log P n =0 z z z j z j (2–38) 20

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Hence 0 isclosed.Moreover, 0 iscompatiblewiththecomplexstructure.Theinduced metricon C P n iscalledtheFubini-Studymetric. Denition9. Let C [ x 1 ,..., x n ] betheringofallhomogeneouspolynomialsover C in n variables.Foreachset S ofhomogeneouspolynomialsdenethezero-locusofStobe theset Z ( S )= f x 2 C P n : f ( x )=0 forall f 2 S g (2–39) Asubset V of C P n iscalledaprojectivealgebraicsetif V = Z ( S ) forsomeS.A projectivevarietyisanirreduciblealgebraicset.Analge braicvarietywithoutsingularities iscalledanalgebraicmanifold. Forexample,accordingtotheChow'stheorem[ 23 ]everycomplexsubmanifoldof thecomplexprojectivespace C P n isanalgebraicmanifold. Denition10. Asymplecticmanifold ( M ) issymplecticallyasphericalif Z S 2 f =0 (2–40) foreverysmoothmap f : S 2 M Clearly,if 2 ( M )=0 thenasymplecticmanifold ( M ) issymplecticallyaspherical. Howevertherearesymplecticallyasphericalmanifoldswit hnontrivial 2 ,[ 17 26 ]. Remark2.1. Thecohomologyclass [ ] inasymplecticallyasphericalmanifold ( M ) is aspherical.ItfollowsfromclassicalresultsofHopf,[ 25 ](seealso[ 9 ,Theorem8.17],[ 2 Theorem5.2]). InviewofthisremarkandProposition 2.1 ,wehavethefollowingcorollary. Corollary1. Everyclosedsymplecticallyasphericalmanifoldisration allyessential. Toproceed,weneedthefollowingtheorems,seee.g[ 28 ,p.41]. Theorem2.6 (LefschetzHyperplaneTheorem) Let V beacomplexprojectivealgebraic varietyofcomplexdimensionkwhichliesinthecomplexproj ectivespace C P n ,andlet P beahyperplanein C P n whichcontainsthesingularpoints ( ifany ) of V .Thenthe relativehomotopygroups r ( V V \ P ) areequaltozeroforall r < k 21

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Notethat V \ P isamanifold(i.e.non-singularvariety)if V is. Theorem2.7 (Donaldson[ 10 ]) Let L V beacomplexlinebundleoveracompact symplecticmanifold ( V ) withcompatiblealmost-complexstructure,andwiththers t Chernclass c 1 ( L )= 2 # .Thenthereisaconstant C suchthatforalllarge k thereisa section s of L n k with j @ s j < C p k j @ s j (2–41) onthezerosetof s Theorem2.8 (Donaldson[ 10 ]) Let W k bethezero-setofasection s of L n k V satisfyingtheconditionsofTheorem 2.7 .When k issufcientlylargetheinclusion i : W k V inducesanisomorphismonhomotopygroups p for p n 2 anda surjectionon n 1 InviewofTheorem 2.7 andTheorem 2.8 weobtainthefollowingcorollary Corollary2. Let ( M ) beaclosedsymplecticmanifoldofdimension 2 n suchthatthe cohomologyclass [ ] isintegral.Thenthereexistsasymplecticsubmanifold V of M of codimension2suchthatinclusion i : V M inducesanisomorphismonhomotopy groups p for p n 2 andasurjectionon n 1 .Furthermore,thehomologyclass [ V ] in M isthePoincar edualtoaclass r [ ] forsomeinteger r Proof. TheprooffollowsfromTheorem 2.7 andTheorem 2.8 with normalizedsuch that c 1 ( L )=[ ] .Let V bethezero-setofasection s of L n k M asinTheorem 2.7 Theninequality( 2–41 )guaranteestheexistenceofsymplecticstructureon V .So V is asymplecticsubmanifoldof M ofcodimension2.Thehomologyclassof V isPoincar e dualtotherstChernclassof L n k uptoamultiplicativeconstant r .Finally,accordingto Theorem 2.8 theinclusion i : V M inducesanisomorphismonhomotopygroups p for p n 2 andasurjectionon n 1 22

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2.3SymplecticEssentialManifolds Theorem2.9. Assumethat M isacomplexprojectivealgebraicmanifoldof ( real ) dimension 2 k whichliesinthecomplexprojectivespace C P N .Supposealsothat M issymplecticallyaspherical.Thenforeveryinteger m with 2 m k thereexists arationallyessentialalgebraicmanifold V ofdimension 2 m withfundamentalgroup isomorphicto 1 ( M ) Proof. Thecase m = k istheCorollary 1 .Byinduction,itsufcestoprovethetheorem for m = k 1 .Indeed,assumethat dim M =2 k > 4 andlet V = M \ C P N 1 .Ifweprove that V isarationallyessentialcomplexalgebraicmanifoldwith dim V =2 k 2 > 4 and thefundamentalgroup = 1 ( M ) ,weapplythepreviousargumentfor V insteadof M BecauseoftheTheorem 2.6 r ( M V )=0 for r < k 1 .Fromtheexactnessofthe homotopysequence 2 ( M V ) 1 ( V ) 1 ( M ) 1 ( M V ) (2–42) itfollowsthat 1 ( M ) 1 ( V ) since 2 ( M V ) 1 ( M V ) 0. (2–43) Hence V isacomplexalgebraicmanifoldwithfundamentalgroupisom orphicto ,and dim V =dim M 2 .Itremainstoprovethat V isrationallyessential.Butthisfollowsfrom Corollary 1 becausetheinducedK ¨ ahlerformon V isaspherical. Theorem2.10. Let ( M ) beaclosedsymplecticallyasphericalmanifoldofdimensio n 2 n > 2 withfundamentalgroup .Thenforevery k suchthat 2 k n there existsasymplecticallyasphericalmanifold V ofdimension 2 k withfundamentalgroup isomorphicto Proof. Weprovethetheorembyinduction.SimilarlytotheproofofT heorem 2.9 ,it sufcestoprovethecase k = n 1 .Withoutlossofgenerality,wecanassumethat thecohomologyclass [ ] isintegral(see[ 26 ,Prop.1.5]).Let M beamanifoldasin 23

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Corollary 2 .Then,for n > 2 ,theinclusion i : V M inducesanisomorphismonthe fundamentalgroups 1 ( V ) 1 ( M ) .Now, V isasymplecticmanifoldwithsymplectic structure i inducedfrom M .Itisclearthat Z S 2 g i =0 (2–44) foreverymap g : S 2 V .Thus ( V i ) isasymplecticallyasphericalmanifoldof dimension 2 n 2 with 1 ( V )= Denition11. Asymplecticmanifold ( M 2 n ) hasthehardLefschetzproperty(HLP)if themap L k[ ] : H n k DR ( M 2 n ) H n + k DR ( M 2 n ), L k[ ] ([ x ])=[ k ^ x ] (2–45) isanisomorphismforall k =0,..., n Forexample,theHardLefschetzTheoremsaysthateveryK ¨ ahlermanifoldhasHLP, see[ 23 ,p.122]. Theorem2.11. Let ( M ) beasymplecticallyasphericalmanifoldofdimension 2 n > 2 withfundamentalgroup andhavingHLP.Thenforevery m suchthat 2 m n there existsasymplecticallyasphericalmanifold ( V ) ofdimension 2 m withfundamental groupisomorphicto andhavingHLP. Proof. WefollowtheproofofTheorem 2.10 andmustprovethatthemanifold V asin Theorem 2.10 hasHLP. First,weneedtoshowthat L k[ ] : H n 1 k ( V ) H n 1+ k ( V ) isanisomorphismfor all k =0,..., n 1 where isthepullbackof underinclusion i : V M .Weneedto considerseparatelythecasewhen k =0 .Soxany k suchthat 0 < k n 1 .Consider thefollowingcommutativediagram H n 1 k ( M ) L k[ ] H n 1+ k ( M ) ^! H n +1+ k ( M ) i 1 ??y ??y i 2 H n 1 k ( V ) L k[ ] H n 1+ k ( V ) (2–46) 24

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where L k[ ] isamonomorphismbecause L k +1 [ ] isanisomorphism.Since H n 1 k ( V ) and H n 1+ k ( V ) havethesamedimension,itsufcestoshowthat L k[ ] isamonomorphism. ItfollowsfromCorollary 2 ,andWhiteheadtheorem(see[ 39 ,p.399])that i 1 isan isomorphism.Henceitsufcestoshowthat i 2 isamonomorphismontheim ( L k[ ] ) Assumethat 2 H n 1 k ( M ) isnontrivialand i 2 ( ^! k )=0 .Then 0 6 = r ([ M ] ( ^! k +1 ))= r ([ M ] ( ^! k )) _! = r ([ M ] _! ) ( ^! k )= i ([ V ]) ( ^! k ) = i ([ V ] i 2 ( ^! k ))=0. (2–47) Thisisacontradiction.So L k[ ] isanisomorphismforall k =1,..., n 1 .If k =0 thenit isobviousthat L 0[ ] : H n 1 ( V ) H n 1 ( V ) isanisomorphism.Thus V isasymplectically asphericalmanifoldofdimension 2 n 2 withfundamentalgroup havingtheHLP.Now wecanapplytheaboveprocedureto V ,andtheresultfollowsbyinduction. Denition12 (Lupton-Oprea[ 27 ]) Amanifold M ofdimension 2 n iscohomologically symplectic(or,briey,c-symplectic)ifthereexistsaclo seddifferential2-form on M suchthat [ ] n 6 =0 Notallc-symplecticmanifoldsaresymplectic.Forexample C P 2 # C P 2 isc-symplectic butisnotsymplectic,[ 16 ]. Theorem2.12. Let ( M ) beac-symplecticmanifoldofdimension 2 n > 2 with fundamentalgroup andwithasphericalc-symplecticform.Thenforevery m suchthat 2 m n thereexistsac-symplecticmanifold ( V ) ofdimension 2 m withfundamental groupisomorphicto andwithasphericalc-symplecticform. Proof. Let f : M K ( ,1) beaclassifyingmapfor M .Then = f a forsome a 2 H 2 ( K ( ,1)) .Thereexistsa (2 n 2) -dimensionalsubmanifold N of M suchthat [ N ]= r forsome r 2 Z ,where = PD ([ ])=[ M ] _! .Let i : N M betheinclusion of N into M .Wewanttoshowthat ( i ) n 1 6 =0 .Supposethat ( i ) n 1 =0 .Then 25

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0 6 = r ([ M ] _! n )= r ([ M ] _!_! n 1 )= = i ([ N ]) _! n 1 = i ([ N ] ( i ) n 1 )=0. (2–48) Thisisacontradiction.Hence ( i ) n 1 6 =0 .Byusingsurgerywecanconstructa manifold N 0 andamap i 0 : N 0 M thatinducesanisomorphismonthefundamental groups.Moreover,thereexistamanifold W with @ W = N t N 0 andamap g : W M thatextends i and i 0 .Inotherwords,thesingularmanifolds i : N M and i 0 : N 0 M arebordant: N j W j 0 N 0 & i ??y g i 0 M (2–49) where j and j 0 aretheinclusions.Thus i 0 ([ N 0 ])= i ([ N ]) .Now h ( i 0 ) n 1 ,[ N 0 ] i = h n 1 i 0 ([ N 0 ]) i = = h n 1 i ([ N ] i = h ( i ) n 1 ,[ N ] i6 =0, (2–50) so ( i 0 ) n 1 6 =0 .Thus ( N 0 i 0 ) isac-symplecticmanifoldofdimension 2 n 2 withfundamentalgroupisomorphicto .Clearly, i 0 isanasphericalformbecause i 0 =( f i 0 ) a .Theresultfollowsbyinduction. Recallthataclosed,connected n -dimensionalmanifold M isorientableifandonly ifthe n thintegralhomologygroup H n ( M Z ) isisomorphictotheinnitecyclicgroup Z Ageneratorof H n ( M Z ) iscalledafundamentalclassof M .Notethatthechoiceofa fundamentalclassdeterminestheorientationofamanifold M Theorem2.13 (Poincar eduality) Let M beaclosedorientable n -manifoldwithfundamentalclass [ M ] 2 H n ( M Z ) .Thenthemap D : H k ( M Z ) H n k ( M Z ) denedby D ( )=[ M ] isanisomorphismforall k 26

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Let M N beclosed,connected n -manifolds.Weconstructtheconnectedsumof M and N asfollows.Take M and N tobedisjoint.Let f : S 0 D n M t N beanembedding with f (1 D n ) M and f ( 1 D n ) N .Let W =( M t N ) = (1, x ) ( 1, x ) .The diffeomorphismclassof W isindependentof f .Thenwepretend W isawell-dened manifoldandwrite W = M # N .Wecanalsoview M # N asformedbygluingtogether M intB and N intD byadiffeomorphisms @ D @ B ,where B M and D N are the n -dimensionaldisks.Forexample,anorientablesurfaceofg enus g istheconnected sumof g tori.Itisclearthat M # N isorientableifandonlyif M and N areorientable. Let M N beanorientable n -dimensionalmanifolds.Wecanobtainanewmanifold V from W = M # N byglueinginan n -dimensionaldisk D n sothat V ishomotopy equivalenttothewedgeproduct M W N .Considerthehomologyexactsequence H n +1 ( V W ) H n ( W ) H n ( V ) H n ( V W ) (2–51) Itisclearthat H i ( V W ) H i ( S n ), i 0 .Hence,itfollowsthat H i ( V ) H i ( W ) H i ( M ) H i ( N ) (2–52) forall 1 i n 2 because H i ( V W ) H i +1 ( V W ) 0 Thefollowingdescriptionofasymplecticblow-upcanbefou ndin[ 7 ],[ 33 ].LetM beacompactsymplecticmanifold.Considerasymplecticemb edding i :( M 2 d ) ( X 2 n ) .Thenthecomplexstructureinthetangentbundle TX restrictstothecomplex structureinthetangentbundle TM ,andthereforetothenormalbundle : E M Hence E isacomplexbundleover M andwecanperformitsprojectivization C P k 1 e M M (2–53) where k = n d .Considerthetautologicallinebundle e E over e M :thesubbundleof e M E whosebersaretheelements f ([ v ], v ), 2 C g .Wehavethefollowingcommutative diagram 27

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e E 0 e E q e M ??y ??y ??y p E 0 ( E ) ( M ) (2–54) where E 0 isthecomplementofthezerosectionin E e E 0 isthecomplementofthezero sectionin e E and q aretheprojectionsover e M and E respectively. Itiseasytoseethat E 0 and e E 0 arediffeomorphicvia .Alsoifweconsidera sufcientlysmalldiscsubbundle V in E withitscanonicalsymplecticstructure ,then itissymplectomorphictoaneighborhoodof M X andwemayidentifythem.Let e V = 1 ( V ) .Wecanformthemanifold e X = X V [ @ V e V (2–55) Thenthemap canbeextendedtoamap f : e X X ,beingtheidentityinthe complementof e V .Themanifold e X istheblow-upof X along M and f : e X X isthe projectionoftheblow-uportheblow-downmap.Theorem2.14. [ 31 ]Ifthecodimensionof M isatleast4,thefundamentalgroupsof X andtheblown-upmanifold e X areisomorphic.Further,thereisashortexactsequence 0 H ( X ) H ( e X ) A 0, (2–56) where A isthefreemoduleover H ( M ) withgenerators a ,..., a k 1 Moreover,thereis arepresentative of a withsupportinthetubularneighborhood V suchthat,for small enough,theform e = f ( )+ isasymplecticformin e X Remark2.2. [ 7 ]Thesymplecticstructureintheblow-up e X maydependon andis notdeterminedbytheonein X .Moreover,foreach thesymplecticstructurein e X alsodependsonotherchoicesmadeduringtheconstruction, suchasthealmost complexstructuretaming andtheidenticationofthenormalbundlewiththetubular neighborhood. 28

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Denition13. Thesignature ofacompact,orientedmanifold M n isdenedtobe zeroifthedimensionisnotamultipleof 4 ,andasfollowsfor n =4 k .Chooseabasis a 1 ,..., a r for H 2 k ( M 4 k ; Q ) sothatthesymmetricmatrix [ h a i ^ a j ,[ M ] 4 k i ] (2–57) isdiagonal,where [ M 4 k ] isthefundamentalclassof M 4 k .Then ( M 4 k ) isthenumber ofpositivediagonalentriesminusthenumberofnegativeon es.Inotherwords isthe signatureoftherationalquadraticform a 7!h a ^ a ,[ M ] i WecangiveaninductivedenitionofChernclassesforacomp lex n -planebundle : E B .Notethattheunderlyingrealvectorbundle R isorientable,sowecantalk abouttheEulerclass e ( R ) .Firstweconstructacanonical ( n 1) -planebundle 0 over thedeletedtotalspace E 0 ,where E 0 = E 0 ( ) denotesthesetofallnon-zerovestorsin thetotalspace E .FixanHermitianmetricon .Notethatapointin E 0 isdetermined byaber F andanon-zerovector v in F .Thentheberof 0 over v isdenedtobe theorthogonalcomplementof v inthevectorspace F .Thisisacomplexvectorspace ofdimension n 1 ,andthesevectorspacescanbeconsideredasthebersofa newvectorbundle 0 over E 0 .ConsideranexactGusinsequenceforanorientedreal 2 n -bundle R H i 2 n ( B ; Z ) ^ e H i ( B ; Z ) 0 H i ( E 0 ; Z ) H i 2 n +1 ( B ; Z ) ... (2–58) Thenitfollowsthatfor i < 2 n 1 thegroups H i 2 n ( B ; Z ) and H i 2 n +1 ( B ; Z ) arezero. Hence 0 : H i ( B Z ) H i ( E 0 Z ) isanisomorphismfor i < 2 n 1 TheChernclasses c i ( ) 2 H 2 i ( B ; Z ) aredenedasfollows,byinductiononthe complexdimension n of .ThetopChernclass c n ( ) isequaltotheEulerclass e ( R ). For i < n wedene c i ( )= 1 0 c i ( 0 ). (2–59) 29

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Notethatthisexpressionmakessensebecause 0 : H i ( B Z ) H i ( E 0 Z ) isan isomorphismfor i < n .For i > n theclass c i ( ) isdenedtobezero.Theformalsum c ( )=1+ c 1 ( )+ + c n ( ) iscalledthetotalChernclassof Recallthatifasmoothmanifold M ofarealdimension n iscompactthenthe cohomologygroups H i ( M ; R ) arenitelydimensionalvectorspaces.Thenwedene the i thBettinumber i of M tobeequaltotherealdimensionof H i ( M ; R ) .TheEuler characteristic ( M ) of M isdenedby ( M )= X ( 1) k dimH k ( M ; R )= X ( 1) k i (2–60) Proposition2.2. Thereisanexampleofarationallyessential4-dimensional csymplecticmanifold M whichisnotsymplectic. Proof. Let beanaspherical4-dimensionalhomologysphere(see[ 36 ]).Weconsider theconnectedsum M = C P 2 # C P 2 # andshowthatitdoesnotadmitanalmost complexstructure.AccordingtotheresultofEhresmannand Wu,acompact4-manifold M hasanalmostcomplexstructurewithrstChernclass c 1 2 H 2 ( M Z ) ifandonlyif c 1 reducesmodulo2tothesecondStiefel-Whitneyclass w 2 and c 2 1 ([ M ])=3 +2 (2–61) where istheEulercharactericticof M and isitssignature([ 33 ,p.119]).According tothePoincar eduality i = 4 i ,for i =0,...,4 .Hence,inordertondtheEuler characteristicitsufcestocomputeBettinimbers 0 1 2 .Since M isaconnected manifoldthen 0 =1. Itisnotdifculttond 1 2 because H i ( M R ) = H i ( C P 2 ) H i ( C P 2 ) ,for i =1,2 .Itfollowsthat 1 =0, 2 =2 .ThustheEulercharacteristicof M is equalto 4 .NotethattherstChernclass c 1 ( M ) ofamanifold M canberepresentedas thesum c 1 ( M )= a + b oftwoelements a b 2 H 2 ( M Z ) suchthat a ^ b = b ^ a =0 Thus c 2 1 ([ M ]) isthesumofsquaresoftwointegers.Alsowecanndabasis f a 1 a 2 g of H 2 ( M Z ) suchthatthematrix [ h a i ^ a j ,[ M ] i ] istheidentitymatrix.Sothesignature 30

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( M ) of M isequalto 2 .Itfolowsthat 14 canberepresentedasthesumofsquares oftwointegers.But14cannotberepresentedinsuchform.He nce M doesnotadmit analmostcomplexstructureandthereforeisnotasymplecti cmanifoldbecauseevery symplecticmanifoldadmitsacompatiblealmostcomplexstr ucture.Furthermore, = K ( 1 (),1) ,andthecollapsingmap f : M hasdegree1.Thus M isa rationallyessentialmanifoldsincethehomomorphismindu cedby f onthe4thhomology groups f : H 4 ( M ; Q ) H 4 (; Q ) isnontrivial.Since isahomologysphere,the collapsingmap i : M C P 2 # C P 2 inducestheisomorphism i : H 2 ( C P 2 ; R ) H 2 ( C P 2 ; R ) H 2 ( M ; R ). (2–62) Let f [ 1 ],[ 2 ] g beabasisof H 2 ( C P 2 ; R ) H 2 ( C P 2 ; R ) .Then i ([ 1 ]+[ 2 ]) 2 6 =0 in H 4 ( M ; R ) .Hence M isac-symplecticmanifold. Remark2.3. NotethattheDranishnikov-Rudyakconjectureisnottruefo rc-symplectic manifolds.Considerarationallyessentialc-symplecticm anifold M = C P 4 # C P 4 #( ) withfundamentalgroup 1 ( M ) 1 () 1 () .Since istheEilenbergMacLanespace K ( 1 ( M ),1) and H 6 ( ; Q ) istrivialthentheredoesnotexista rationallyessential6-manifoldwithfundamentalgroupis omorphicto 1 ( M ) Theorem2.15. Alltheinclusionsofclasses EAEHSESECS (2–63) areproper.Proof. Firstweprovethattheinclusion EAEHS isproper.Let H betheHeisenberg manifold.Thentheblow-up M of H H alongatorusisasymplecticmanifoldthat satisesthehardLefschetzpropertyandhasnontrivialtri pleMasseyproduct,[ 6 ]. Since H isanasphericalmanifoldthen H H istheEilenberg-MacLanespace.So M isarationallyessentialmanifoldbecausethereexistsade gree1(classifying)map f : M H H .Notethat M isnotalgebraicsinceithasnon-trivialMasseyproduct, 31

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whileallK ¨ ahler(andthereforealgebraic)manifoldsareformalspace s,[ 11 ],andhence alltheirMasseyproductsaretrivial. Nowweprovethattheinclusion EHSES isproper.ConsidertheKodaira-Thurston manifoldKTobtainedbytakingtheproductoftheHeisenberg manifold H andthecircle S 1 .Itiswell-knownthatKTisasymplecticmanifold.TheKodai ra-Thurstonmanifold isrationallyessentialbecauseitisanilmanifoldanditca nnothavethehardLefschetz propertybecauseasymplecticnilmanifoldofLefschetztyp eisdiffeomorphictoatorus, [ 3 ]. Wehavealreadyshownthattheinclusion ESECS isproper,seeProposition 2.2 above. TheDranishnikov-Rudyakconjecturecannotbereducedtoth easphericalcasein viewofthefollowingProposition2.3. Theblowupofa4-torusatasinglepoint M = T 4 # C P 2 isan algebraicmanifoldwhichdoesnotadmitanasphericalsympl ecticform. Proof. Let beasymplecticformon M .Then Z M 2 6 =0 .Wecanobtainaform 0 on C P 2 thatextendstherestrictionof on C P 2 n D suchthat Z C P 2 0 2 6 =0 where D isa smallenoughdisk.Thenthereexistsamap f : S 2 C P 2 n D with Z S 2 f 0 6 =0 because ifweassumethat Z S 2 f 0 =0 forallmaps f : S 2 C P 2 n D then [ 0 ]=0 in H 2 ( C P 2 ; R ) Therefore [ 0 ] 2 =0 and Z C P 2 0 2 =0 whichcontradictstothechoiceof 0 .Consider f : S 2 C P 2 n D suchthat Z S 2 f 0 6 =0. Since and 0 coincideon C P 2 n D then Z S 2 f 6 =0 .Thus isnotanasphericalsymplecticform. ItisnaturaltoconsidertheclassofK ¨ ahlermanifolds K andaskwhetherthe inclusions EAEKES areproper.Itisknownthatinclusions AKS areproper 32

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[ 44 ],[ 6 ]andmanifold M inTheorem 2.15 showsthatinclusion EKES isalsoproper. Notethat M isnotK ¨ ahlerbecauseitisnotformal. Question2.1. DoesthereexistanessentialK ¨ ahlermanifoldthatisnotalgebraic? Question2.2. Inviewofthetheoremsprovedabovewemayaskwhetherthe Dranishnikov-Rudyakconjectureholdstruefortheclassof K ¨ ahlermanifoldswith asphericalK ¨ ahlerform. 33

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CHAPTER3 INVARIANTCONTACTSTRUCTURES 3.1Nilmanifolds Inthischapterwedescribeall7-dimensionalnilmanifolds admittinganinvariant contactstructure,seeTheorem 3.7 .Toachievethisgoal,weclassifyall7-dimensional realnilpotentLiealgebrasthatadmitacontactstructure, seeCorollary 5 .First,werecall somedenitions.Denition14. ALiealgebra g overaeld F ofcharakteristic 0 isavectorspaceon F withabilinearmapping g g g denoted ( x y ) 7! [ x y ] andcalledthebracketof g and satisfying: ( i )[ x x ]=0, 8 x 2 g (3–1) ( ii )[[ x y ], z ]+[[ y z ], x ]+[[ z x ], y ]=0, 8 x y z 2 g (3–2) Identity ( ii ) iscalledtheJacobiidentity. Example3.1. Everyvectorspace V withthebracket [ x y ]=0 ,forall x and y in V ,isa LiealgebracalledanabelianLiealgebra.Example3.2. Let M n ( F ) bethespaceof n n matriceson F .Themultiplication [ A B ]= AB BA (3–3) satisesconditions ( i ) and ( ii ) .Thenthevectorspace M n ( F ) withthisbracketisaLie algebra,anddenotedby gl ( n F ) Example3.3. Let V bea 2 k +1 -dimensionalvectorspaceand f e 1 ,..., e 2 k +1 g abasisof V .Thebracketsdenedby [ e 2 i e 2 i +1 ]= e 1 for i =1,..., k (3–4) (otherbracketsexceptthoseobtainedbyanticommutativit yare 0 )endows V withthe structureofaLiealgebra,calledtheHeisenbergalgebra. 34

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Denition15. ForaLiealgebra g theuppercentralseriesistheincreasingsequenceof idealsdenedasfollows: C 0 ( g )=0, C i +1 ( g )= f x 2 g j [ x g ] C i ( g ) g (3–5) Inotherwords C i +1 ( g ) istheinverseimageunderthecanonicalmappingof g onto g = C i ( g ) ofthecenterof g = C i ( g ) .TheLiealgebra g iscallednilpotentifthereisaninteger k suchthat C k ( g )= g Theminimalsuch k iscalledtheindexofnilpotency,[ 2 ]. AconnectedLiegroupisnilpotentifandonlyifitsLiealgeb raisnilpotent. Denition16. Anilmanifold M isacompacthomogeneousspaceoftheform M = N = where N isasimplyconnectednilpotentLiegroupand isadiscretecocompact subgroupin N ,[ 43 ]. Forexample,an n -dimensionaltorus T n = R n = Z n isobviouslyanilmanifold.Ifwe considerthegroup U n ( R ) ofuppertriangularmatriceshaving1salongthediagonal U n ( R )= 0BBBBBBB@ 1 a 12 ... a 1 n 01... a 2 n ... ... ... 00...1 1CCCCCCCA (3–6) thenthequotient M = U n ( R ) = U n ( Z ) isanilmanifold,calledtheHeisenbergnilmanifold, where U n ( Z ) U n ( R ) isthesetofmatriceshavingintegralentries. Let N = beanilmanifoldandlet n betheLiealgebraoftheLiegroup N .Itis well-knownthattheexponentialmap exp: n N isaglobaldiffeomorphismandthe quotientmap N N = istheuniversalcoveringmap.Henceeverynilmanifoldisth e Eilenberg-MacLanespace K (,1) .ByMalcev'stheoremadiscretegroup canbe realizedasthefundamentalgroupofanilmanifoldifandonl yifitisanitelypresented nilpotenttorsionfreegroup,[ 29 ]. Let A beanabeliangroupandlet G beagroupwithahomomorphism : G Aut ( A ) .Inothherwords, G actson A through andwedenotetheactionby ( g )( a )= 35

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g a .Denealowercentralseriesof by 0 ( A )= A andfor i 1 i +1 ( A ) isagroup generatedby f g a a j g 2 G a 2 i ( A ) g (3–7) Thenwesaythattheactionof G on A isnilpotentif N ( A )= f 1 g forsome N .Notethat if A isnotabelianthenwemustuseslightlymorecomplicatedexp ressionfor i +1 ( A ) Namely, i +1 = h ( g a ) ba 1 b 1 j g 2 G a 2 i ( A ), b 2 A i (3–8) Itisclearthatif A isanabeliangroupthenthisexpressionreducestotheprevi ousone. Notethatif A = G andtheactionisbyconjugationthen i ( A )= i ( A ) forall i ,where i ( A ) istheusuallowercentralseriestermfor A .Inthiscase,anilpotentactionissimply sayingthatthegroup A isnilpotent. Recallthataspace X iscallednilpotentifthefundamentalgroup 1 ( X ) isnilpotent andtheactionof 1 ( X ) onallhomotopygroups n ( X ), n 1 isnilpotent.The rationalizationofanilpotentspace X isarationalspace X Q with k ( X Q ) = k ( X ) n Q for k 1 andamap X X Q inducingisomorphism k ( X ) n Q k ( X Q ) ,for k 1 Therationalhomotopytypeofanilpotentspace X isthehomotopytypeof X Q .Itisclear thateverynilmanifold M = N = isanilpotentspacesince 1 ( M ) isnilpotentandall higherhomotopygroups n ( M ), n 2 aretrivial.Hence,itfollowsfromthefundamental theoremoftherationalhomotopytheory[ 11 ]thattwonilmanifoldshavethesame rationalhomotopytypeifandonlyifthecorrespondingChev alley-Eilenbergcomplexes areisomorphic.Theorem3.1. [ 29 ]AsimplyconnectednilpotentLiegroup N admitsadiscretecocompactsubgroup ifandonlyifthereexistsabasis f e 1 e 2 ,..., e n g oftheLiealgebra n of N suchthatthestructuralconstants c k ij arisinginbrackets [ e i e j ]= X k c k ij e k (3–9) arerationalnumbersforall i j k 36

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Let g beaLiealgebrawithabasis f X 1 ,..., X n g .Denoteby f x 1 ,..., x n g thebasisfor g dualto f X 1 X n g .Weobtainadifferential d ontheexterioralgebra g bydening itondegree1elementsas dx k ( X i X j )= x k ([ X i X j ]) (3–10) andextendingto g asagradedbilinearderivation.Then [ X i X j ]= X l c l ij X l (3–11) where c l ij arethestructureconstantsof g ,anditfollowsfromdualitythat dx k ( X i X j )= c k ij (3–12) Henceongeneratorsthedifferentialisexpressedas dx k = X i < j c k ij x i x j (3–13) Notethatthecondition d 2 =0 isequivalenttotheJacobiidentityintheLiealgebra.We callthedifferentialgradedalgebra ( g d ) (3–14) theChevalley-EilenbergcomplexoftheLiealgebra g Acontactstructureonamanifold M ofodddimension 2 n +1 isacompletely non-integrable 2 n -dimensionaltangentplanedistribution .Inthecoorientablecase thedistributionmaybedenedbyadifferential1-form as =ker .Recallthatthe Frobeniusintegrabilitytheoremstatesthat isintegrableifandonlyifthesectionsof areclosedundertheLiebracket.Notethatavectoreld X isasectionof ifandonlyif ( X )=0 ,andso isintegrableifandonlyif ([ X Y ])=0 whenever ( X )= ( Y )=0. Inviewoftheidentity d ( X Y )= L X ( ( Y )) L Y ( ( X ))+ ([ X Y ]) (3–15) 37

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thisamountstorequiringthat d tobezerowhenrestrictedtothevectorsin or, equivalently,that ^ d =0 .Thecontactconditionisasfarfromthisaspossible.It requiresthat d restrictstoanondegenerateformon .Notethat d isnondegenerate on ifandonlyif ^ ( d ) n 6 =0, (3–16) i.e.theform ^ ( d ) n isnowherezero.Suchdifferentialform iscalledacontact differentialform.Acontactstructurecanbeviewedasaneq uivalenceclassofcontact differentialforms,wheretwoforms and 0 areequivalentifandonlyif 0 = f where f isanowherezerosmoothfunctionon M .Acontactmanifoldisapair ( M 2 n +1 ) where M isasmoothmanifoldofdimension 2 n +1 and isacontactstructureon M ,[ 14 33 ]. Example3.4. Thebasicexampleisthecontactstructureon R 2 n +1 givenbythe 1 -form 0 = dz X j y j dx j (3–17) whereweusethecoordinates x 1 ,..., x n y 1 ,..., y n z on R 2 n +1 .Thisiswhatisusually takentobethestandardstructureon R 2 n +1 Example3.5. Let = S 2 n +1 betheunitspherein R 2 n +2 = C n +1 .Ateachpoint q 2 thecomplexpart q = T q \ J 0 T q (3–18) ofitstangentspacehascodimension 1 .Thesehyperplanesformacontactstructureon withcontactform 0 = 1 2 n X j =1 ( x j dy j y j dx j ). (3–19) Thisstructureonthesphereiscalledthestandardstructur e.Itcanbeprovedthatthe inducedstructureon S 2 n +1 ptiscontactomorphictothestandardstructureon R 2 n +1 Denition17. WesaythataLiealgebra g ofdimension 2 n +1 admitsacontact structureifthereisanelement ofdegree 1 intheChevalley-Eilenbergcomplex ( g d ) of g suchthat ^ ( d ) n 6 =0. 38

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Denition18. Wesaythatacontactstructure(resp.acontactform)onanil manifold N = isleft(right)invariantifthecontactstructure(resp.ac ontactform)isinvariantwith respecttotheleft(resp.right) N -actionon N = Remark3.1. Sinceeveryinvariantdiferentialformonanilmanifold N = iscompletely determinedbyitsvaluesontheLiealgebra n of N ,weconcludethatanilmanifold N = admitsaninvariantcontactstructureifandonlyif n admitsacontactstructure. Notethatanyinvariantcontactstructureon N = iscoorientable. Nowwedescribetheaveragingprocess.Let G beaLiegroupandlet bea smoothdifferentialformon G .Wedeneadifferentialform I on G asfollows ( I )( p )= Z G g ( p ) d (3–20) where g isthepull-backoftheform undertherightmultiplication R g and isan invariantmeasureon G .Then R h ( I )( p )= R h Z G g ( p ) d = Z G ( R h g )( p ) d = Z G hg ( p ) d = Z G g ( p ) d = I ( p ), (3–21) forall h 2 G p 2 G .Hence I isarightinvariantdifferentialform. Remark3.2. Clearly,thekernelofinvariantcontactformgivesusaninv ariantcontact structure.Conversely,foranycontactform on N = wecanperformtheaveragingof theliftof to N rescaleddownbyanowherezerofunctionwiththeniteinteg raland getaninvariantcontactform .Furthermore, and yieldthesamecontactstructure provided denesaninvariantcontactstructure.Inparticular,anil manifold M admitsan invariantcontactstructureifandonlyif M admitsaninvariantcontactform. 3.2ClassicationofNilpotentRealLieAlgebras Tondall7-dimensionalnilmanifoldsthatadmitaninvaria ntcontactstructurewe usedtheclassicationof7-dimensionalindecomposableni lpotentLiealgebras.Many attemptshavebeendoneonthistopic.(Weshallseebelowtha tthedecomposable 39

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nilpotentLiealgebrasdonotadmitacontactstructure.)Th ereareafewlistsavailable: Saullina(1964,over C )[ 39 ],Romdhani(1985,over R and C )[ 35 ],Seeley(1988,over C )[ 40 ],AncocheaandGoze(1989,over C )[ 1 ].Thelistsaboveareobtainedbyusing differentinvariants.Carles[ 6 ]introducedanewinvariant-theweightsystem,compared thelistsofSaullina,RomdhaniandSeeley,andfoundmista kesandomissionsin allofthem.Laterin1993Seeleyincorporatedalltheprevio usresultsandpublished hislistover C ,[ 41 ].In1998GongusedtheSkjelbred-Sundmethodtoclassifyal l 7-dimensionalnilpotentLiealgebrasover R ,[ 18 ].WewilluseGong'sclassicationwith somecorrectionsfromthelistofMagnin,[ 28 ]. Notethatthereareonlynitelymanyisomorphismclassesof nilpotentLiealgebras ofdimensionlessorequalto 6 .Butinhigherdimensionsthereareinnitefamilies ofpairwisenonisomorphicnilpotentLiealgebras.Indimen sion 7 thereare 9 innite familieswhichcanbeparametrizedbyasingleparameter.In 1992Seeleyhassolved theproblemofestimatingthenumberofparameters F n neededtoclassifythelaws of n -dimensionalcomplexnilpotentLiealgebras,andcameupwi ththeestimation F n +2 1 6 n ( n 1)( n +4) 3 .Henceitisverydifculttowriteacompletelistfordimens ions greaterthan 7 .Forexamplefordimensions 8 and 10 thenumberofparametersinvolved willberespectively 4 and 13 .Soitbecomesverydesirabletoobtainacomplete andnon-redundantlistfor 7 -dimensionalnilpotentLiealgebras. In 1977 SkjelbredandSundhavedevelopedamethodofconstructinga llnilpotent Liealgebrasofdimension n giventhosealgebrasofdimensionlessthan n and theirautomorphismgroups.Gongusedthismethodtoconstru ctallnon-isomorphic 7 -dimensionalnilpotentLiealgebrasovertherealeld R NowwegivesomepreliminariesonthecohomologyofLiealgeb ras.Let g bea nilpotentLiealgebraofdimension n overaeld F .Recallthatamapping f : g g F iscalledmultilinearif f sendsan i -tuple ( x 1 ,..., x i ) into f ( x 1 ,..., x i ) insuchawaythat forxedvalues ( x 1 ,..., x q 1 x q +1 ,..., x i ) themapping x q 7! f ( x 1 ,..., x i ) isalinear 40

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mappingof g into F .Amultilinearmapping f isskewsymmetricifittakesvalue 0 when anytwo x q arethesame.Let ( g ) betheChevalley-Eilenbergcomplexof g .Thenan element f 2 g ofdegree i isaskewsymmetricmultilinearmapping f : g g F Theelementsof ( g ) ofdegree i arecalledthe i -cochains.Theset C i ( g F ) ofall i -cochainsisavectorspacewiththeusualadditionandscala rmultiplicationoffunctions. Thenthedifferential : C i ( g F ) C i +1 ( g F ) isgivenbytheformula ( f )( x 1 ,..., x i +1 )= X m < l ( 1) m + l f ( x 1 ,...,^ x m ,...,^ x l ,..., x i +1 ,[ x m x l ]), (3–22) wherethe ^ overanargumentmeansthatthisargumentisomitted.An i -cochain f is calledacocycleif df =0 andacoboundaryif f = g forsome ( i 1) -cochain g Theset Z i ( g F ) of i -cocyclesisthekernelofthehomomorphism from C i to C i +1 soitisasubspaceof C i .Similarlytheset B i ( g F ) of i -coboundariesisasubspace of C i becauseitistheimageof C i 1 under .When i =0 wedene B 0 ( g F )=0 Since 2 =0 thecoboundariesformasubspaceofthecocycles.Thefactor space H i ( g F )= Z i ( g F ) = B i ( g F ) iscalledthe i -dimensionalcohomologygroupof g Nowwedescribesomepropertiesof H i ( g F ) for i 2 .For i =0 wehave Z 0 = F and B 0 =0 sothat H 0 ( g F )= F .For i =1 wehave B 1 =0 sothat H 1 ( g F )= Z 1 ( g F ) Notethatif f 2 C 1 ( g F ) then ( f )( x 1 x 2 )= f ([ x 1 x 2 ]) .Therefore f isa 1 -cocycleifand onlyifitvanisheson [ g g ] .Hence H 1 ( g F ) isisomorphictothedualspaceof g = [ g g ] For i =2 ,if f 2 C 2 ( g F ) then ( f )( X 1 X 2 X 3 )= f ( X 3 ,[ X 1 X 2 ])+ f ([ X 2 X 3 ], X 1 )+ f ([ X 3 X 1 ], X 2 )=0. (3–23) Therefore, f =0 or f 2 Z 2 ifandonlyiftheJacobiidentityholds: Jac ( X 1 X 2 X 3 )= f ([ X 1 X 2 ], X 3 )+ f ([ X 2 X 3 ], X 1 )+ f ([ X 3 X 1 ], X 2 )=0. (3–24) Let B 2 ( g F ) bethesetofall 2 -coboundaries,i.e.elements f forwhichthereexist h 2 Hom ( g F ) suchthat f ( X Y )= h ([ X Y ]) forany X Y 2 g .Itimmediatelyfollows 41

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that dimB 2 ( g F )= dim [ g g ]. (3–25) Let g beaLiealgebraoveraeld F andlet Aut g betheatomorphismgroupof g .Then foreach B 2 C 2 ( g F k ) and 2 Aut g wedene B 2 C 2 ( g F k ) by B = B ( X Y ) forany X Y 2 g (3–26) Wecandenetheactionof Aut g on H 2 ( g F k ) because Z 2 ( g F k ) and B 2 ( g F k ) are invariantunderthisaction.Ifwedenoteby e B thecorrespondingelementfor B 2 Z 2 ( g F k ) in H 2 ( g F k ) ,thenwemaywritetheactionof Aut g on e B as e B = f B .For B 2 C 2 ( g F k ) thekernelof B willbedenedas g ?B ,with g ?B = f x 2 g : B ( x g )=0 g (3–27) Notethat C 2 ( g F k )= C 2 ( g F ) k and H 2 ( g F k )= H 2 ( g F ) k (3–28) Forevery B 2 C 2 ( g F k ) wewillwrite B =( B 1 ,..., B k ) 2 C 2 ( g F ) k (3–29) Thenwehave g ? B = g ? B 1 \\ g ? B k Let G k ( H 2 ( g F )) betheGrassmannianofsubspaces ofdimension k in H 2 ( g F ) .Thereisanaturalactionoftheautomorphismgroup Aut g on G k ( H 2 ( g F )) .Let e B 1 F e B k F 2 G k ( H 2 ( g F )) .Then ( e B 1 F e B k F )= f B 1 F f B k F .Itiswell-dened.If e B 1 F e B k F 2 G k ( H 2 ( g F )) thenwewillwrite B =( B 1 ,..., B k ) .Denotethecenterof g by Z ( g ) .Then U k ( g )= f e B 1 F e B 2 G k ( H 2 ( g F )): g ?B \ Z ( g )=0 g (3–30) iswell-denedandisalso Aut g stable.Let U k ( g ) = Aut g bethesetof ( Aut g ) -orbitsof U k ( g ) 42

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Theorem3.2. Let g beaLiealgebraoveraeld F .TheisomorphismclassesofLie algebras e g withcenter e ofdimension k e g = e = g ,andwithoutabeliandirectfactors,are inbijectivecorrespondencewiththeelementsin U k ( g ) = Aut g Denition19. AcentralextensionofaLiealgebra g isanexactsequenceofLie algebras 0 ~ g g 0 (3–31) suchthat isinthecenterof ~ g UsingTheorem 3.2 onecanconstructallthenilpotentLiealgebrasofdimensio n n giventhosealgebrasofdimensionlessthan n bycentralextension.Theprocedureof constracting 6 and 7 dimensionalLiealgebrascanbecarriedoutinthefollowing way, citeGo. (1)Foragivenalgebraofsmallerdimensionwelistatrstth egeneratorsofits centertoidentifythe 2 -cocyclessatisfying g ?B \ Z ( g )=0 (2)Wealsolistthegeneratorsofitsderivedalgebraneeded incomputing coboundaries B 2 ( g F ) (3)Thenwecomputeallthe 2 -cocycles Z 2 ( g F ) .Foreachxedalgebra g withbase f x 1 ,..., x n g wemayrepresenta 2 -cocycle B byaskew-symmetricmatrix B = X 1 i < j n C ij ij ,where ij isthe n n matrixwith ( i j ) elementbeing 1 ( j i ) element being 1 andallothers 0 .Henceinordertondmatrix B itsufcestodetermineall coefcients C ij (4)Wecanrepresent Z 2 ( g ) asadirectsum B 2 ( g F ) W ,where B 2 ( g F ) W = f df j f 2 C 1 ( g F )= g g d isthecoboundaryoperatorand W isasubspaceof Z 2 ( g F ) complementaryto B 2 ( g F ) .Oneeasywaytoobtain W isasfollows.Let f x 1 ,..., x r x r +1 ,..., x r + s g beabasisofanilpotentLiealgebra g ofdimension n = r + s where f x 1 ,... x r g arethegenerators,and f x r +1 ,..., x r + s g formsabasisforthederived algebra [ g g ] with x r + t =[ x i t x j t ] ,where 1 i t < j t r + t and 1 t s .Consider 43

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C 1 ( g )= g generatedbythedualbasis h f 1 ,..., f r g 1 ,..., g r i (3–32) of h x 1 ,..., x r x r +1 ,..., x r + s i (3–33) Then B 2 ( g F )= f dh j h 2 g g = h df 1 ,..., df r dg 1 ,..., dg s i (3–34) Then B 2 ( g F )= h dg 1 ,..., dg s i because df i ( x y )= f i ([ x y ])=0 .Nowwehave Z 2 ( g F )= h dg 1 ,..., dg s i W (3–35) Withoutlossofgeneralitywemayassumethat B ( x i t x j t )=0, t =1,..., s ,because if B ( x i t x j t ) 6 =0 forsome t thenwechoose B + u i t j t dg t instead.Whenwecarryout thegroupactionon W ,wecandoitasifitweredonein H 2 ( g F ) ,andwecanidentify H 2 ( g F ) with W (5)Wecanlistthedimensionofthesecondcohomologygroup.(6)Let f x 1 ,..., x n g beabasisof g .Thenabasisfor W isgivenasin (4) ,andwe maysimplyregarditasabasisfor H 2 ( g F ) withoutcausinganyconfusion. (7)Wecanndanarbitraryelementofthesecondcohomologyg rouptogether withtheactionofthegenericautomorphismonit.Notethate lementsarechosenfrom W Z 2 ( g F ) ,butweregardthemaselementsfrom H 2 ( g F ) andthegroupactionon theseelementsiscarriedoutasiftheywerein H 2 ( g F ) (8)Wedeterminealltherepresentativesoftheorbitsinthe Grassmanian G k ( H 2 ( g F )) undertheactionoftheautomorphismgroupthatsatisfythec onditionmentionedin (1) (9)Withtherepresentativesobtainedin (8) ,wecanobtainthelistofnonisomorphic centralextensionalgebrasof g withoutabelianfactors.If B isagivenrepresentative, 44

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thenwecandeneaLiealgebrastructureon g F k asfollows [( x u ),( y v )]=([ x y ], B ( x y )). (3–36) Wehavethefollowingtheoremthatdescribestheautomorphi smgroupofthenew algebra g ( B )= g F k obtainedbytheabovemethodthrougha 2 -cocycle B from g Theorem3.3. Let g beanilpotentLiealgebra,and 0 2 Aut g .Let B 2 H 2 ( g F k ) and g ?B \ Z ( g )=0 .Thentheautomorphismgroup Aut g ( B ) oftheextendedalgebra g ( B ) consistsofalllinearoperatorsofthematrixform = 0B@ 0 0 1CA (3–37) where 0 2 Aut g 2 GL k 2 Hom ( g F k ), and B ( 0 X 0 Y )= B ( X Y )+ [ X Y ], X Y 2 g (3–38) Notethatfromthemethoddescribedaboveitispossibletoge tdecomposableLie algebrasbycentralextensions.Thefollowingtheoremisdu etoSeeley. Theorem3.4. InadecompositionofanitedimensionalLiealgebraasadir ectsum ofindecomposableideals,theisomorphismclassesoftheid ealsareunique.If L = A 1 A r and L = C 1 C s aretwosuchdecompositions,then r = s Afterreorderingtheseindicesthederivedparts D 1 ( A i ) and D 1 ( C i ) areequal, A i = C i andasetofgeneratorsfor A i equalsasetofgeneratorsfor C i moduloaddingtoeach generatoravectorin Z ( L ) Tondaninvariantcontactstructureonanilmanifold,wer stapplyRemark 3.1 So,wemustcheckwhetherthenilpotentLiealgebrasfromthe Gong'slistadmita contactstructure.Toachievethisgoal,weusetheSullivan minimalmodeltheory andCorollary 3 (seesections 3.3 and 3.5 ).Next,foragiven7-dimensionalnilpotentLie algebrahavingacontactstructure,wemustcheckwhetherth ecorrespondingsimply 45

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connectedLiegroupcontainsadiscretecocompactsubgroup .Forthisweusethe Malcev'scriterion(Theorem 3.1 ). 3.3Preliminaries:MinimalModels Referencesfortheminimalmodeltheoryare[ 11 15 43 ]. Denition20. Adifferentialgradedalgebra(DGA)overaeld k ofcharacteristic 0 isagradedvectorspace A = 1i =0 A i withmultiplications A p n A q A p + q whichare associativeandcommutativeingradedsense a b =( 1) j a jj b j b a (3–39) where j a j and j b j denotethedegreesof a and b respectivelyintheunderlyinggraded vectorspace.TheDGA A alsopossessesadifferential d : A p A p +1 whichisagraded derivation d ( a b )=( da ) b +( 1) j a j a ( db ) (3–40) satisfying d 2 =0. AmorphisminthecategoryoftheDGA'sisanalgebrahomomorphismwhichcommuteswiththedifferentialandrespectsgra ding. Denition21. ADGA ( M d ) iscalledamodelforaDGA ( A d A ) ifthereexistsa DGA-morphism f :( M d ) ( A d A ) (3–41) thatinducesanisomorphismoncohomology.If ( M d ) isfreelygeneratedinthe sensethat M = V foragradedvectorspace V = V i thenitiscalledafreemodel for ( A d A ) ,wherethenotation V meansthat,asagradedalgebra, M isthetensor productofapolynomialalgebraonevenelements V even andanexterioralgebraonodd elements V odd Denition22. ADGA ( M d ) iscalledaminimalmodelof ( A d A ) ifthefollowing conditionsaresatised: (i) ( M d )=( V d ) isafreemodelfor ( A d A ) ; 46

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(ii) d isindecomposableinthefollowingsense:thereexistsabas is f x : 2 I g for somewell-orderedindexset I suchthat deg(x ) deg(x ) if < ,andeach d ( x ) is expressedintermsof x < .Itfollowsthat dx doesnothavealinearpart. Denition23. Aminimalmodelofasmoothmanifold M isaminimalmodelofitsde RhamDGA. Weneedthefollowingfact,[ 8 ].Let M = N = beanilmanifold.Thenthecomplexof differentialformson M canbeidentiedwiththethecomplexofdifferentialformso n N whicharerightinvariantbytheelementsof Theorem3.5. [ 34 ]Let M = N = beanilmanifold.Thenaturalinclusionofthecomplex ofrightinvariantdifferentialformson N intothecomplexofthedifferentialformson N = n invDR ( N ) n DR ( N = ) (3–42) inducesanisomorphismonthecohomologylevel.Corollary3. [ 43 ]Theminimalmodelofacompactnilmanifold N = isisomorphictothe Chevalley-EilenbergcomplexoftheLiealgebra n of N 3.4ContactStructuresonDecomposableNilpotentLieAlgeb ras Denition24. ALiealgebra n iscalleddecomposableifitcanberepresentedasa directsumofitsideals.Theorem3.6. Let N = beanilmanifoldofdimension 2 n +1, n 0 .IftheLiealgebra n oftheLiegroup N isdecomposablethenthenilmanifold N = doesnotadmitaninvariant contactstructure.Proof. Let n = V W where V and W areidealsin n sothat [ X Y ]=0 forall X 2 V Y 2 W .Let X 1 ,..., X s Y 1 ,..., Y t bethebasisof V and W ,respectively. Denoteby x 1 ,..., x s y 1 ,..., y t thedualbasisof V and W ,respectively.Considerthe Chevalley-Eilenbergcomplex ( n ) oftheLiealgebra n .Then x p = X i < j c p ij x i x j j < p p =1,..., s (3–43) 47

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y q = X l < m c q lm y l y m m < q q =1,..., t (3–44) because [ X p Y q ]=0 forall p =1,..., s q =1,..., t .Hence,withoutlossofgeneralitywe mayassumethat f x 1 x 2 ,..., x s y 1 y 2 ,..., y t g isthebasisof n thatisorderedtosatisfy theindecomposabilityconditionoftheminimalmodel ( n ) .Consideranarbitrary 1 -form = s X i =1 a i x i + t X j =1 b j y j a i b j 2 R (3–45) on n .Thenthederivative willcontainnotermswith x s or y t .Thus,the 2 n -form ( ) n vanishesbecauseitisalinearcombinationoftermseachofw hichisaproductof 2 n 1 -formsfromtheset f x 1 x 2 ,..., x s 1 y 1 y 2 ,..., y t 1 g (3–46) ofcardinality 2 n 1 .Itfollowsthatthenilmanifold N = cannotadmitaninvariant differentialform suchthat ^ ( d ) n 6 =0 Proposition3.1. Let n bea7-dimensionalnilpotentLiealgebrawiththebasis f X i : i =1,...,7 g andthecorrespondingdualbasis f x i : i =1,...,7 g .Let ( n ) bethe Chevalley-EilenbergcomplexoftheLiealgebra n .Ifonecannotndnonzerostructural constants c k 1 i 1 j 1 c k 2 i 2 j 2 c k 3 i 3 j 3 suchthat f i 1 j 1 i 2 j 2 i 3 j 3 g = f 1,2,3,4,5,6 g then n doesnot admitacontactstructure.Proof. Consideranarbitrary 1 -form = X a l x l a l 2 R l =1,...,7 (3–47) on n .Thenthe 6 -form ( ) 3 containsatmostvedistinctelementsfromthebasis f x 1 ,..., x 7 g andtherefore ( ) 3 =0 .Hence n doesnotadmitacontactstructure. Corollary4. Thefollowing 7 -dimensionalindecomposablenilpotentLiealgebrasdonot admitacontactstructure. 48

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UpperCentralSeriesDimensions (37) (37A) [ X 1 X 2 ]= X 5 ,[ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 (37B) [ X 1 X 2 ]= X 5 ,[ X 2 X 3 ]= X 6 ,[ X 3 X 4 ]= X 7 (37B 1 ) [ X 1 X 2 ]= X 5 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 7 [ X 2 X 4 ]= X 6 ,[ X 3 X 4 ]= X 5 (37C) [ X 1 X 2 ]= X 5 ,[ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 [ X 3 X 4 ]= X 5 (37D) [ X 1 X 2 ]= X 5 ,[ X 1 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 [ X 3 X 4 ]= X 5 (37D 1 ) [ X 1 X 2 ]= X 5 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 7 [ X 2 X 3 ]= X 7 ,[ X 2 X 4 ]= X 6 ,[ X 3 X 4 ]= X 5 UpperCentralSeriesDimensions (357) (357A) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 4 ]= X 7 [ X 2 X 4 ]= X 6 (357B) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 4 ]= X 7 [ X 2 X 3 ]= X 6 (357C) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 4 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 5 UpperCentralSeriesDimensions (27) (27A) [ X 1 X 2 ]= X 6 ,[ X 1 X 4 ]= X 7 ,[ X 3 X 5 ]= X 7 (27B) [ X 1 X 2 ]= X 6 ,[ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 7 [ X 3 X 4 ]= X 6 UpperCentralSeriesDimensions (257) (257A) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 4 ]= X 6 (257B) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 7 [ X 2 X 5 ]= X 7 49

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(257C) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 2 X 4 ]= X 6 [ X 2 X 5 ]= X 7 (257D) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 7 [ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 (257E) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 [ X 4 X 5 ]= X 6 (257F) [ X 1 X 2 ]= X 3 ,[ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 [ X 4 X 5 ]= X 6 (257G) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 4 ]= X 7 ,[ X 4 X 5 ]= X 6 (257H) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 2 X 4 ]= X 6 [ X 4 X 5 ]= X 7 (257I) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 6 [ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 7 (257J) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 7 ,[ X 2 X 4 ]= X 6 (257J 1 ) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 6 [ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 7 ,[ X 2 X 5 ]= X 6 (257K) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 2 X 3 ]= X 7 [ X 4 X 5 ]= X 7 (257L) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 6 ,[ X 2 X 3 ]= X 7 [ X 2 X 4 ]= X 6 ,[ X 4 X 5 ]= X 7 UpperCentralSeriesDimensions (247) (247A) [ X 1 X i ]= X i +2 i =2,3,4,5. (247B) [ X 1 X i ]= X i +2 i =2,3,4,[ X 3 X 5 ]= X 7 (247C) [ X 1 X i ]= X i +2 i =2,3,4,[ X 1 X 5 ]= X 7 ,[ X 3 X 5 ]= X 6 50

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(247D) [ X 1 X i ]= X i +2 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 (247E) [ X 1 X i ]= X i +2 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 (247E 1 ) [ X 1 X i ]= X i +2 i =2,3,4,[ X 2 X 4 ]= X 7 ,[ X 3 X 5 ]= X 7 (247F) [ X 1 X i ]= X i +2 i =2,3,[ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 (247F 1 ) [ X 1 X i ]= X i +2 i =2,3,[ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 ,[ X 3 X 5 ]= X 6 (247G) [ X 1 X i ]= X i +2 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 4 ]= X 6 [ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 ,[ X 3 X 5 ]= X 6 (247H) [ X 1 X i ]= X i +2 i =2,3,4,[ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 ,[ X 3 X 5 ]= X 6 (247H 1 ) [ X 1 X i ]= X i +2 i =2,3,4,[ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 ,[ X 3 X 5 ]= X 6 (247I) [ X 1 X i ]= X i +2 i =2,3,[ X 2 X 5 ]= X 6 ,[ X 3 X 4 ]= X 6 [ X 3 X 5 ]= X 7 (247J) [ X 1 X i ]= X i +2 i =2,3,[ X 1 X 5 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 ,[ X 3 X 5 ]= X 6 (247K) [ X 1 X i ]= X i +2 i =2,3,4,[ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 [ X 3 X 5 ]= X 6 (247L) [ X 1 X i ]= X i +2 i =2,3,4,5,[ X 2 X 3 ]= X 6 (247M) [ X 1 X i ]= X i +2 i =2,3,4,[ X 2 X 3 ]= X 6 ,[ X 3 X 5 ]= X 7 (247N) [ X 1 X i ]= X i +2 i =2,3,[ X 1 X 5 ]= X 6 ,[ X 2 X 3 ]= X 7 [ X 2 X 4 ]= X 6 (247O) [ X 1 X i ]= X i +2 i =2,3,4,[ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 7 [ X 3 X 5 ]= X 6 51

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(247P) [ X 1 X i ]= X i +2 i =2,3,[ X 2 X 3 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 (247P 1 ) [ X 1 X i ]= X i +2 i =2,3,[ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 [ X 3 X 5 ]= X 7 (247Q) [ X 1 X i ]= X i +2 i =2,3,4,[ X 2 X 3 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 (247R) [ X 1 X i ]= X i +2 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 3 ]= X 6 [ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (247R 1 ) [ X 1 X i ]= X i +2 i =2,3,4,[ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 [ X 3 X 5 ]= X 7 UpperCentralSeriesDimensions (2457) (2457A) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X i ]= X i +2 i =4,5. (2457B) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 2 X 5 ]= X 6 (2457C) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X i ]= X i +2 i =4,5,[ X 2 X 5 ]= X 6 (2457D) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X i ]= X i +2 i =4,5,[ X 2 X 3 ]= X 6 [ X 2 X 5 ]= X 6 (2457E) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 2 X 3 ]= X 6 [ X 2 X 5 ]= X 6 (2457F) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X i ]= X i +2 i =4,5,[ X 2 X 3 ]= X 6 (2457G) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 1 X 5 ]= X 6 [ X 2 X 3 ]= X 6 (2457H) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 2 X 3 ]= X 6 [ X 2 X 5 ]= X 7 (2457I) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 2 X 3 ]= X 6 [ X 2 X 5 ]= X 7 (2457J) [ X 1 X i ]= X i +1 i =2,3,[ X 2 X 3 ]= X 6 + X 7 ,[ X 1 X 4 ]= X 6 [ X 2 X 5 ]= X 7 52

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(2457K) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 1 X 5 ]= X 6 [ X 2 X 3 ]= X 6 ,[ X 2 X 5 ]= X 7 (2457L) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 7 ,[ X 2 X 5 ]= X 6 (2457L 1 ) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 7 ,[ X 2 X 5 ]= X 6 (2457M) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 1 X 5 ]= X 6 [ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 6 UpperCentralSeriesDimensions (2357) (2357A) [ X 1 X 2 ]= X 4 ,[ X 1 X 4 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 + X 6 ,[ X 3 X 4 ]= X 7 (2357B) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 5 [ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 5 ,[ X 3 X 4 ]= X 7 (2357C) [ X 1 X 2 ]= X 4 ,[ X 1 X 4 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 6 ,[ X 3 X 4 ]= X 7 (2357D) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 5 [ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 6 [ X 3 X 4 ]= X 7 (2357D 1 ) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 4 ]= X 5 [ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 6 [ X 3 X 4 ]= X 7 UpperCentralSeriesDimensions (23457) (23457A) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 3 ]= X 7 (23457B) [ X 1 X i ]= X i +1 i =2,3,4,[ X 2 X 3 ]= X 7 ,[ X 2 X 5 ]= X 6 [ X 3 X 4 ]= X 6 (23457C) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 53

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(23457D) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 3 ]= X 6 [ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (23457E) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 3 ]= X 5 + X 7 [ X 2 X 4 ]= X 6 (23457F) [ X 1 X i ]= X i +1 i =2,3,4,[ X 2 X 3 ]= X 5 + X 7 ,[ X 2 X 5 ]= X 6 [ X 3 X 4 ]= X 6 (23457G) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 6 ,[ X 2 X 3 ]= X 5 [ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 UpperCentralSeriesDimensions (1457) (1457A) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 5 X 6 ]= X 7 UpperCentralSeriesDimensions (137) (137A) [ X 1 X 2 ]= X 5 ,[ X 1 X 5 ]= X 7 ,[ X 3 X 4 ]= X 6 [ X 3 X 6 ]= X 7 (137A 1 ) [ X 1 X 3 ]= X 5 ,[ X 1 X 4 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 5 ,[ X 2 X 6 ]= X 7 (137C) [ X 1 X 2 ]= X 5 ,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 3 X 5 ]= X 7 UpperCentralSeriesDimensions (1357) (1357B) [ X 1 X 2 ]= X 4 ,[ X 1 X 4 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 3 X 4 ]= X 7 ,[ X 3 X 6 ]= X 7 (1357E) [ X 1 X 2 ]= X 3 ,[ X 2 X i ]= X i +2 i =3,4,[ X 2 X 5 ]= X 7 [ X 4 X 6 ]= X 7 (1357G) [ X 1 X 2 ]= X 3 ,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 5 ]= X 7 (1357I) [ X 1 X 2 ]= X 3 ,[ X 1 X 4 ]= X 6 ,[ X 2 X 3 ]= X 5 [ X 2 X 5 ]= X 7 ,[ X 4 X 6 ]= X 7 (1357O) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 5 ,[ X 2 X 5 ]= X 7 54

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(1357Q) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 6 ,[ X 2 X 6 ]= X 7 (1357Q 1 ) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 6 ,[ X 2 X 6 ]= X 7 UpperCentralSeriesDimensions (13457) (13457A) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 7 ,[ X 2 X 6 ]= X 7 (13457B) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 7 [ X 2 X 6 ]= X 7 (13457D) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 5 [ X 2 X 4 ]= X 7 ,[ X 2 X 6 ]= X 7 (13457F) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 6 [ X 2 X 6 ]= X 7 UpperCentralSeriesDimensions (12457) (12457A) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 5 ]= X 6 ,[ X 3 X 5 ]= X 7 (12457B) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 5 ]= X 6 + X 7 ,[ X 3 X 5 ]= X 7 (12457C) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 2 X 5 ]= X 6 [ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]= X 7 (12457F) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 2 X 3 ]= X 6 [ X 2 X i ]= X i +1 i =5,6,[ X 3 X 4 ]= X 7 (12457H) [ X 1 X i ]= X i +1 i =2,3,5,6,[ X 2 X j ]= X j +2 j =3,4,[ X 3 X 4 ]= X 7 (12457K) [ X 1 X i ]= X i +1 i =2,3,5,6,[ X 1 X 4 ]= X 7 ,[ X 2 X 3 ]= X 5 [ X 2 X 4 ]= X 6 ,[ X 3 X 4 ]= X 7 (12457L 1 ) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 5 ]= X 6 ,[ X 3 X 5 ]= X 7 55

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(12457N 1 ) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 5 ]= X 6 + X 7 ,[ X 3 X 5 ]= X 7 UpperCentralSeriesDimensions (12357) (12357A) [ X 1 X i ]= X i +1 i =4,5,6,[ X 1 X 2 ]= X 4 ,[ X 2 X 3 ]= X 5 [ X 3 X 4 ]= X 6 ,[ X 3 X 5 ]= X 7 (12357B) [ X 1 X i ]= X i +1 i =4,5,6,[ X 1 X 2 ]= X 4 ,[ X 2 X 3 ]= X 5 + X 7 [ X 3 X 4 ]= X 6 ,[ X 3 X 5 ]= X 7 (12357B 1 ) [ X 1 X i ]= X i +1 i =4,5,6,[ X 1 X 2 ]= X 4 ,[ X 2 X 3 ]= X 5 X 7 [ X 3 X 4 ]= X 6 ,[ X 3 X 5 ]= X 7 UpperCentralSeriesDimensions (123457) (123457A) [ X 1 X i ]= X i +1 ,2 i 6. (123457B) [ X 1 X i ]= X i +1 ,2 i 6,[ X 2 X 3 ]= X 7 (123457D) [ X 1 X i ]= X i +1 ,2 i 5[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 6 [ X 2 X 4 ]= X 7 (123457E) [ X 1 X i ]= X i +1 ,2 i 5[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 6 + X 7 [ X 2 X 4 ]= X 7 (123457H) [ X 1 X i ]= X i +1 ,2 i 5[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 5 + X 7 [ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 (123457H 1 ) [ X 1 X i ]= X i +1 ,2 i 5[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 5 + X 7 [ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 Thus,ifa7-dimensionalsimplyconnectednilpotentLiegro up N admitsadiscrete cocompactsubgroup andtheLiealgebra n of N fallsintothislistof 1037 -dimensional indecomposablenilpotentLiealgebrasthenthenilmanifol d N = doesnotadmitan invariantcontactstructure. 3.5ContactStructuresonIndecomposableNilpotentLieAlg ebras Nowwediscusstheexistenceofcontactstructuresonindeco mposableLie algebras.Wehaveprovedthattherearenocontactstructure sondecomposable nilpotentLiealgebras,seeTheorem 3.6 above.All7-dimensiomalindecomposable 56

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nilpotentLiealgebrasovertherealeldhavebeenclassie d,[ 18 28 ].Overrealsthere are138non-isomorphic7-dimensiomalindecomposablenilp otentLiealgebrasand, inaddition,9one-parameterfamilies.Fortheone-paramet erfamilies,aparameter isusedtodenoteastructureconstantthatmaytakeonarbitr aryvalues(withsome exceptions)in R Aninvariant I ( ) isgivenforeachfamilyinwhichmultiplevaluesof yieldisomorphicLiealgebras,i.e.,if I ( 1 )= I ( 2 ) ,thenthetwocorrespondingLie algebrasareisomorphic,andconversely.Allone-paramete rfamiliesadmitacontact structure(withsomerestrictionsonaparameter )andonly 35 Liealgebrasadmit acontactstructure.Weconductadetailedcomputationfort healgebra(1357C)and theone-parameterfamily(147E).Hereweusethesamenotati onfortheLiealgebras asin[ 18 ],wheretheLiealgebrasarelistedinaccordancewiththeir uppercentral seriesdimensions.Forinstance,theLiealgebrasthathave theuppercentralseries dimensions1,4,7arelistedasfollows:(147A),(147B),(14 7C),etc.Forthesakeof brevitywewilldropthesignofthewedgeproduct(forinstan ce, x i x j means x i ^ x j ). NowweshowthattheLiealgebra(1357C)andtheone-paramete rfamily(147E) admitacontactstructure.FirstweconsidertheLiealgebra (1357C).Let f X i : i =1,...,7 g beabasisoftheLiealgebra(1357C)and f x i : i =1,...,7 g be thecorrespondingdualbasis.Notethat (1357) standsfortheuppercentralseries dimensions.ThenontrivialLiebracketsoftheLiealgebra( 1357C)aredenedas follows: [ X 1 X 2 ]= X 4 ,[ X 1 X 4 ]= X 5 ,[ X 1 X 5 ]= X 7 ,[ X 3 X 6 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 7 ,[ X 3 X 4 ]= X 7 ThenwecanndthedifferentialoftheChevalley-Eilenberg modeloftheLiealgebra (1357C): dx i =0, i =1,2,3, dx 4 = x 1 x 2 dx 5 = x 2 x 3 + x 1 x 4 dx 7 = x 1 x 5 + x 2 x 4 x 3 x 4 + x 3 x 6 57

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Let = x 7 .Astraightforwardcomputationshowsthat ^ ( d ) 3 =6 x 1 x 2 x 3 x 4 x 5 x 6 x 7 6 = 0. Hencethealgebra(1357C)admitsacontactstructure.Simil arlyonecanseethat thereare 34 otheralgebrasthatadmitacontactstructureandthecontac tform isgiven by = x 7 ,exceptforthealgebra(12457L)forwhich = x 7 + x 6 Considertheone-parameterfamily(147E)withinvariant I ( )= (1 + 2 ) 3 2 ( 1) 2 6 =0,1 andtheLiebracketdenedasfollows: [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]=(1 ) X 7 ThenwecanndthedifferentialoftheChevalley-Eilenberg modeloftheone-parameter family(147E): dx i =0, i =1,2,3, dx 4 = x 1 x 2 dx 7 = x 2 x 6 +(1 ) x 3 x 4 x 1 x 5 dx 6 = x 1 x 3 dx 5 = x 2 x 3 Let = x 7 .Then ^ ( d ) 3 =6 (1 ) x 1 x 2 x 3 x 4 x 5 x 6 x 7 6 =0, for 6 =0,1. Hencethe one-parameterfamily(147E)admitsacontactstructurefor any 6 =0,1. Similarlyone canseethattheremaining 8 familiesadmitacontactstructureandthecontactform is givenby = x 7 .Eventually,wegetthefollowingresult. Theorem3.7. Thefollowingisacompleteandnon-redundantlist L ofall 7 -dimensional nilpotentrealLiealgebrasthatadmitacontactstructure: UpperCentralSeriesDimensions (17) (17) [ X 1 X 2 ]= X 7 ,[ X 3 X 4 ]= X 7 ,[ X 5 X 6 ]= X 7 UpperCentralSeriesDimensions (147) (147A) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 6 ]= X 7 [ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (147A 1 ) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 6 ]= X 7 [ X 2 X 4 ]= X 7 ,[ X 3 X 5 ]= X 7 (147B) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 4 ]= X 7 [ X 2 X 6 ]= X 7 ,[ X 3 X 5 ]= X 7 58

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(147D) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 5 ,[ X 2 X 6 ]= X 7 [ X 3 X 4 ]= 2 X 7 (147E) [ X 1 X 2 ]= X 4 ,[ X 1 X 3 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]=(1 ) X 7 I ( )= (1 + 2 ) 3 2 ( 1) 2 6 =0,1. (147E 1 ) [ X 1 X 2 ]= X 4 ,[ X 2 X 3 ]= X 5 ,[ X 1 X 3 ]= X 6 [ X 1 X 6 ]= X 7 ,[ X 2 X 5 ]= X 7 ,[ X 2 X 6 ]=2 X 7 [ X 3 X 4 ]= 2 X 7 > 1. UpperCentralSeriesDimensions (157) (157) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 7 ,[ X 2 X 4 ]= X 7 [ X 5 X 6 ]= X 7 UpperCentralSeriesDimensions (1457) (1457B) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 7 ,[ X 2 X 3 ]= X 7 [ X 5 X 6 ]= X 7 UpperCentralSeriesDimensions (137) (137B) [ X 1 X 2 ]= X 5 ,[ X 1 X 5 ]= X 7 ,[ X 2 X 4 ]= X 7 [ X 3 X 4 ]= X 6 ,[ X 3 X 6 ]= X 7 (137B 1 ) [ X 1 X 3 ]= X 5 ,[ X 1 X 4 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 5 ,[ X 2 X 6 ]= X 7 [ X 3 X 4 ]= X 7 (137D) [ X 1 X 2 ]= X 5 ,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 ,[ X 3 X 5 ]= X 7 UpperCentralSeriesDimensions (1357) (1357A) [ X 1 X 2 ]= X 4 ,[ X 1 X 4 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]= X 7 59

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(1357C) [ X 1 X 2 ]= X 4 ,[ X 1 X 4 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 7 ,[ X 3 X 4 ]= X 7 [ X 3 X 6 ]= X 7 (1357D) [ X 1 X 2 ]= X 3 ,[ X 2 X 3 ]= X 5 ,[ X 1 X 6 ]= X 7 [ X 2 X 5 ]= X 7 ,[ X 2 X 4 ]= X 6 ,[ X 3 X 4 ]= X 7 (1357F) [ X 1 X 2 ]= X 3 ,[ X 2 X i ]= X i +2 i =3,4,[ X 1 X 3 ]= X 7 [ X 2 X 5 ]= X 7 ,[ X 4 X 6 ]= X 7 (1357F 1 ) [ X 1 X 2 ]= X 3 ,[ X 2 X i ]= X i +2 i =3,4,[ X 1 X 3 ]= X 7 [ X 2 X 5 ]= X 7 ,[ X 4 X 6 ]= X 7 (1357H) [ X 1 X 2 ]= X 3 ,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 5 ,[ X 2 X 5 ]= X 7 ,[ X 2 X 6 ]= X 7 [ X 3 X 4 ]= X 7 (1357J) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 7 ,[ X 1 X 4 ]= X 6 [ X 2 X 3 ]= X 5 ,[ X 2 X 5 ]= X 7 ,[ X 4 X 6 ]= X 7 (1357L) [ X 1 X 2 ]= X 3 ,[ X 1 X i ]= X i +2 i =3,4,5,[ X 2 X 3 ]= X 7 [ X 2 X 4 ]= X 5 ,[ X 2 X 6 ]= 1 2 X 7 ,[ X 3 X 4 ]= 1 2 X 7 (1357M) [ X 1 X 2 ]= X 3 ,[ X 1 X i ]= X i +2 i =3,4,5,[ X 2 X 4 ]= X 5 [ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]=(1 ) X 7 6 =0,1. (1357N) [ X 1 X 2 ]= X 3 ,[ X 1 X i ]= X i +2 i =3,4,5,[ X 2 X 3 ]= X 7 [ X 2 X 4 ]= X 5 ,[ X 3 X 4 ]= X 7 ,[ X 4 X 6 ]= X 7 (1357P) [ X 1 X 2 ]= X 3 ,[ X 1 X i ]= X i +2 i =3,5,[ X 2 X 3 ]= X 6 [ X 2 X 4 ]= X 5 ,[ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]= X 7 (1357P 1 ) [ X 1 X 2 ]= X 3 ,[ X 1 X i ]= X i +2 i =3,5,[ X 2 X 3 ]= X 6 [ X 2 X 4 ]= X 5 ,[ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]= X 7 60

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(1357QRS 1 ) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 4 ]= X 6 [ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 5 [ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]=(1 ) X 7 I ( )= + 1 6 =0, 1. (1357R) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 (1357S) [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 5 ,[ X 1 X 5 ]= X 7 [ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 6 [ X 2 X 5 ]= X 7 ,[ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]= X 7 6 =0,1. UpperCentralSeriesDimensions (13457) (13457C) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 6 ]= X 7 ,[ X 2 X 5 ]= X 7 [ X 3 X 4 ]= X 7 (13457E) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 5 [ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (13457G) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 6 [ X 2 X 4 ]= X 7 ,[ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (13457I) [ X 1 X i ]= X i +1 i =2,3,4,[ X 1 X 5 ]= X 7 ,[ X 2 X 3 ]= X 6 [ X 2 X 5 ]= X 7 ,[ X 2 X 6 ]= X 7 ,[ X 3 X 4 ]= X 7 UpperCentralSeriesDimensions (12457) (12457D) [ X 1 X i ]= X i +1 i =2,3,[ X 2 X 6 ]= X 7 ,[ X 2 X 5 ]= X 6 [ X 1 X i ]= X i +2 i =4,5,[ X 3 X 4 ]= X 7 (12457E) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 6 ]= X 7 [ X 2 X 3 ]= X 6 ,[ X 2 X 4 ]= X 7 ,[ X 2 X 5 ]= X 6 [ X 3 X 5 ]= X 7 (12457G) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 2 X i ]= X i +1 i =5,6,[ X 2 X 3 ]= X 6 ,[ X 3 X 4 ]= X 7 61

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(12457I) [ X 1 X i ]= X i +1 i =2,3,5,6,[ X 3 X 4 ]= X 7 ,[ X 2 X 5 ]= X 7 [ X 2 X j ]= X j +2 j =3,4. (12457J) [ X 1 X i ]= X i +1 i =2,3,5,6,[ X 1 X 4 ]= X 7 ,[ X 2 X 3 ]= X 5 [ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (12457J 1 ) [ X 1 X i ]= X i +1 i =2,3,5,6,[ X 1 X 4 ]= X 7 ,[ X 2 X 5 ]= X 7 [ X 2 X j ]= X j +2 j =3,4,[ X 3 X 4 ]= X 7 (12457L) [ X 1 X i ]= X i +1 i =2,3,5,6,[ X 3 X 4 ]= X 7 ,[ X 2 X 6 ]= X 7 [ X 2 X j ]= X j +2 j =3,4,[ X 3 X 5 ]= X 7 (12457N) [ X 1 X i ]= X i +1 i =2,3,5,6,[ X 1 X 4 ]= X 7 ,[ X 2 X 3 ]= X 5 [ X 2 X 4 ]= X 6 ,[ X 2 X 5 ]= X 7 ,[ X 2 X 6 ]= X 7 [ X 3 X 4 ]= X 7 ,[ X 3 X 5 ]= X 7 I ( )= + 1 6 =0. (12457N 2 ) [ X 1 X i ]= X i +1 i =2,3,[ X 1 X 4 ]= X 6 ,[ X 1 X 5 ]= X 7 [ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 5 ,[ X 2 X 4 ]= X 7 [ X 2 X 5 ]= X 6 + X 7 ,[ X 3 X 5 ]= X 7 0, 6 =1. UpperCentralSeriesDimensions (12357) (12357C) [ X 1 X 2 ]= X 4 ,[ X 1 X i ]= X i +1 i =4,5,6,[ X 2 X 3 ]= X 5 [ X 2 X 4 ]= X 7 ,[ X 3 X 4 ]= X 6 ,[ X 3 X 5 ]= X 7 UpperCentralSeriesDimensions (123457) (123457C) [ X 1 X i ]= X i +1 ,2 i 6,[ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (123457F) [ X 1 X i ]= X i +1 ,2 i 5,[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 6 [ X 2 X 4 ]= X 7 ,[ X 2 X 5 ]= X 7 ,[ X 3 X 4 ]= X 7 (123457I) [ X 1 X i ]= X i +1 ,2 i 5,[ X 1 X 6 ]= X 7 ,[ X 2 X 3 ]= X 5 [ X 3 X 4 ]=(1 ) X 7 ,[ X 2 X 5 ]= X 7 ,[ X 2 X 4 ]= X 6 6 =0,1. Corollary5. Fora 7 -dimensionalnilpotentLiealgebra n wedenoteby C ( n ) thesetof allnilmanifolds N = suchthat n isisomorphictotheLiealgebraofaLiegroup N .Then 62

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S n 2L C ( n ) isthesetofall 7 -dimensionalnilmanifoldsthatadmitaninvariantcontact structure.ToseewhichoftheseLiealgebrasyieldanilmanifold,wemus tcheckwhetheraLie algebrafromthelistadmitsabasiswithrespecttowhichall structureconstantsare rational.Fromtheabovelist,weseethat,denitely,thefo llowingLiealgebrassatisfythis condition: (17),(157),(147( A A 1 B D )),(1457),(137),(12457( D E G I J J 1 L )), (13457),(12357),(123457( C F )) (1357( A C D F F 1 H J L P P 1 R )) Concerningtheremainingcases,notethattheone-paramete rfamiliesofLie algebraswitharationalparameter giveusthenilmanifolds.Thecaseofanirrational parameter isnotcleartous.Inprinciple,itcanhappenthatsuchaLiea lgebrahas abasiswithrespecttowhichallstructureconstantsarerat ional.Thisneedsmore research. 3.6TheCaseofDim < 7 Forcompletenesswediscusstheexistenceofinvariantcont actstructureson nilmanifoldsofdimension 3 and 5 .NotethatallnilpotentLiealgebrasofdimensionup to6havebeenclassied,[ 19 ].Indimension 5 thereare 9 nonisomorphicnilpotentLie algebrasandthereare 3 ofthemthatadmitacontactstructure = x 5 .Weshallusethe samenotationasin[ 19 ]. L 5,1 [ X 1 X 2 ]= X 5 ,[ X 3 X 4 ]= X 5 L 5,3 [ X 1 X 2 ]= X 4 ,[ X 1 X 4 ]= X 5 ,[ X 2 X 3 ]= X 5 L 5,6 [ X 1 X 2 ]= X 3 ,[ X 1 X 3 ]= X 4 ,[ X 1 X 4 ]= X 5 [ X 2 X 3 ]= X 5 Indimension 3 thereareonly twonon-isomorphicnilpotentLiealgebras.Oneofthemisab elianandthereforedoes notadmitacontactstructure.Henceatorus T = R 3 = Z 3 doesnotadmitaninvariant contactstructure.Another 3 -dimensionalnilpotentLiealgebrahasnontrivialLiebrac ket [ X 1 X 2 ]= X 3 andthereforeadmitsacontactstructure = x 3 .HencetheHeisenberg manifoldistheonly 3 -dimensionalnilmanifoldthatadmitsaninvariantcontact structure. 63

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BIOGRAPHICALSKETCH SergiiKutsakwasborninChernivtsi,Ukraine.Hegraduated fromChernivtsiHigh School27in2000.HeearnedhisMasterofSciencedegreeinma thematicsfrom ChernivtsiNationalUniversityin2005.In2008heenteredt hegraduateschoolatthe UniversityofFloridatocontinuehisstudies.Hereceivedh isPh.D.inmathematicsfrom theUniversityofFloridain2013. 67