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Non compact manifolds proper homotopy equivalent to geometrically simply connected polyhedra and proper realizability

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Niveau: Supérieur, Doctorat, Bac+8
Non-compact 3-manifolds proper homotopy equivalent to geometrically simply connected polyhedra and proper 3-realizability of groups Louis Funar1, Francisco F. Lasheras2 and Dusˇan Repovsˇ3 ? 1Institut Fourier BP 74, UFR Mathematiques, Univ.Grenoble I 38402 Saint-Martin-d'Heres Cedex, France 2Departamento de Geometria y Topologia, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain 3Faculty of Mathematics and Physics, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia May 14, 2009 Abstract The principal result of this paper is a homotopy criterion for detecting the tameness of non-compact 3-manifolds which extends the one worked out by L.Funar and T.L.Thickstun for open 3-manifolds. A group is properly 3-realizable if it is the fundamental group of a compact 2-polyhedron whose universal covering is proper homotopy equivalent to a 3-manifold. As a consequence of the main result a properly 3-realizable group which is also quasi-simply filtered has pro-(finitely generated free) fundamental group at infinity and semi-stable ends. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups. AMS MOS Subj.

  • compact manifold

  • actually any standard

  • group

  • dimensional subset

  • p1 ?

  • simply connected

  • manifold

  • fundamental pro


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Non-compact3-manifoldsproperhomotopyequivalenttogeometricallysimplyconnectedpolyhedraandproper3-realizabilityofgroupsLouisFunar1,FranciscoF.Lasheras2andDuˇsanRepovsˇ31InstitutFourierBP74,UFRMathe´matiques,Univ.GrenobleI38402Saint-Martin-d’H`eresCedex,France2DepartamentodeGeometriayTopologia,UniversidaddeSevilla,Apdo1160,41080Sevilla,Spain3FacultyofMathematicsandPhysics,UniversityofLjubljana,P.O.Box2964,Ljubljana1001,SloveniaMay14,2009AbstractTheprincipalresultofthispaperisahomotopycriterionfordetectingthetamenessofnon-compact3-manifoldswhichextendstheoneworkedoutbyL.FunarandT.L.Thickstunforopen3-manifolds.Agroupisproperly3-realizableifitisthefundamentalgroupofacompact2-polyhedronwhoseuniversalcoveringisproperhomotopyequivalenttoa3-manifold.Asaconsequenceofthemainresultaproperly3-realizablegroupwhichisalsoquasi-simplyfilteredhaspro-(finitelygeneratedfree)fundamentalgroupatinfinityandsemi-stableends.Conjecturallythequasi-simplyfiltrationassumptionissuperfluous.Usingtheserestrictionsweprovidethefirstexamplesoffinitelypresentedgroupswhicharenotproperly3-realizable,forinstancelargefamiliesofCoxetergroups.AMSMOSSubj.Classification(1991):57M50,57M10,57M30.Keywordsandphrases:Properly3-realizable,geometricsimpleconnectivity,quasi-simplefilteredgroup,missingboundary3-manifold,Coxetergroup.1IntroductionIn[18,21]theauthorsprovedthatanopen3-manifoldissimplyconnectedatinfinityifithastheproperhomotopytypeofaweaklygeometricallysimplyconnectedpolyhedron.Thesimpleconnectivityatinfinityisastrongtamenessconditionforopen3-manifoldswhich,roughlyspeaking,expressesthefactthateachendiscollaredbya2-sphere.Themainconcernofthispaperistogiveasimilarhomotopycriterionfordetectingthetamenessinthecaseof3-manifoldswithboundary.Therelevanttamenessconditionshavetobechangedaccordingly,inordertotakeintoaccounttheboundarybehavior.Insteadofthesimpleconnectivityatinfinitywewillconsidertheso-calledmissingboundarymanifoldconditionintroducedbySimon(see[40]),whiletheweakgeometricsimpleconnectivityhastobereplacedbythestrongerpl-geometricsimpleconnectivity,tobedefinedbelow.Despitethefactthattherearesimilaritiesintheproofswith[18,21],thecasewheremanifoldshavenon-compactboundarycomponentspresentssomenewandinterestingfeatures.Workinginthismoregeneralcontextopensthepossibilitytofindapplicationstogeometricgrouptheory.Specifically,weobtainnecessaryconditionsforafinitelypresentedgrouptoactfreelyco-compactlyonasimplyconnected2-complexhavingtheproperhomotopytypeofa3-manifold.Inparticularwefindexplicitexamplesofgroupswhichdonothavethisproperty.Emails:funar@fourier.ujf-grenoble.fr(L.Funar),lasheras@us.es(F.F.Lasheras),dusan.repovs@guest.arnes.si(D.Repovˇs)1