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1、0leftactionG-spaceG-mapequivariantCHAPTER1G-equivalenceG-isomorphismweaklyG-equivariantTransformationGroupsweakG-equivalenceweakG-isomorphismrightactionTOP(X)FriJun1611:30:382000action!smooth1.1.Introduction1.1.1.OurspacesXarepath-connected,completelyregularandHausdor sothatthevariousslicethe
2、oremsarevalid.Forcoveringspacetheory,weneedandtacitlyassumethatourspacesarelocallypath-connectedandsemi-simplyconnected.1.1.2.AleftactionofatopologicalgroupGonatopologicalspaceXisacontinuousfunction':GX !Xsuchthat(i)'(gh;x)='(g;'(h;x))forallg;h2Gandx2X,(ii)'(1;x)=x,forallx2X,where1istheident
3、ityelementofG.Weshallusuallywrite'(g;x)simplyasgx,g(x),orsometimesgx.Clearly,eachelementg2GcanbeviewedasahomeomorphismofXontoitself.Wemaydenotethisactionby(G;X;'),ormoresimplysuppressthe'andcallXaG-space.IfXandYareG-spaces,thenaG-mapisacontinuousfunctionf:X!Ywhichisequivariant;i.e.,f(gx)=gf(
4、x)forallg2Gandx2X.IffisaG-mapandahomeomorphism,thenfiscalledaG-equivalenceorG-isomorphism(intherelevantcategory).Amapf:X!YisweaklyG-equivariantifthereexistsacontinuousautomorphismfofGsuchthatf(gx)=f(g)f(x),forallg2G,x2X.Iffisahomeomorphism,thenfisaweakG-equivalenceoraweakG-isomorphism.Thereis
5、ananalogousnotionofarightaction,:GX !Xwhichwedenoteby(x;g)=xgorxg.Then(x;gh)=(xg)h=xgh.AnyrightG-action(x;g)canbeconvertedtoaleftG-action'(g;x)by'(x;g)=(x;g 1)andviceversa.Notethat,forareasonablynicespaceX(e.g.,locallycompactHaus-dor ),thesetofself-homeomorphismsofXbecomesatopologicalgroupT
6、OP(X)(seesection1.2.5),andaleftG-actionisequivalenttohavingagrouphomomor-phismG!TOP(X).ArightG-actionbecomesananti-homomorphism.Wewillalwaysassumewehavealeftactionunlesswespecifyotherwise.Anaction(G;X;')iscalledasmoothactionifGisaLiegroup,Xisasmoothmanifold,andthefunction'issmooth.121.TRANSFO
7、RMATIONGROUPSorbitofGthroughx1.1.3.IfXisaG-spaceandx2X,thenorbitspaceorbitmapGx=fy2X:y=gxforsomeg2GgpropermappingiscalledtheorbitofGthroughx.ItcanbedenotedbyGx,GxorG(x).isotropysubgroupstabilizerThecollectionofallorbitsofXformsapartitionofXi
