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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 so
2、thatthevariousslicetheoremsarevalid.Forcoveringspacetheory,weneedandtacitlyassumethatourspacesarelocallypath-connectedandsemi-simplyconnected.1.1.2.AleftactionofatopologicalgroupGonatopologicalspaceXisacontinuousfunction':GX !Xsuchthat(i)'(gh;x)='(g;'(h;x))forallg;h2Gan
3、dx2X,(ii)'(1;x)=x,forallx2X,where1istheidentityelementofG.Weshallusuallywrite'(g;x)simplyasgx,g(x),orsometimesgx.Clearly,eachelementg2GcanbeviewedasahomeomorphismofXontoitself.Wemaydenotethisactionby(G;X;'),ormoresimplysuppressthe'andcallXaG-space.IfXandYareG-spaces,the
4、naG-mapisacontinuousfunctionf:X!Ywhichisequivariant;i.e.,f(gx)=gf(x)forallg2Gandx2X.IffisaG-mapandahomeomorphism,thenfiscalledaG-equivalenceorG-isomorphism(intherelevantcategory).Amapf:X!YisweaklyG-equivariantifthereexistsacontinuousautomorphismfofGsuchthatf(gx)=f(g)f(x)
5、,forallg2G,x2X.Iffisahomeomorphism,thenfisaweakG-equivalenceoraweakG-isomorphism.Thereisananalogousnotionofarightaction,: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,
6、forareasonablynicespaceX(e.g.,locallycompactHaus-dor ),thesetofself-homeomorphismsofXbecomesatopologicalgroupTOP(X)(seesection1.2.5),andaleftG-actionisequivalenttohavingagrouphomomor-phismG!TOP(X).ArightG-actionbecomesananti-homomorphism.Wewillalwaysassumewehavealeftacti
7、onunlesswespecifyotherwise.Anaction(G;X;')iscalledasmoothactionifGisaLiegroup,Xisasmoothmanifold,andthefunction'issmooth.121.TRANSFORMATIONGROUPSorbitofGthroughx1.1.3.IfXisaG-spaceandx2X,thenorbitspaceorbitmapGx=fy2X:y=gxforsomeg2GgpropermappingiscalledtheorbitofGthrough
8、x.ItcanbedenotedbyGx,GxorG(x).isotropysubgroupstabilizerThecollectionofallorbitsofXformsapartitionofXi
