cellular gradients. condition (c)

《cellular gradients. condition (c)》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。

1、CHAPTER7Condition(C)Letf:W→[a,b]beaMorsefunctiononacobordismWandvanf-gradient.Thegradientdescentfrom∂1Wto∂0Wyieldsadiffeomorphism(−v):∂1WD(−v)→∂0WD(v);bydefinition(−v)(x)isthepointwherethetrajectoryγx,τ(x,−v);−vintersects∂0W.Themap(−v)doesnotadmitacontinu

2、ousextensionto∂1Wanditspropertiesingeneralareratherdifficulttostudy.InthispartweshowthatforaC0-genericgradientvonecanderivefrom(−v)ageometricstructure(whichcanbecalledagraded-continuousmap)closelyresemblingacellularmapbetweenCWcomplexes.Thecomponentsofthisstruc

3、tureare:aMorse-Smalefiltration∂1Wkof∂1W,†aMorse-Smalefiltration∂0Wkof∂0W,andasequenceofcontinuousmapsφ:∂Wk/∂Wk−1→∂Wk/∂Wk−1k1100suchthatforeveryk:(A)(−v)(∂Wk)⊂∂Wkand10(B)thefollowingdiagramiscommutative(wherethehorizontalarrowsaretheprojectionmaps).kkk−1∂1W∂1W/

4、∂1W(−v)φkkkk−1∂0W∂0W/∂0WEachφkisacontinuousmapbetweenspaceshomotopyequivalenttowedgesofk-dimensionalspheres,andtheinvariantsofthesemapsarequiteusefulforthestudyofthegradientdescent.†k(k)Toabbreviatethenotationwewrite∂1Winsteadof∂1Winthispart.232Chapter7.Co

5、ndition(C)1.ThegradientdescentrevisitedBeforeconstructingthegraded-continuousmap,mentionedintheintro-duction,letusdiscussthequestioninasomewhatgreatergenerality.Letf:W→[a,b]beaMorsefunctiononacobordismWandvanf-gradient.ThesubsetD(−v)∩∂1Wcanbeconsideredastheset

6、ofsin-gularitiesofthepartiallydefinedmap(−v).Letusquotientoutthissingularityset.Namely,putS1=D(−v)∩∂1W,S0=D(v)∩∂0W.Wewillconstructacontinuousmapφ:∂1W/S1→∂0W/S0betweenthequotientspaces.Letω1denotethepointof∂1W/S1corre-spondingtothecollapsedsubsetS1,andω0denotet

7、hepointof∂0W/S0correspondingtothecollapsedsubsetS0.Putφ(ω1)=ω0.Foreverypointx∈(∂1W/S1){ω1}=∂1WS1defineφ(x)tobetheimageof(−v)(x)∈∂0Wviathequotientmap∂0W→∂0W/S0.Proposition1.1.Themap(1)φ:∂1W/S1→∂0W/S0iscontinuous.Proof.Continuityonthecomplementtothepointω1foll

8、owsfromthecontinuityof(−v).Asforthepointω1,itsufficestoshowthatforeveryopenneighbourhoodUof∂0W∩D(v)thereisaneighbourhoodVof∂1W∩D(−v)suchthat(−v)(V)⊂U.Pickδ>0suchthatBδ(v)∩∂0W⊂U(thi