lesson9(商映射，流形)

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1、111µµµûûûNNN!!!444¡¡¡-ûN->6/ûûûNNNûm±^ÊÜóEm§ûN±^ÊÜó®m#£ã"½½½ÂµµµeëY÷f:X→Y÷v?B⊆Y,XJf−1(B)mKB½m§K¡fûN"···KKKµµµ(1)ÊÜN´ûN¶(2)÷v/f−1(B)4KB½40ëY÷Ñ´ûN¶(3)ëY÷mN½4NÑ´ûN§ÏéuëY÷kf(f−1(B))=B.(4)ûNEÜ´ûN"½½½nnnµµµef:X→Y´ûN§K?¿g:Y→ZëY¿^´g◦f:X→ZëY"X↓&Y→Z···

2、KKKµµµef:X→Y´ûN§K3X¥½Âd'X∼:x∼x0=f(x)=f(x0),¿∼kÓX/∼=Y.£é''uû+nؤyyy²²²µµµg:X/∼→Y,g()=f(x),éeãüã¦^½n(Øg,g−1ÑëY"XX↓&.↓X/∼→YX/∼←Y½½½nnnµµµ;8HausdorffmëY÷´4N§l´ûN"yyy²²²µµµX;"eA⊆X4§KA;§f(A)⊆Y;"YHausdorff,f(A)4"~~~µµµf:[0,1]→E2,t7→(cos(2πt),sin(2πt)).u´S1=f([0,1]

3、)=∼[0,1]/∼f,=ò[0,1]üà:ÊܱS1.~~~µµµÓnyò/mü>^ÓÊܤmÓuE3¥/IO0²½M¨obius"~~~µµµÄNf:(x,y,z)7→(x2−y2,xy,xz,yz),K?E3¥k.4f8U,f

4、U:U→f(U)´ûN"AO/§±±d½Â/3E4¥ÊÜ¥¡é»:0ö§,3E3¥ù´ØU¢y"¥¡ÊÜé»:¤ûm¡K²¡"~~~µµµ3En+1{O}¥Äd'X∼:x∼y=3"~êλ¦y=λx,Kda´L:"ûmRPn=En+1{O}/∼¡n¢Km"Ä4N

5、i:Sn→En+1,þ¡ÊÜNEÜûNf:Sn→RPn.¿f(x)=f(y)=x=±y.ÏdRPnÓuSnÊÜé»:£c~½Â¤"~~~µµµòE2n+2¥:w´EþmCn+1¥¿Äd'X∼:x∼y=3"Eêλ¦y=λx"ûmCPn=E2n+2{O}/∼¡nEKm"gKµCPn´2nÿÀ6/"CP1=∼S2.>>>666///½½½ÂµµµeHausdorffmMz:kÓuEn={(x1,...,xn)∈En

6、xn≥0},K¡X+n£>¤6/"∂M={x∈M

7、xvkÓuEn}¡M>.§Ù¥:

8、¡>.:§¡M∂MMSܧ٥:¡S:"ÚÚÚnnnµµµx∈∂M=xkÓuEn,¿+TÓòxC¤(0,...,0).5¿µù(ئ^êÿÀâUy²§6k«@§~~~µµµ∂En={(x1,...,xn)∈En

9、xn=0}.++íííØØصµµn>16/M>.∂M´>.8n−16/"µ6/¶¡µ6/¶46/µ>.8;6/§ØAO(²·ob46/ëÏ"gggKKKµµµ;>6/Ýþz"5¿µ¢Sþ?Û÷v1êún6/þÝþz§¿?Ûn;>6/±iE2n+1.~~~µµµ¥¡S2§¡T2=∼S1×S1Ñ´4¡"

10、²S1×[0,1]´±ü±>.¡"M¨obius´±±>.¡"gµP86:9,13,P87:14.