lesson12(映射的同伦)

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1、111nnnµµµNNNÓÓÓÔÔÔ-NÓÔ-'u+~^£NNNÓÓÓÔÔÔ½½½ÂµµµPC(X,Y)¤kXYëYN8Ü"f,g∈C(X,Y),ekëYNH:X×I→Y,¦H(x,0)=f(x),H(x,1)=g(x)K¡§£gd¤ÓÔ§¡H:f'gÔ£§¡ht:X→Y,x7→H(x,t)Ô£t-¡"Ô£þÒ´Nm¥´,zéAN¡¡",«EÔ£g´´Ñ,§=H(x,·)/¤´"ézxxÑf(x)g(x)´§Ü3å±ÏEÔ£"~~~µµµef,g∈C(X,Y),Y´En¥àf8§K±ÓÔH(x,t)=(1−t

2、)f(x)+tg(x).~~~µµµef,g∈C(X,Sn),∀x∈X,f(x)6=−g(x),(1−t)f(x)+tg(x)KÓÔH(x,t)=.

3、

4、(1−t)f(x)+tg(x)

5、

6、~~~µµµef,g∈C(X,S1),∀x∈X,f(x)=−g(x),KÓÔH(x,t)=eiπtf(x).5¿òS1¤S2,·KÒؤá"···KKKµµµÓÔ´C(X,Y)þd'X§da8[X,Y]¥¡Na"yyy²²²µµµ5µH(x,t)=f(x),KH:f'f.é¡5µeH:f'g,H(x,t)=H(x,1−t),KH:g'f.D45µeH0:g'h,H(x,2t),et≤

7、1H00(x,t)=2H0(x,2t−1),et≥12KdÊÚnHëY§lH00:f'h.···KKKµµµef0'f1:X→Y,g0'g1:Y→Z,Kg0◦f0'g1◦f1:X→Z.yyy²²²µµµF:f0'f1,G:g0'g1,H(x,t)=G(F(x,t),t),KH´¤¦Ô£"ùpNX×I→Y×I,(x,t)7→(F(x,t),t)¡FÎz"efÓÔu~N§K¡f"Ô"½½½ÂµµµA⊆X,f,g∈C(X,Y),ekëYNH:X×I→Y,¦H(x,0)=f(x),H(x,1)=g(x),¿∀a∈A,t∈I,H(a,t)=f(a)=g(a)K¡§éuf8AÓ

8、Ô§¡H:f'grelAéuf8AÔ£"···KKKµµµA⊆X,KéuAÓÔ´C(X,Y)þd'X£y²ÓgdÓÔ¤"···KKKµµµef0'f1:X→YrelA,g0'g1:Y→ZrelB,¿f0(A)⊆B,Kg0◦f0'g1◦f1:X→ZrelA£y²ÓgdÓÔ¤"~~~µµµ½x∈X9y∈Y,·òC(X,Y)¥3xþyNéuf8{x}£é¤ÓÔaP[(X,x),(Y,y)].Ä"¥¡S0={−1,1},K?x0∈X,[(S0,1),(X,x0)]X´ëÏ©

9、éA"3Sn¥½:P,ÏL3[(Sn,P),(X,x0)]þE+(¿±5ïÄX{Ò´ÓÔØ

10、"n=1ÿùÒ´Ä+"5¿zr1xx0ëYNa∈C(S1,X)éA±x0Ä:4´ae:I→X,÷vei2πea(t)=a(t),^4´Ø´(S1,1)5?Ø"'''uuu+++~~~^^^£££Ä+nØÌ{´r'uÿÀmÚëYN¯K=z'u+ÚÓ¯K"ÿÀméAVg´+§=38ÜGþ5½$/¦{0§÷v-(ÜƵ∀x,y,z∈G,(xy)z=x(yz);-küµ∃1∈G¦∀x∈G,1x=x1=x;-±¦_µ∀x∈G,∃x∈G¦xx=xx=1.ëYNéAVg´£+¤Ó§=++Nf:G→H,÷vf(xy)=f(x)f(y).Óé

11、AVg´£+¤Ó§=++éAf:G→H,¦f9f−1Ñ´Ó"d∼PG=H.XJ+¥/¦{0÷vÆxy=yx,K¡+"+¥¦{U¡/{0§Ù¥üU¡/"0"~~~µµµkÌ+Zn={0,1,...,n−1},ùpx,y/¦È0½/{0½Âx+y£x+y

12、f¦È¤G5f+G0,¡f+"û+G/G0´+§¡Gz"¡f+N5´∀x∈N,y∈G,okyxy−1∈N.dPyN={yx

13、x∈N}={xy

14、x∈N}=Ny,KG¤kL«¤yN/ªf8¤+G/N,¡û+"kkk)))¤¤¤+++ÚÚÚ©©©)))½½½nnnµµµ?¿k)¤+HÓuÌ+Ú§=k©)ªMMMMM∼H=Z···ZZk1···Zkn

15、{z}rankH£Ù¥zki∈N