lesson15 (van kampen 定理)

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1、111888-ûm/CÂ-+LÑ-VanKampen½nûûûmmm///CCC···KKKµµµf:X→Y´ûN§H´XA⊆X£r¤/C§¿f(x)=f(x0)§∀t∈I,f(H(x,t))=f(H(x0,t)),K3Yf(A)£r¤/CÂ"yyy²²²µµµÄG:Y×I→Y,(y,t)7→f(H(x,t))£ùpx∈f−1(y)¤"KG◦(f×idI)=f◦H,lGëY"´yG´£r¤/CÂ"~~~µµµM¨obiusXdÝ/ò

2、é>ÊÜѧ±ddXÙr/

3、CÂ"~~~µµµ¡T2´ò/I2é>Êܤûm"ddÑK:¡/o^>ÊÜûmS1∨S1r/CÂ"Ón4¡MS2K:r/CÂW2−χåS1,Ù¥χÙEulerê"i=1~~~µµµn>0,A=Dn×{0}∪Sn−1×I´X=Dn×Ir/CÂØ"ÂN´'u(0,2)¥%ÝK"~~~µµµf'g:Sn−1→X,H:Sn−1×I→X´fgÔ£§KX∪fDnÚX∪gDnÑ´X∪H(Dn×I)r/CÂØ"ÏdX∪fDn'X∪gDn.+++LLLÑÑÑ)¤XCµG

4、zÑ´C¥9Ù_1k¦È¤µg=c···c,ci∈C,i=±1.1k1kiwµ/ªLªw=c···c.1kzihwiµ3w¥ØäíØc−1−1ic½cciii¤(J£3¤"C)¤gd+µzi¤+§iü§Ä2z¦{"eN´Gf+§∀x∈G,8xN=Nx,K¡N5f+§Ï~PNCG.R⊆G,P¹R5f+[R],§z1x−1···xn−1Ñäkx1r11nrnxn/ª§Ù¥xi∈G,ri∈R,i=±1.eNCG,K¤k88G/N={xN

5、x∈G}

6、´+§¡û+"+LѵF´±C)¤Xgd+"R⊂F´

7、zi§N´F¥¹R∼5f+"eG=F/N,K¡

8、R>GLѧC¥¡)¤§R¥¡'X"C,Rþk¡GkLÑ+"~~~µµµkgd+Fn=

9、>,n¡∼Ù§5¿ØF1=ZÙ§þ"~~~µµµkÌ+Zn=

10、cn>.5¿µØJÑ+Lѧ´Ó+±kéõ«ØÓLѧ~X

11、a2,b3,aba−1b−1>=∼

12、a6>=∼Z6.~~~µµµe+GkLÑ

13、R>,KGzkLÑ

14、R∪C0>,

15、Ù¥C0={[a,b]

16、a,b∈C}.ü+G1=

17、R1>,G2=

18、R2>gdȵG1∗G2=

19、R1∪R2>.VanKampen½½½nnnvanKampen½½½nnnµµµX,Y´Zmf8§Z=X∪Y,A=X∩Y´ëÏ"i:A,→X,j:A,→Y´¹N"3A¥½Ä:x0,¿N´gdÈπ1(X,x0)∗π1(Y,x0)¥¹¤k/Xiπ(α)jπ(α−1)(α∈π1(A,x0))5f+§Kπ1(Z,x0)=π1(X,x0)∗π1(Y,x0)/N.`²µ(1)

20、X,YZ¹pÓφ:π1(X,x0)∗π1(Y,x0)→π1(Z,x0).(2)Z¥´þ¤a=a1···ak,zai´X½Y¥´à:3A¥"ddφ÷"(3)?X,Y9A¥´aX,aY9a÷vaX∈iπ(hai),aY∈jπ(hai),3Z¥ww,a−1Xa"Ô"ddkerφ⊇N.Y(4)kerφ⊆N,E,"ëN¹B.kkkLLLÑÑÑ+++///µµµXJ∼π1(X,x0)=

21、r1,...,rp>,∼π1(Y,x0)=

22、s1,...,sq>

23、,π1(A,x0)±c1,...,ck)¤,K∼π1(Z,x0)=

24、r1,...,rp,s1,...,sqiπ(c1)jπ(c1)−1,...,iπ(ck)jπ(ck)−1>AAA~~~1:cãúª¥XJAüëϧK∼π1(Z,x0)=

25、r1,...,rp,s1,...,sq>AAA~~~2:cãúª¥XJYüëϧK∼π1(Z,x0)=

26、r1,...,rp,iπ(c1),...,iπ(ck)>½½½nn

27、nµµµòX,Y^4f8§¿?Ú¦A=X∩Y´§3Z¥mr/CÂاKvanKampen½n(ØE¤á"~~~µµµS1∨S1©)¤üÓÔduS1mf8X,Y¿§i/A=X∩Y§π111(S∨S,z0)=

28、>.~~~µµµ?ÛkLÑ+

29、r1,...,rp>Ñ´,ÿÀmÄ+"yyy²²²µµµÄkåX110=S

30、∨···∨{zS},P´nú
:§¿π1(X0,P)=