lesson14(圆周的基本群，同伦等价)

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3、p:E1→S1´þãnz§K?S1þ´a:I→S1,9x∈p−1(a(0)),3±x©:J,´˜a:I→E1.J,´35±dE1S1ÛÜÓÑ"ekü÷v¦J,˜a,˜b,Kd:[0,1]→Z,t7→˜a(t)−˜b(t)ëY§Ïdð""½½½ÂµµµéuS1þ?Û±1Ä:4´a,˜a´±0©:J,§¡q(a)=˜a(1)−˜a(0)∈Zaê"ÚÚÚnnnµµµS1þêÓ4´½àÓÔ"yyy²²²µµµeq(a)=q(b),Ka,b±0©:J,´˜a,˜bª:Ó"E1üëÏÏdh˜aih˜bi−1"Ô§¤±haihbi−1"Ô"ÚÚ

4、ÚnnnµµµS1þ½àÓÔ4´êÓ"yyy²²²µµµhsÚht¿©C§J,A¿©C§lJ,êA¿©CÖ§=XJk½àÓÔht:a'b,K4´htê(∈Z)·A´tëY¼ê§Ïd´~"5µî`{A´µnzÓÔJ,5`²§HkJ,H˜,Ïdq(t)=H˜(1,t)−H˜(0,t)´[0,1]ZëY¼ê"½½½nnnµµµ±S1Ä+´gdÌ+§êNq:π1(S1,1)→Z´Ó"yyy²²²µµµIÑJ,´§y4´¦ÈêuêÚ"5¿Z¥/¦{0´¦Ú"~~~µµµòaq{A^nzp:E2→S1×S1þ´y

5、¡Ä+π2∼1(T))=Z⊕Z.~~~µµµòaq{A^Ýp:S2→RP2þ´y¢K²¡Ä+π2∼1(RP))=Z2.ÄÄÄ+++ÓÓÓÔÔÔØØØCCC555f'g:X→Y,Ô£´H.½X¥Ä:x0,¡Y¥´w:I→Y,t7→H(x0,t)x03Hel½,"Pγ=.½½½nnnµµµüÓÔN¤pÄ+ÓmÓµgπ=γ#◦fπ:π1(X,x0)→π1(Y,g(x0)).yyy²²²µµµIy?X¥±x0Ä:´a,g◦a'w(f◦a)w.ëYNF:I2→Y,·F(t,s)=H(a(t),s),¿3I2¥½´(

6、0,1−3t),t≤13u(t)=(3t−1,0),1≤t≤2v(t)=(t,1).33(1,3t−2),t≥23ÏI2E2àf8§¤±u'v,lg◦a=·F◦u'F◦v=w(f◦a)w.·½½½ÂµµµXYÿÀm§XJ3ëYNf:X→Y,g:Y→X,¦g◦f'idX:X→X,f◦g'idY:Y→Y,K¡f:X'YÓÔd§¡gfÓÔ_"Ów,´ÓÔd§´ÓÔdÓÔ_ا§¤Na£SK3¤"½½½nnnµµµÓÔd´ëÏmÄ+Ó'.Óµef:X−→Y´ÓÔd§Kfπ:π1(X,x0)→π1(Y,f(x0))´Ó"yyy²²²µµµ

7、g´fÓÔ_§Kg◦f'idX,lgπ◦fπ=(g◦f)π:π1(X,x0)→π1(X,g◦f(x0))(idX)π=idπ=Ó§ù`1(X,x0)²gπ◦fπ´V§lfπügπ÷"Óndf◦g'idYgπüfπ÷"fπ´Ó"///CCC½½½Âµµµi:A,→X´¹N§XJ3ÂNr:X→A£=r◦i=idA¤§¦i◦r'idX,K¡AX/CÂØ"eÔ£H:i◦r'idX÷v∀a∈A,H(a,t)≡a,K¡AXr/CÂØ"/CÂ'Ó