1701044725
1701044726
4.求下列空间的基本群:
1701044727
1701044728
(1)E2中去掉3个点;
1701044729
1701044730
(2)S2中去掉3个点;
1701044731
1701044732
(3)T2上去掉3个点.
1701044733
1701044734
5.求下列空间的基本群:
1701044735
1701044736
(1)E3中去掉2条不相交直线;
1701044737
1701044738
(2)E3中去掉3条坐标轴;
1701044739
1701044740
(3)“田”字形.
1701044741
1701044742
6.把三角形的三条边按图4-30所示方式粘接在一起.求所得商空间的基本群.
1701044743
1701044744
7.证明:如果曲面M与N同胚,则它们的边界也同胚.并由此说明Möbius带与平环不同胚.
1701044745
1701044746
1701044747
1701044748
1701044749
图4-30
1701044750
1701044751
8.设f:D2→E2连续.证明在下列条件之一成立时,f有不动点:
1701044752
1701044753
(1)f(S1)⊂D2;
1701044754
1701044755
(2)∀x∈S1,f(x),x与原点不共线;
1701044756
1701044757
1701044758
(3)∀x∈S1,线段过原点.
1701044759
1701044760
9.设f:D2→D2连续,并且S1上每一点都不动,证明f是映满的.
1701044761
1701044762
1701044763
1701044764
1701044765
1701044766
10.记是E3中以(i,0,0)为球心,为半径的球面,(图4-31).证明X单连通.
1701044767
1701044768
1701044769
1701044770
1701044771
图4-31
1701044772
1701044774
*§6 Jordan曲线定理
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