Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/ium/postscript/s12/T1-Sem3.pdf Дата изменения: Fri Mar 9 10:26:14 2012 Дата индексирования: Mon Feb 4 10:50:05 2013 Кодировка: Windows-1251 Поисковые слова: south pole

identication of points of the square 0x, y1 belonging to its sides: (x; 0) (x; 1) and (0; y) (1; y ). (This space is called the torus.) III.3. Prove that the following spaces are homeomorphic: (a) the set of lines in Rn+1 passing through the origin; (b) the sphere S n with identi ed diametrically opposite points (every pair of diametrically opposite points is identi ed); (c) the disc Dn with diametrically opposite points of the boundary sphere S n-1 = @ Dn identi ed. III.4. Prove that the following spaces are homeomorphic: (a) the set of complex lines in Cn+1 passing through the origin; (b) the sphere S 2n+1 Cn+1 with identi ed points of the form x for every C, || = 1 (for any xed point x S 2n+1 ); (c) the disc D2n Cn with points of the boundary sphere S 2n-1 =@ D2n of the form x for every C, ||=1 identi ed (for any xed point xS 2n-1 ). III.5. Prove that C Dn Dn+1 and Dn Dn+1 . III.6. Prove that C S n Dn+1 and S n S n+1 . III.7. Prove that RP 1 S 1 and CP 1 S 2 . III.8. Prove that S n S m S n+m+1 . III.9. Prove that Rn \ Rk S n-k-1 Ч Rk+1 , ЗДЕ Rk Rn is the set {(x1 ; : : : ; xk ; 0; : : : ; 0)}. III.10. Prove that S n+m-1 \ S n-1 Rn Ч S m-1 where S n-1 S n+m-1 is standard: S n+m-1 = {(x1 ; : : : ; xn+m ) | x2 + ћ ћ ћ + x2 +m = 1} and S n-1 = {(x1 ; : : : ; xn ; 0; : : : ; 0) | x2 + ћ ћ ћ + x2 = 1}. 1 n 1 n p Ч S q )=(S p S q ) S p+q . III.11. Prove that (S III.12. Prove that T 2 #RP 2 3RP 2 . III.13. (a) Prove that Kl#Kl is homeomorphic to the Klein bottle with one handle attached. (b) Prove that RP 2 #Kl is homeomorphic to the pro jective plane with one handle attached. III.14. Prove that if a surface M1 is nonorientable, then for any surface M2 the surface M1 #M2 is nonorientable. III.15. (a) Prove that the two surfaces-with-holes obtained from the same closed triangulated surface by removing two di erent open 2-simplices from it are homeomorphic. (b) Show that the connected sum of surfaces is well de ned. III.16. Let I = [0; 1]. Prove that the space S 1 ЧI is not homeomorphic to the M band. obius

III.1. Prove that Dn =@ Dn S n . III.2. Prove that the space S 1 ЧS 1 is homeomorphic to the space obtained by the following

Problems III (22.02.2012)

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