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Problems IV (29.02.2012) The following facts are considered known in this problem set:
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The Euler characteristic of a compact surface S is de ned as V - E + F where V is the number of vertices and E is the number of edges of a graph embedded into S so that S \ is homeomorphic to a disjoint union of F disks (faces). The Euler characteristics depends on the surface S only and does not depend on the graph . A closed surface (i.e. a compact surface without boundary) is homeomorphic to either a sphere with g handles (a connected sum of a sphere with g tori) or to a sphere with g M obius bands attached (a connected sum of a sphere with g copies of RP 2 ). A compact surface with boundary is homeomorphic to a closed surface with a nite number of disks deleted. Find (Kl), where Kl is the Klein bottle. Prove that (M1 #M2 ) = (M1 ) + (M2 ) - 2. Prove that (mT 2 ) = 2 - 2m and (nRP 2 ) = 2 - n. Determine the type of the surface shown in Fig. 16 [page 31 of the Prasolov{Sossinsky \Topology-I"

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book]. (a) for n = 3; (b) for arbitrary n 2. IV.5. Prove that the surface shown in Fig. 17 [page 31 of the Prasolov{Sossinsky \Topology-I" book] is homeomorphic to the torus from which a disk has been removed. IV.6. Consider the quotient space (S 1 в S 1 )=(x; y) (y; x). Prove that this space is a surface. Which one? IV.7. (a) Prove that any closed nonorientable surface is homeomorphic to one of the surfaces in the following list: RP 2 (pro jective plane), RP 2 #RP 2 (Klein bottle), . . . RP 2 #RP 2 # : : : #RP 2 ; . . . . (b)* Any two distinct surfaces in the list are not homeomorphic. IV.8. Prove that a closed orientable surface is not homeomorphic to a closed nonorientable surface. IV.9. What orientable surfaces can be obtained by identifying the sides of a regular octagon? For a given g nd how many are there ways to identify the sides so as to obtain a sphere with g handles. IV.10. Prove that on the sphere with g handles the maximal number of nonintersecting closed curves not dividing this surface is equal to g. IV.11. Draw four closed curves issuing from a common point on the sphere with two handles so that cutting along these curves produces an octagon (a topological disk). IV.12. Prove that the standard circle can be spanned by a M obius band, i.e. there exists a subset M R3 of the 3-space homeomorphic to the M obius band and such that its boundary @ M is a circle (lying in some plane). IV.13. Prove that the boundary of Mb в [0; 1], where Mb is the M obius band, is the Klein bottle. IV.14. Give an example of two surfaces-with-boundary M and N that are not homeomorphic but M в [0; 1] and N в [0; 1] are homeomorphic. IV.15. Two three-dimensional disks D3 , i = 1; 2, are glued together by a homeomorphism h of their boundary i spheres. Find the resulting quotient space D3 h D3 if 1 2 (a) the homeomorphism h is the identity; (b)* the homeomorphism h is the symmetry w.r.t. the equatorial plane? IV.16. * Two solid tori D2 в S1 and S1 в D2 with coordinates (r; '; ) and ( ; r; '), where (r; ') are polar coordinates in D2 , are glued together by identifying their boundaries according to the rule (1; '; ) ('; 1; ). Prove that the quotient space obtained is S3 .

IV.1. IV.2. IV.3. IV.4.

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