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INIDEPENDENT UNIVERSITY OF MOSCOW

5. HOMOTOPIES

TOPOLOGY-1, SPRING 2012 Г.

Problem 7. Let Xn be a wedge product of n circles, f
gn;k : Xk S 1

(b) the union of the sphere S 2 and one of its diameters; (c) S 1 S 2 . Problem 2. (a) Prove that the spaces R3 \ S 1 and S 2 S 1 are homotopy equivalent; here S 1 R3 is the standard circle lying in the xy coordinate plane. (b) Let X be the space R3 from which n copies of S 1 lying in parallel planes are deleted. Prove that X is homotopy equivalent to the wedge product of n copies of the space S 2 S 1 . Problem 3. Let L = {(x; y; 0) | x2 + y2 = 1} {(0; y; z ) | (y - 1)2 + z 2 = 1} R3 be a union of two circles linked in the simplest way. Prove that the spaces R3 \ L and S 2 T2 are homotopy equivalent. For every continuous map f : S 1 S 1 one can de ne an integer deg(f ) so that the following statements are true: · Maps f1 and f2 are homotopic if and only if deg(f1 ) = deg(f2 ). · If fn : S 1 S 1 (n Z) is the map de ned by the formula f (') = n' (where ' is the central angle on the circle) then deg(f ) = n. In particular, if f is a constant map then deg(f ) = 0; if f is the identity map then deg(f ) = 1. Problem 4. Let f ; g : S 1 S 1 be continuous maps. Prove that deg(f g) = deg(f ) deg(g). Problem 5. (a) Prove that if a map f : X S n is not onto, then f is homotopic to a constant map. (b) Prove that if for a map f : S 1 S 1 there exists c S 1 such that f -1 (c) consists of n points then |deg(f )| n. Problem 6 (a hint for Problem 7). (a) Let f0 f1 and g0 g1 where f0 ; f1 : X Y and g0 ; g1 : Y Z are continuous maps. Prove that g0 f0 g1 f1 . (b) Let X1 X2 and Y1 Y2 . De ne a one-to-one correspondence EqY11;Y22 between the homotopy classes of continuous maps X1 Y1 and X2 Y2 . Check the following properties of X ;X the correspondence de ned: if [f2 ] = EqY11;Y22 ([f1 ]) and [f3 ] = EqY22;Y33 ([f2 ]) then [f3 ] = EqY11;Y33 ([f1 ]) ([f ] means a X ;X X ;X X ;X 1 ;Z homotopy class of the map f ), and if [f2 ] = EqY11;Y22 ([f1 ]) and [g2 ] = EqZ1 ;Y22 ([g1 ]) then [g2 f2 ] = EqZ11;Z22 ([g1 f1 ]). X ;X Y X ;X : S 1 Xn be a map sending S 1 homeomorphically to the k-th circle of the wedge, and be a map sending the k-th circle of Xn homeomorphically to S 1 and all the other circles, to a point. (a) Prove that fn;k and gn;k are not homotopic to the map to a point. (b) Prove that Xn for n > 1 is not homotopy equivalent to a circle. (c) Prove that Xn1 is not homotopy equivalent to Xn2 is n1 = n2 . Hint. Consider a composition gn;k1 fn;k2 : S 1 S 1 and use the results of Problems 4 and 6. Problem 8. (a) Prove that the map f : [0; 1) {z C | |z | = 1} = S 1 de ned by the formula f (t) = exp(2it) is continuous and one-to-one, but is not a homeomorphism. (b) Prove that a continuous one-to-one map of compact topological spaces is a homeomorphism. Problem 9. Prove that the sphere with g handles from which a point has been removed is homotopy equivalent to the wedge product of n copies of the circle and nd n. (a) Prove that the sphere with g1 handles is not homeomorphic to a sphere with g2 handles if g1 = g2 . Problem 10. Prove that A X is a retract of X if and only if any continuous map f : A Y can be extended to X . Problem 11. Prove that if any continuous map X X has a xed point and A X is a retract of X , then any continuous map A A also has a xed point. Brouwer Fixed Point Theorem says that any continuous map f : Dn Dn has a xed point. Problem 12. Prove that the following assertions are equivalent to the Brouwer theorem: (a) There is no retraction r : Dn S n-1 . (b) Let v(x) be a continuous vector eld on Dn such that v(x) = x for any point x @ Dn = S n-1 . Then v(x) = 0 for some point x Dn . Problem 13. Is there a continuous map f : X X without xed points if X is (a) a 2-sphere with n holes; (b) a 2-sphere with g handles and a hole?
n;k

Problem 1. Prove that the following spaces are homotopy equivalent: (a) the sphere S 2 with two points identi ed;

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