Download 2. Homeomorphisms and homotopy equivalent spaces. (14 October

2. Homeomorphisms and homotopy equivalent spaces. (14 October)
Problem 1. Prove that the following spaces are homeomorphic:
a) the set of lines in Rn+1 passing through the origin (this topological space is called real
projective space and denoted by RP n );
b) the set of hyperplanes in Rn+1 passing through the origin;
c) the sphere S n with identified diametrically opposed points (every pair of diametrically
opposed points is identified);
d) the disc Dn with identified diametrically opposed points of the boundary sphere S n−1 =
∂Dn .
Problem 2. Prove that RP 1 ≈ S 1 .
Problem 3. Prove that the space S 1 × I 1 is not homeomorphic to S 1 .
Problem 4. Classify (without proof) the capital letters of Latin alphabet up to homeomorphism
equivalence.
Problem 5. (Extra.) Prove that the space S 1 × I 1 is not homeomorphic to the Möbius band.
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We say that two topological spaces are homotopy equivalent if there exist continuous maps
f : X → Y and g : Y → X such that g ◦ f is homotopic to the identity map of X and f ◦ g is
homotopic to the identity map of X.
Problem 6. Prove that the spaces S 1 ∨ I and S 1 are homotopy equivalent.
Problem 7. a) Prove that if the image of a map f : X → S 1 is not the whole space S 1 (i.e.,
the map is not onto) then f is homotopic to a constant map.
b) Prove that if a map f : X → S n is not onto then f is homotopic to a constant map.
Problem 8. Prove that the following spaces are homotopy equivalent: a) the sphere S 2 with
two points identified; b) the sphere S 2 with one diameter; c) S 1 ∨ S 2 .
Problem 9. Classify (without proof) the capital letters of Latin alphabet up to homotopy
equivalence.
Problem 10. Prove that the Möbius band is homotopy equivalent to the circle.
Problem 11. Prove that the space S 1 × I 1 is homotopy equivalent to the Möbius band.
Problem 12. (Extra.) Prove that the spaces R3 \ S 1 and S 2 ∨ S 1 are homotopy equivalent.