Middle East Technical University Fall 2000 Math 537 Algebraic Topology I

HOMEWORK I

HOMOTOPY AND CW-COMPLEXES

Problem 1. Prove that the following conditions are equivalent (1) X is contractible,

(2) any map Y → X is null-homotopic,

(3) any map X → Y is null-homotopic.

Problem 2. Consider space X = S^{2}/p = q obtained by gluing together a pair of points, p and q, on a sphere S^{2}. Then an arc, a ⊂ S^{2} connecting p and q becomes a loop ā ⊂ X. Attaching a 2-cell along ā we obtain a 2-complex Y. Describe a simplest (minimal) homotopy model of Y (i.e., a CW complex homotopic to Y with the minimal number of cells).

Observe the relation between Y and S^{1} ∧ S^{1}.

Problem 3. Given any map f: X → Y and a space Z, (1) define the induced map f_{∗}: π(Z, X) → π(Z, Y).

(2) Prove that f_{∗} = g_{∗} if f and g are homotopic.

(3) Deduce from the latter that f_{∗} is a bijection if f is a homotopy equivalence.

(4) What can be said about f_{∗} if f is a retraction ?

Problem 4. Prove that X ∗Y is homotopy equivalent to S(X ∧Y). Hint: construct a homeomorphism between X ∗ Y/X ∗ {y_{0}} ∪ {x_{0}} ∗ Y and S(X ∧ Y)/S({x_{0}} ∧ {y_{0}}) using that both X ∗ Y and S(X ∧ Y) are quotients of X × Y × I.

Problem 5. Prove that Z = X ∪ Y is contractible provided CW-complexes X, Y as well as their common CW-subcomplex X ∩ Y are contractible.

Problem 6. By definition, the infinitely-dimensional sphere S^{∞} is the union ⋃_{n>0} S^{n}. Prove that S^{∞} is contractible. Define the contraction either by formulae, or describing the trajectories of the points of S^{∞} in the process of contraction.

Problem 7. Prove homotopy invariance of the constructions of a product X × Y, join X ∗Y, wedge of spaces X ∨Y, smash-product X ∧Y and suspension SX. More precisely, prove that

(1) The constructions involving basepoints do not depend of its choice within a path connected component of the space. (So, it makes sense to omit the basepoint in the notation, at least for path connected spaces.)

(2) If X, Y are replaced by homotopy equivalent spaces, then the results of all the constructions will be the same up to homotopy equivalence.

Problem 8. Assume that CW complexes X, Y are n-connected. Prove that (1) X × Y is n-connected,

(2) X ∨ Y is n-connected,

(3) the suspension S(X) is (n + 1)-connected,

(4) X ∧ Y is (2n)-connected,

(5) X ∗ Y is (2n + 1)-connected. Hint: you may use the results of the previous problem.

Problem 9. Assume that X, Y are CW complexes, A ⊂ X is a subcomplex, dim A ⩽ n and Y is n-connected. Prove that X ∪_{f} Y ≅ (X/A) ∨ Y for any mapping f: A → Y.