Math 872 Algebraic Topology Homework # 2 Starred (*) problems due Thursday, February 1

6. Show that if there are maps f : X → Y and g, h : Y → X such that f ∘ g : Y → Y and h∘f : X → X are homotopic to I_{Y} and I_{X}, respectively, then f is a homotopy equivalence. Show that the same is true if f ∘ g and h ∘ f are homotopy equivalences.

(*) 7. Show that if X is a CW complex, A, B ⊆ X are subcomplexes, A ∪ B = X, and A, B and A ∩ B are contractible, then X is contractible.

8. Show that if f : ∂D^{n} → X is the attaching map of an n-cell D^{n}, with n ≥ 3, then the inclusion X ↪ X ∪_{f} D^{n} induces an isomorphism on π_{1}. Show that the same is true if we attach any (finite or infinite) collection of n ≥ 3 cells.

(*) 9. Let X = the space obtained from a cube J^{3} = J × J × J, J = [−1, 1], by gluing opposite square faces to one another with a 90-degree righthand twist (e.g., glue J × J × {−1} to J × J × {1} by the map (x, y, −1) ↦ (y, −x, 1). Describe a CW structure for X and compute a presentation for π_{1}(X).

10. Starting with a 2-disk, D^{2}, with two small sub-disks deleted, show that there are essentially only two spaces obtained by identifying each of the two interior boundary circles to ∂D^{2} by homeomorphisms. Compute presentations for the fondamental groups of each, and (by abelianizing) show that the two groups are not isomorphic, so the two spaces are not homotopy equivalent.