Middle East Technical University Fall 2002 Math 744 Fundamental Techniques in Differential Topology

PROBLEM LIST III

Pontryagin-Thom construction

Problem 1. A regular fiber p^{−1}(y) for a map p: S^{n+2} → S^{n}, n ⩾ 4, is homeomorphic to S^{2}. Deduce from the Pontryagin theorem that p is null-homotopic.

Problem 2. Let f: S^{n+3} → S^{n}, n ⩾ 2, and M = f^{−1}(S^{1}), a regular pull-back of a standard circle. Show that M is not homeomorphic to (a) ℝP^{4}, (b) ℂP^{2}.

Problem 3. Given f: S^{n+4} → S^{n}, n ⩾ 1, show that a regular fiber f^{−1}(y), y ∈ S^{n}, cannot be (a) ℝP^{4}, (b) ℂP^{2}, (c) ℂP^{2}#\overline{ℂP}^{2}.

Problem 4. Given f: S^{n+1} → S^{n}, S^{n+2} → S^{n} and a degree 2 map h: S^{n} → S^{n}, prove that h ∘ f and h ∘ h are null-homotopic for sufficiently big n. Specify the estimate on n.

Problem 5. Let H_{n}: S^{3} → S^{2} denotes a map with the Hopf invariant n. Show that a CWcomplex X obtained by gluing a pair of 3-cells to S^{2} along H_{2} and H_{3}, is homotopy equivalent to ℂP^{2} ∧ S^{3}.

Problem 6. Let f_{n} = Σ^{n−2}f_{2} : S^{n+1} → S^{n} be the suspension over the Hopf map f_{2} and g_{n} = f_{n} ∘ f_{n+1} : S^{n+1} → S^{n}, h_{n} = f_{n} ∘ f_{n+1} ∘ f_{n+2} : S^{n+3} → S^{n}.

(1) Describe regular fibers of f_{n}, g_{n} and h_{n}.

(2) Show that g_{n} represent a generator of π_{n+2}(S^{n}) = ℤ/2 for sufficiently big n (specify how big should be n).

Obstruction Theory

Problem 7. Determine an obstruction to extend a degree n map f: ℂP^{1} → ℂP^{1} to a map F : ℂP^{2} → ℂP^{1}.

Problem 8. Let f_{n}: ℂP^{2} → ℂP^{2}, [z_{0} : z_{1} : z_{2}] ↦ [z_{0}^{n} : z_{1}^{n} : z_{2}^{n}].

(1) Determine f_{n}^{∗}: H^{∗}(ℂP^{2}) → H^{∗}(ℂP^{2}).

(2) Describe the first obstruction o(f_{n}, f_{m}).

(3) determine the Euler number of the bundle f_{n}^{∗}ξ|_{ℂP^{1}}, induced from the canonical bundle ξ over ℂP^{2}.

(4) Describe the bundles f_{n}^{∗}(T)|_{ℂP^{1}} and f_{n}^{∗}(det)|_{ℂP^{1}}, where T and det are respectively the tangent and the determinant bundles on ℂP^{2}.

Typeset by AMS-TEX