Middle East Technical University Spring 1996 Math 505 Differentiable Manifolds

FINAL EXAM

JUNE 25

TIME: 4 HOURS

Problem 1. (4 pts) Prove that a continuous map f: ℝp^{n} → M is smooth if and only if the product f ∘ p: S^{n} → M with the projection S^{n} → ℝp^{n} is smooth.

Problem 2. (4 pts) Write down in cylidrical coordinates in ℝ^{3} (1) 1-form ω = z dx − y dy,

(2) vector field V (x, y, z) = z \frac{d}{dx} − y \frac{d}{dy}.

Problem 3. (3 pts) Let ω ∈ Ω^{1}(S^{2}) be the restriction of dx + dy − dz ∈ Ω^{1}(ℝ^{3}) to the unit sphere. Determine at which points ω = 0.

Problem 4. (3 pts) Consider a proper submersion f: M → N and a form ω ∈ Ω^{m−n}(N), where m = dim(M), n = dim(N). Assuming that M is orientable,

prove that ∫_{f^{−1}(x)} f^{∗}ω = 0 for any x ∈ N.

Problem 5. (3 pts) Consider a n-form ω on a n-dimentional manifold M and assume that H ⊂ M is the zero set of ω. Prove that M − H is an orientable n-dimentional manifold.

Problem 6. (20 pts) Assume that τ: M → M is a smooth fixed point free involution (i.e. τ ∘ τ = id) on a manifold M. Consider the quotient space N = M/τ.

(1) (2 pts) Prove that N is a manifold.

(2) (3 pts) Provide N with a differential structure so that the quotient map f: M → N is smooth.

(3) (2 pts) Prove that f^{∗}: H_{DR}^{0}(N) → H_{DR}^{0}(M) is isomorphism.

(4) (6 pts) Prove that the map f^{∗}: H_{DR}^{k}(N) → H_{DR}^{k}(M) is monomorphism for any k ⩾ 0. Hint: Show that if ω = dη ∈ Ω^{k}(M) is τ^{∗}-invariant (that is τ^{∗}(ω) = ω), then η can be made τ^{∗}-invariant. Then show that for any τ^{∗}-invariant form η in M there exist a form α in N such that η = f^{∗}(α).

(5) (3 pts) Assuming that M is oriented and τ is orientation preserving, provide N with an orientation so that f is orientation preserving.

(6) (4 pts) Prove that f^{∗}: H_{DR}^{n}(N) → H_{DR}^{n}(M) is an isomorphism if n = dim M and the assumption of (5) is satisfied.

Problem 7. (3 pts) Prove that H_{DR}^{1}(ℝp^{2}) = 0. Hint: use one of the results of the previous problem.

Problem 8. (10 pts) Assume that M is a compact smooth orientable manifold of dimension m, and f: M → N is a smooth map. Determine in which of the following cases we can guarantee that ∫_{M} f^{∗}ω = 0 for any closed form ω ∈ Ω^{m}(N). Make your answer argumented.

(1) (3 pts) N = ℝ^{n},

(2) (3 pts) M is a ball B^{m} ⊂ ℝ^{m}, (3) (4 pts) f can be extended to a map f̂: W → N for some compact manifold

W bounded by M.

Problem 9. (14 pts) Assume that n-manifold M has an atlas consisting of two charts, ϕ_{1} : U_{1} → ℝ^{n}, ϕ_{2} : U_{2} → ℝ^{n}.

(1) (3 pts) Prove that for any closed form ω ∈ Ω^{k}(M), k ⩾ 1, there exist forms η_{i} ∈ Ω^{k−1}(U_{i}), i = 1, 2, such that ω|_{U_{i}} = dη_{i}.

(2) (5 pts) Let η ∈ Ω^{k−1}(U) be the difference between the restrictions η_{i}|_{U} of the above forms to U = U_{1} ∩ U_{2}. Prove that [ω] ↦ [η] is a well-defined map from f: H_{DR}^{k}(M) → H_{DR}^{k−1}(U).

(3) (6 pts) Prove that f is isomorphism.