PROBLEM SET 1, PART 2: SMOOTH MANIFOLDS/MAPS/FUNCTIONS DUE: SEP. 26 (1) [Construct smooth manifolds by gluing Euclidean open sets] Let M be a smooth manifold with atlas A = {(ϕα, Uα, Vα)}. (a) Prove: the transition maps ϕαβ := ϕβ ◦ ϕ −1 α satisfy the cocycle conditions (i) ϕβγ ◦ ϕαβ = ϕαγ on ϕα(Uα ∩ Uβ ∩ Uγ). (ii) ϕαα = IdVα . (iii) ϕαβ = (ϕβα) −1 . (b) Now on the disjoint union Mf := F α Vα we define an equivalence relation via x ∼ y ⇐⇒ ∃α, β s.t. x ∈ Vα, y ∈ Vβ and y = ϕαβ(x). (i) Check: ∼ is an equivalence relation on Mf. (ii) Prove: the quotient M / f ∼ is homeomorphic to M. (iii) Define a natural smooth structure on M / f ∼ (2) [Product of manifolds] (a) Prove: The product of two topological manifolds is a topological manifold. (b) Let M, N be smooth manifolds with atlases {(φα, Uα, Vα)} and {(ψβ, Xβ, Yβ)} respectively. Define an atlas on the product space M × N to be {(φα × ψβ, Uα × Xβ, Vα × Yβ)}. Check that this gives a smooth structure on M × N. (c) Explicitly construct local charts on T 2 = S 1 × S 1 ⊂ R 4 to make it into a smooth manifold. (You need to prove that your charts are compatible.) (3) [The need of global continuity in the definition of smoothness] (a) Define two atlases A and B on R as follows: A = {(ϕ, R, R)}, ϕ(x) = x, B = {(ψi , Xi , Yi) : i ∈ Z}, where Xi = Yi = (i − 1, i + 1) and ψi(x) = x. Define a map f : (R, A) → (R, B) via f(x) = 2, x > 0 0, x ≤ 0 Prove: For each i, the map ψi ◦ f ◦ ϕ −1 : ϕ(R ∩ f −1 (Xi)) → ψi(Xi) is smooth. [So we can’t drop the continuity assumption in Definition 1.1 in Lecture 4.] (b) Let (M, A) and (N, B) be smooth manifolds. Suppose for each point p, there exists (ϕα, Uα, Vα) ∈ A and (ψβ, Xβ, Yβ) ∈ B so that p ∈ Uα and f(Uα) ⊂ Xβ. Prove: Under this assumption we can drop the continuity assumption of f in Definition 1.1 in Lecture 4. 1
2 PROBLEM SET 1, PART 2: SMOOTH MANIFOLDS/MAPS/FUNCTIONS DUE: SEP. 26 (4) [Orientability of smooth manifolds] Let M be a smooth manifold of dimension n. We say M is orientable if there exists an atlas {(ϕα, Uα, Vα)} of M so that det(dϕαβ)p > 0 for all p ∈ ϕα(Uα ∩ Uβ), where dϕαβ is the Jacobian matrix of ϕαβ. (a) Prove: S n is orientable. (b) Prove: For RPn , the atlas we constructed in Lecture 2 makes RPn orientable for odd n but not for even n. (c) [Not required] Prove: CPn is orientable. (d) Prove: If M, N are orientable, so is M × N. Conversely, if M × N are orientable, so are M and N. (5) [Smooth Urysohn’s Lemma] Suppose A and B are two disjoint closed subsets of a smooth manifold M. (a) Show that there exists a smooth function f on M so that 0 ≤ f(x) ≤ 1 for all x ∈ M, f|A = 0 and f|B = 1. (b) [Not required] For any closed subset K in M, there is a smooth nonnegative function f : M → R so that f −1 (0) = K. [Hint: First work out this problem for M = R n , in which case R n \ K can be written as the union of countably many open balls. Try to construct a smooth function on each such ball which is positive inside the ball and equals 0 outside the ball. Then add these functions (each multiplied by suitable factors to guarantee smoothness). ] (c) Show that there exists a smooth function f on M so that 0 ≤ f(x) ≤ 1 for all x ∈ M, f −1 (0) = A and f −1 (1) = B. (6) [Exhaustion function] A real-valued continuous function f on M is called an exhaustion function for M if for any c ∈ R, the sublevel set f −1 ((−∞, c]) is compact. Prove: There exists a positive smooth exhaustion function on any smooth manifold