PROBLEM SET 3, PART 1: EMBEDDING AND NEIGHBORHOODS DUE: OCT. 31 (1) [Embedding T 2 into R 3 ] (a) Recall that T 2 = S 1 × S 1 = {(x 1 , x2 , x3 , x4 ) | (x 1 ) 2 + (x 2 ) 2 = 1,(x 3 ) 2 + (x 4 ) 2 = 1}. Let ι : T 2 ,→ R 4 be the canonical embedding and g : R 4 → R 3 be the map g : R 4 → R 3 , (x 1 , x2 , x3 , x4 ) 7→ (x 1 (2 + x 3 ), x2 (2 + x 3 ), x4 ). (i) For any p = (x 1 , x2 , x3 , x4 ) ∈ T 2 , write down a basis of Im(dι)p. (ii) Write down the matrix of dg. (iii) Prove: The map f = g ◦ ι : T 2 → R 3 is an embedding. (b) (Not required) Prove: S m × S n can be embedded into R m+n+1 . (Hint: First embed S n × R into R n+1 ) (2) [Embedding of RP2 into R 4 ] (a) Consider the map f : S 2 → R 3 , (x, y, z) 7→ (yz, zx, xy). Prove: It induces an immersion of RP2 outside several (how many?) points. (b) Prove: The map f : S 2 → R 3 , (x, y, z) 7→ (x 2 − y 2 , yz, zx, xy) is an embedding of RP2 to R 4 . (3) [The sphere bundle] (a) Prove: The “sphere bundle” SM described in the proof of Theorem 2.5.8 is a smooth submanifold of TM of dimension 2m − 1. [Hint: Find a smooth function on M so that SM is a regular level set.] (b) Write down a complete proof of Theorem 2.5.8. (4) [The normal bundle] (a) Prove Proposition 2.6.6. (b) (Not required) More generally, Proposition 2.6.6 for the normal bundle N(X, M), where X is a smooth submanifold of M. (c) (Not required) Suppose f ∈ C∞(R K) and 0 is a regular value of f. Let X = f −1 (0). Prove: The normal bundle N(X, R K) is diffeomorphic to X × R. (d) (Not required) Let ∆ = {(x, x) |x ∈ M} ⊂ M × M be the diagonal. Prove: Φ : TM → N(∆, M × M), (x, v) 7→ ((x, x),(v, −v)) is a diffeomorphism. 1
2 PROBLEM SET 3, PART 1: EMBEDDING AND NEIGHBORHOODS DUE: OCT. 31 (5) [Restriction of smooth map to smooth submanifold] Let f : M → N be a smooth map, ι : X ,→ M be a smooth submanifold, Y be a smooth submanifold of N, and f(X) ⊂ Y . (a) Prove: ˜f = f ◦ ι : X → Y is a smooth map. (b) Prove: If f is an immersion on X, then ˜f is an immersion. (c) Prove: The condition “f : X → f(X) is a diffeomorphism” in the statement of Theorem 2.6.4 can be weakened to “f is injective on X”. (6) [Immersions/embeddings are dense] (a) Let U ⊂ R m be open, and f : U → R n be smooth. Suppose n ≥ 2m. Prove: For any ε > 0, there exists an m × n matrix A = (aij ) with |aij | 0, there is an immersion fε : M → R K with |f − fε| < ε. (c) (Not required) State and prove a similar theorem for embedding. (7) [Isotopy] (Not required) Let f, g : M → N be two embeddings. An smooth isotopy between f and g is a smooth map F : M × R → N so that • F(x, 0) = f(x), F(x, 1) = g(x); • Ft := F|M×{t} is an embedding for 0 ≤ t ≤ 1. (a) Prove: Smooth isotopy is an equivalence relation. (b) Prove: If the induced level-preserving map G : M × R → N × R, G(x, t) = (F(x, t), t) is an embedding, then F : M × R → N is an isotopy. (c) Prove: If each Ft is a proper embedding, then the induced level-preserving map G is an embedding. [In particular, the conclusion holds if M is compact.]