Differential Topology: Homework Set # 3.

1) Basic on tangent bundles. Recall that the tangent bundle TM to an extrinsicly defined manifold M k ⊂ Rn is defined to be: n n k n ∼ n TM := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } This exercise is designed to give you some practice working with tangent bundles. (i) Show that for a manifold M, the tangent bundle TM is well-defined, i.e. does not depend on the embedding M k ⊂ Rn. (ii) If M, N are a pair of smooth manifolds, show that T (M × N) is diffeomorphic to TM × TN.

(iii) If f : M → R is a smooth, positive function, show that the corresponding scaling map φ : TM → TM defined by φ(p, v) = (p, f(p) · v) is a diffeomorphism.

(iv) Show that the tangent bundle to S1 (the unit sphere) is diffeomorphic to the cylinder S1 × R. (v) Show that if Sn is an arbitrary sphere, then there exists a diffeomorphism between (TSn) × R and Sn × Rn+1.

2) Relatives of the tangent bundle. Closely associated to the tangent bundle are some other manifolds:

(i) For an extrinsicly defined manifold M k ⊂ Rn, define the normal bundle as follows: n n k ⊥ n ∼ n N(M) := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } Show that N(M) is an n-dimensional manifold.

(ii) Show that for the standard n-sphere Sn ⊂ Rn+1, the normal bundle N(Sn) is diffeomorphic to Sn × R. (iii) Consider the subset T 1M of TM consisting of pairs (p, v) ∈ TM with ||v|| = 1 (where || · || denotes the standard norm on Rn). Show that T 1M is a (2k − 1)-dimensional manifold, called the sphere bundle of M (or the unit tangent bundle of M). [Hint: construct a suitable submersion from TM → R.]

(iv) Recall that a vector field on M was defined to be a smooth map v : M → TM satisfying π ◦ v = IdM . Let ρ : Rn × Rn → Rn be the projection onto the second factor, and say that a vector field v has a zero at p if (ρ ◦ v)(p) = 0 ∈ Rn. Show that if Sn is an odd dimensional sphere, then there exists a non-vanishing vector field v on Sn (i.e. v has no zeros). (v) Show that if Sn supports a non-vanishing vector field v, then its antipodal map (i.e. p 7→ −p) is smoothly homotopic to the identity map. [Hint: normalize the v so that the image lies in T 1Sn, and rotate p to −p along the direction given by v(p).]

3) Intrinsic Whitney embedding. In class, I outlined an argument for showing the intrinsic version of Whit- ney embedding, using some notions from differential geometry. This exercise outlines an alternate, “bare hands” construction of an embedding of the compact manifold M k into some RN (with large N). (i) Show that the function f : R → R defined below is smooth:

( 2 e−1/x x > 0 f(x) = 0 x ≤ 0. (ii) Given a < b, show that the associated function g(x) = f(x − a)f(b − x) is smooth, positive on the interval (a, b) ⊂ R and identically zero elsewhere. Show that the corresponding function h : R → R defined by: R t −∞ g dx h(t) = R ∞ −∞ g dx is a smooth function satisfying (1) h(x) = 0 for x < a, (2) h(x) = 1 for b < x, and (3) 0 < h(x) < 1 for a < x < b.

k (iii) Now given positive real numbers a < b, construct a smooth function ψa,b on R which is (1) identically one on the ball of radius a, (2) identically zero outside the ball of radius b, and (3) is strictly between 0 and 1 at intermediate points.

(iv) For r > 0, let B(r) denote the open ball of radius r centered at the origin in Rk. Show that there exists a ˆ k finite open cover {Ui} (1 ≤ i ≤ m) of M by coordinate charts, with maps φi : Ui → Ui ⊂ R , chosen to have the following two properties: ˆ k • Ui ⊃ B(2), i.e. each Ui contains an open ball of radius 2 centered at the origin in R , −1 • the open subsets {φi (B(1))} cover M. k (v) Let ψ be the map ψ1,2 constructed in (iii). Verify that for each index i, the map fi : M → R defined below is smooth: ( ψ(φi(x)) · φi(x) x ∈ Ui fi = 0 x∈ / Ui.

k (vi) Define the maps gi : M → R × R by gi(p) = (fi(p), ψi(p)), and show that the associated map G : M → k+1 m (R ) defined by p 7→ (g1(p), . . . , gm(p)) yields an injective immersion (and hence an embedding) of the manifold M into m(k + 1)-dimensional Euclidean space.

4) Projective spaces. Define the n-dimensional real projective space, denoted RP n, to be the quotient of Sn by the antipodal map p 7→ −p.

(i) Show that RP n is a compact n-dimensional manifold (use either the intrinsic or extrinsic definition, see exercise (4) below).

(ii) Viewing S1 ⊂ S2 as the equator, we have a natural embedding RP 1 ⊂ RP 2. Show that this submanifold cannot be realized as the pre-image f −1(r) of a regular value r of a smooth function f : RP 2 → R. [Hint: consider connected neighborhoods of RP 1 inside RP 2.] (iii) Show that the sphere bundle of the 2-sphere T 1S2 is in fact diffeomorphic to RP 3. (iv) Show that RP 2 embeds in R4, via the following steps: • show that RP 2 is diffeomorphic to the union of a Moebius band M and a 2-dimensional disk D2 along their common boundary. • show that there exists a 3-dimensional smooth curve γ ⊂ R3 which simultaneously arises as the boundary of an embedded D2, and as the boundary of an embedded M [Hint: find a convenient embedding of M first, then think of “flattening out” the boundary curve.] • the two embeddings of M and D2 into R3 might intersect each other; show that this can be fixed in R4.