PROBLEM SET 3, PART 2: TRANSERSALITY DUE: OCT. 31 (1) [Smooth submanifolds as regular level sets] Here is what we argued in class: (a) Summarize the above as a lemma. (b) Show that the “global version” fails: Let K be the Klein bottle and let S be its central circle. Prove that there is no smooth function f : K → R so that 0 is a regular value and f −1 (0) = S. (c) (Not required) In general, suppose S is a smooth submanifold of M of codimension r, and suppose the normal bundle N(S, M) is trivial in the sense that N(S, M) is isomorphic(You may need to find the exact meaning of “isomorphic” between vector bundles) to S × R r . Prove: There is a smooth map f : M → R r so that 0 ∈ R r is a regular value of f and S = f −1 (0). (d) (Not required) Conversely, suppose f : M → R r is a submersion, and S = f −1 (0) is a submanifold of M. Show that N(S, M) is trivial. (e) Write down a theorem summarizing (c) and (d). (2) [Stability of various properties] We need the following definition: We say a property P concerning maps in C∞(M, N) is a stable property if it is preserved under small deformation, namely, if f ∈ C∞(M, N) satisfies P and F is a smooth homotopy with F(x, 0) = f, then there exists ε > 0 so that for each 0 2. Consider the map F(x, t) = xh(tx) as a homotopy with f0(·) = F(·, 0). 1
2 PROBLEM SET 3, PART 2: TRANSERSALITY DUE: OCT. 31 (3) [Stability of transversal intersection ] (a) Show that if M is compact and X is a smooth submanifold of N, then the property “f ∈ C∞(M, N) intersect X transversally” is a stable property. (b) (Not required) Let F : S × M → N be a smooth map. Suppose M is compact, and X ⊂ N is a closed submanifold. Denote fs(·) = F(s, ·). Prove: the set {s ∈ S | fs intersect X transversally} is an open subset of S. (4) [Lefschetz maps] You will need the following conceptions. Let f : M → M be a smooth map. A point p ∈ M is a called a fixed point of f if f(p) = p. We say f is a Lefschetz map if for each fixed point p of f, 1 is not an eigenvalue of dfp : TpM → TpM. The local Lefschetz number Lp(f) of a Lefschetz map at a fixed point p is the sign of the determinant det(dfp−Id), i.e. Lp(f) := 1 if det(dfp − Id) > 0, and Lp(f) := −1 if det(dfp − Id) < 0. Do the following questions: (1) Let rθ : S 2 → S 2 be the map “rotate S 2 by an angle θ”, (θ 6= 2kπ), defined by rθ(x 1 , x2 , x3 ) = (x 1 cos θ − x 2 sin θ, x1 sin θ + x 2 cos θ, x3 ). Prove: rθ is a Lefschetz map. (2) Let V be a vector space, and L : V → V a linear map. Let ∆ = {(v, v) : v ∈ V } be the diagonal in V × V , and ΓL = {(v, Lv) : v ∈ V } be the graph of L in V × V . Prove: ΓL intersects ∆ transversally if and only if 1 is not an eigenvalue of L. (3) Prove: If M is compact and f : M → M is a Lefschetz map, then f has only finitely many fixed points. (4) The Lefschetz number of a Lefshetz map f is defined to be L(f) = P f(p)=p Lp(f), where the summation is over all fixed points p. Compute L(rθ)for rθ in (1). (5) [Simply connectedness of R n \ M (dim M ≤ n − 3)] (Not required) Let M be a connected smooth manifold of dimension m. Prove: if S ⊂ M is a smooth submanifold of dimension k ≤ m−3, then for any p ∈ M−S, π1(M, p) ' π1(M−S, p)