PROBLEMSET3,PART2:TRANSERSALITYDUE:OCT31[Smooth submanifolds as regular level sets](1Here is what we argued in class:一方面，由子流形的定义可知，对于任意9EX，均存在邻域V以及由坐标卡映射所诱导的光滑映射g：V→R（其中1=dimN-dimX是X在N中的余维数）使得g(0)=XnV。注意由g的构造可知dg是满射，于是0是g的正则值，且它的核为ker(dga)=TgX(a) Summarize the above as a lemma.(b)Showthat the"global version"fails:LetK be theKlein bottle and let S be itscentral circle. Prove that there is no smooth function f : K -→ R so that 0 is aregular value and f-l(O) = S.(c) (Not required) In general, suppose S is a smooth submanifold of M of codi-mension r, and suppose the normal bundle N(S, M) is trivial in the sense thatN(S,M)is isomorphic(You mayneed to find the exactmeaning of"isomorphic"betweenvector bundles) to S × Rr. Prove: There is a smooth map f : M -→ Rr so that0 ERr is a regular value of f and S=f-l(O).(d) (Not required) Conversely, suppose f : M → Rr is a submersion, and S = f-1(0)is a submanifold of M. Show that N(S, M) is trivial.(e)Writedownatheoremsummarizing (c)and (d).[Stability of various properties](2)We need the following definition:We say a propertyPconcerning maps in C(M,N) is a stable property ifit is preserved under small deformation, namely, if f e Co(M, N) satisfiesP and F is a smooth homotopy with F(r, o) = f, then there exists e > 0so that for each 0 < t < e, the map ft() = F(,t) satisfies the property P.(a) Prove: If M is compact, then the following properties of maps in Coo(M,N) arestable:(i) submersion,(i)embedding.(b) (Not required) Show that the conclusion above fails if M is non-compact bystudying the following example.Counterexample:Let h E Co(R) be a smooth bump functionwithh(r) =1 for [<1,h(r) = 0 for [| > 2.Consider the map F(r,t) = rh(ta) as a homotopy with fo() =F(-,0).1
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 < t < ε, the map ft(·) = F(·, t) satisfies the property P. (a) Prove: If M is compact, then the following properties of maps in C∞(M, N) are stable: (i) submersion, (ii) embedding. (b) (Not required) Show that the conclusion above fails if M is non-compact by studying the following example. Counterexample: Let h ∈ C∞(R) be a smooth bump function with h(x) = 1 for |x| < 1, h(x) = 0 for |x| > 2. Consider the map F(x, t) = xh(tx) as a homotopy with f0(·) = F(·, 0). 1
2PROBLEMSET3,PART2:TRANSERSALITYDUE:OCT.31[Stabilityoftransversal intersection](3)(a) Show that if M is compact and X is a smooth submanifold of N, then theproperty"f e Co(M,N) intersect X transversally"is a stable property.(b) (Not required)Let F:S×M Nbe a smooth map. Suppose M is compact,and X N is a closed submanifold. Denote fs() = F(s, ). Prove: the set[s E SI f, 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 pE M is a called a fired point off if f(p) =p. We say f is a Lefschetz map if for each fixed point p of f, 1 isnot an eigenvalue of dfp :TpM -→ TpM. The local Lefschetz number Lp(f) ofa Lefschetz map at a fixed point p is the sign of the determinant det(df,-Id),i.e. Lp(f) := 1 if det(dfp - Id) > 0, and L(f) := -1 if det(dfp - Id) < 0.Do the following questions:(1) Let re : $2 → s? be the map “rotate $2 by an angle 0", (0 + 2k元), defined byre(l,,3) = (cos sino, sin+cos0, 3).Prove:re is a Lefschetz map.(2) Let V be a vector space, and L : V→ V a linear map. Let △ = ((u, u) : u e V)be the diagonal in V × V, and IL = [(u, Lu) : E V) be the graph of L inV× V.Prove: F intersects transversally if and only if 1 is not an eigenvalueof L.(3) Prove: If M is compact and f : M → M is a Lefschetz map, then f has onlyfinitelymanyfixed points.(4) The Lefschetz number of a Lefshetz map f is defined to be L(f) = f(P)=p Lp(f),where the summation is over all fixed points p. Compute L(re)for re in (1).(5)[Simply connectedness of Rn/M (dim M ≤n-3)) (Not required)LetMbeaconnected smoothmanifoldof dimensionm.Prove:if S CMisasmoothsubmanifold of dimension k ≤ m-3, then for any p E M-S, πi(M,p) ~ πi(M-S,p)
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)