PROBLEM SET 7,PART 2:DE RHAM COHOMOLOGY GROUPSDUE:DEC.28(1) [The de Rham cohomology groups of punctured manifolds](a) Compute Hlr(T? - (p)).(b) Suppose n >1. Compute Har(IRn - (p,q)) and H(Rn - (p,q).(c) Let M be a compact oriented connected smooth manifold of dimension m > 1,PEM.(i) Prove: Har(M /(p)) = 0.(ii) For k 0. Prove: f is proper, and isproperly homotopic to fi(z)= zk.(b) Use mapping degree to prove the fundamental theorem of algebra(5)[Orientation covering] (NOT required)Let M be a connected non-orientable manifold of dimension m. LetM - ((p, Op) I pe M, Op is an orientation on TpM).Let π : M→M be the obvious projection map.(a) Show that M admits a natural topological and smooth structure so that it is anm-dimensional connected orientable manifold, and is a smooth double coveringmap.(b) Show that the non-trivial Deck transformation : M -→ M is an orientation-reverting diffeomorphism.(c) Find the relation between Har(M) and Har(M)1
PROBLEM SET 7, PART 2: DE RHAM COHOMOLOGY GROUPS DUE: DEC. 28 (1) [The de Rham cohomology groups of punctured manifolds] (a) Compute H1 dR(T 2 − {p}). (b) Suppose n > 1. Compute Hk dR(R n − {p, q}) and Hk c (R n − {p, q}). (c) Let M be a compact oriented connected smooth manifold of dimension m > 1, p ∈ M. (i) Prove: Hm dR(M \ {p}) = 0. (ii) For k 0. Prove: f is proper, and is properly homotopic to f1(z) = z k . (b) Use mapping degree to prove the fundamental theorem of algebra. (5) [Orientation covering] (NOT required) Let M be a connected non-orientable manifold of dimension m. Let Mf = {(p, Op) | p ∈ M, Op is an orientation on TpM}. Let π : Mf → M be the obvious projection map. (a) Show that Mf admits a natural topological and smooth structure so that it is an m-dimensional connected orientable manifold, and π is a smooth double covering map. (b) Show that the non-trivial Deck transformation σ : Mf → Mf is an orientationreverting diffeomorphism. (c) Find the relation between Hk dR(M) and Hk dR(Mf). 1
2PROBLEMSET7,PART2:DERHAMCOHOMOLOGYGROUPSDUE:DEC.28(6) [The Euler characteristics](a)Let B be a compact smooth manifold with boundary aB.Let 2Bbe the manifoldobtained by"gluing"two copies of B along the boundary.Prove:x(2B) +x(B) = 2x(B).(b) (NOT required) Given the following formula (the answer to a problem in previousPSet) R, k=0,Har(Rp2n) = 0k±0Prove: Rp2n can not be the boundary of any 2n +1 dimensional compact mani-fold.(7) [Hdr v.s. i] (NOT required)Let M be a compact connected smooth manifold.Considerthemap亚: Han(M) →Hom(πi(M),R),[g] → (MI -→ /αwhere we only take smooth representatives [], and the map L : → M is theinclusionmap.(a)Prove: The map is well-defined.[Hint:usethe cochain homotopy property.](b) Prove: The map is injective.[In fact the mapis also surjective, but theproof is hard.In particular, if M is simply connected, then Hdr(M) = 0.(8) [Moser's trick] (NOT required)Let M be a m-dimensional compact orientable manifold, and wo,wi E m(M) aretwo volume forms on M such that JM wo = JM w1. Show that there exists a diffeo-morphism : M -→ M such that p*w2 = wi as follows:(a) For t e [0,1], let wt = (1 -t)w1 + tw2. Prove: wt is a family of volume forms onM, and there exists α e m-1(M) so that wt = dn holds for all t e [0, 1].(b) Let Xt be the time-depending vector field so that x,wt = -Q. Let pt be theflow of Xt. Prove: twt = wo holds for all t e [0, 1]
2 PROBLEM SET 7, PART 2: DE RHAM COHOMOLOGY GROUPS DUE: DEC. 28 (6) [The Euler characteristics] (a) Let B be a compact smooth manifold with boundary ∂B. Let 2B be the manifold obtained by “gluing” two copies of B along the boundary. Prove: χ(2B) + χ(∂B) = 2χ(B). (b) (NOT required) Given the following formula (the answer to a problem in previous PSet) Hk dR(RP2n ) = R, k = 0, 0, k ̸= 0. Prove: RP2n can not be the boundary of any 2n + 1 dimensional compact manifold. (7) [H1 dR v.s. π1] (NOT required) Let M be a compact connected smooth manifold. Consider the map Ψ : H1 dR(M) → Hom(π1(M), R), [ω] 7→ [γ] 7→ Z γ ι ∗ γα where we only take smooth representatives γ ∈ [γ], and the map ιγ : γ ,→ M is the inclusion map. (a) Prove: The map Ψ is well-defined. [Hint: use the cochain homotopy property.] (b) Prove: The map Ψ is injective. [In fact the map Ψ is also surjective, but the proof is hard.] In particular, if M is simply connected, then H1 dR(M) = 0. (8) [Moser’s trick] (NOT required) Let M be a m-dimensional compact orientable manifold, and ω0, ω1 ∈ Ω m dR(M) are two volume forms on M such that R M ω0 = R M ω1. Show that there exists a diffeomorphism φ : M → M such that φ ∗ω2 = ω1 as follows: (a) For t ∈ [0, 1], let ωt = (1 − t)ω1 + tω2. Prove: ωt is a family of volume forms on M, and there exists α ∈ Ω m−1 (M) so that ˙ωt = dη holds for all t ∈ [0, 1]. (b) Let Xt be the time-depending vector field so that ιXtωt = −α. Let φt be the flow of Xt . Prove: φ ∗ tωt = ω0 holds for all t ∈ [0, 1]