《微分流形》课程教学资源（英文讲义）PSet7-1 DE RHAM COHOMOLOGY GROUPS

PROBLEM SET 7.PART1:DE RHAM COHOMOLOGYGROUPSDUE:DEC.28())[An infinite dimensional Hlr](a) Find a closed 1-form on R2(0,O)) that is not exact.(b) Let M =R2-z2.Prove: Har(M) is infinitely dimensional.(2) [The de Rham cohomologies of the union](a) Let M = Mi U M2 be the disjoint union of two smooth manifolds. Find therelation between Hke(M) and Hkr(M).(b) Calculate the de Rham cohomology groups of sl via Mayer-Vietoris sequence.()What if M =U,M, is the disjoint union of countably many smooth manifolds?(3)[Wittendeformation]Let M be a smooth manifold, and e 2'(M) be an exact 1-form.(a) For any k, define de : 2h(M) → 2k+1(M) byde(w) = dw +0 ^w.Prove: de(dgw) = 0 for all w E 2*(M)(b) Consider the complex0 - 2(M) d2(M) e, ... d 0m-1(M) d 2m(M) → 0.Define z(M), B(M) and H(M), and prove: H(M) is isomorphic to Har(M)(4)[Missing parts in the proof of Mayer-Vietoris sequence](a)ProveProposition6.2.1.(b) (NOT required) To prove Theorem 6.2.4, we need to prove six inclusion relations.We proved one in class. Try to prove the rest five relations.[Applications of Mayer-Vietoris sequence](5）(a) Let Mi,M2 be connected manifolds, and M = Mi#M, be their connected sum.Find the relation between Har(M) and Har(Mi), Har(M2).(b) Compute the de Rham cohomology groups of Rp2 and Cp2(c) (NOT required) Compute the de Rham cohomology groups of Ripn and Cpn[Kunneth formulaand its applications](6)(a) Read g6.2.2"Application 3: Kunneth formula"(b) Prove Har(Tn) ~ R(r).1

PROBLEM SET 7, PART 1: DE RHAM COHOMOLOGY GROUPS DUE: DEC. 28 (1) [An infinite dimensional H1 dR] (a) Find a closed 1-form on R 2 − {(0, 0)} that is not exact. (b) Let M = R 2 − Z 2 . Prove: H1 dR(M) is infinitely dimensional. (2) [The de Rham cohomologies of the union] (a) Let M = M1 ∪ M2 be the disjoint union of two smooth manifolds. Find the relation between Hk dR(M) and Hk dR(Mi). (b) Calculate the de Rham cohomology groups of S 1 via Mayer-Vietoris sequence. (c) What if M = ∪∞ i=1Mi is the disjoint union of countably many smooth manifolds? (3) [Witten deformation] Let M be a smooth manifold, and θ ∈ Ω 1 (M) be an exact 1-form. (a) For any k, define dθ : Ωk (M) → Ω k+1(M) by dθ(ω) = dω + θ ∧ ω. Prove: dθ(dθω) = 0 for all ω ∈ Ω k (M). (b) Consider the complex 0 → Ω 0 (M) dθ −→ Ω 1 (M) dθ −→ · · · dθ −→ Ω m−1 (M) dθ −→ Ω m(M) → 0. Define Z k θ (M), Bk θ (M) and Hk θ (M), and prove: Hk θ (M) is isomorphic to Hk dR(M). (4) [Missing parts in the proof of Mayer-Vietoris sequence] (a) Prove Proposition 6.2.1. (b) (NOT required) To prove Theorem 6.2.4, we need to prove six inclusion relations. We proved one in class. Try to prove the rest five relations. (5) [Applications of Mayer-Vietoris sequence] (a) Let M1, M2 be connected manifolds, and M = M1#M2 be their connected sum. Find the relation between Hk dR(M) and Hk dR(M1), Hk dR(M2). (b) Compute the de Rham cohomology groups of RP2 and CP2 . (c) (NOT required) Compute the de Rham cohomology groups of RPn and CPn . (6) [Kunneth formula and its applications ] (a) Read §6.2.2 ”Application 3: Kunneth formula”. (b) Prove Hk dR(T n ) ' R( n k ) . 1

2PROBLEMSET7,PART1:DERHAMCOHOMOLOGYGROUPSDUE:DEC.28(c) For any compact manifold M, define pm(t) to bethe polynomial pm(t)r=ob;(M)t.Prove:If M,N arecompact, thenPMxN(t) = PM(t)pN(t).(In particular, weget x(M × N) =x(M)x(N).)(d)(NOT required)Prove:Sm ×sn ishomeomorphicto Sm×sn'if and only ifm,n) = (m',n'].(7) [Not required] Let G be a compact connected Lie group acting smoothly on asmooth manifold M (from left).(a) For each k, define left-invariant k-forms on M, and then define the"Kth left-invariant de Rham cohomology group" H(M).(b)Prove H(M)=Har(M),as follow:() Let i:h(M) ar(M) be the inclusion map. Prove: iinduces a linearmap i : H(M) → Har(M).(ii) Prove i is injective.Hint:Let dgbea normalized Haar measure on G,in otherwords,dgisthemeasure associated toa volumeform α on G which is bothleftand right invariant, such that Jeα=1.[You maytry to prove theexistence of such dgif youwant.JFor eachwE2(M)definetheaveraging of wwithrespecttoGtobeA(w) =Twdg.Show thatA is a linearmapfrom 2(M)to 2(M),which inducesalinearmapA:Har(M)-→H(M).Moreover,proveAoi+=Id.(ii) Prove: i, is surjective.Hint: It's enough to prove [A(w)] = [w] for any [w] e Hkr(M). First noticethat the map A above can be rewritten asTw^a,A(w) =whereT:GxM-→M isthe action ofG on M,and :GxM-Gistheprojection,and one regardsthe differential form-*wA*aasatop form on G (with M variables as parameters). Take a contractibleneighborhood U ofe in G,and a top formβon Gwhich is supportedin U and satisfies Jβ=1.Then there exists a differential form n onGsothata-β=d.(Why?)LetTu:UxM→MbetherestrictionofT.Then*wΛr*β=TuwΛ*β.FinallyproveTuw=元Mw+dnforsomedifferential formn onUxM,whereM:UxM→M istheprojection(c) Prove: Har(Tn) ~ span[dr A...Ade[1≤i<...<ik≤n.]

2 PROBLEM SET 7, PART 1: DE RHAM COHOMOLOGY GROUPS DUE: DEC. 28 (c) For any compact manifold P M, define pM(t) to be the polynomial pM(t) = n i=0 bi(M)t i . Prove: If M, N are compact, then pM×N (t) = pM(t)pN (t). (In particular, we get χ(M × N) = χ(M)χ(N).) (d) (NOT required) Prove: S m × S n is homeomorphic to S m0 × S n 0 if and only if {m, n} = {m0 , n0}. (7) [Not required] Let G be a compact connected Lie group acting smoothly on a smooth manifold M (from left). (a) For each k, define left-invariant k-forms on M, and then define the “k th left￾invariant de Rham cohomology group” Hk L (M). (b) Prove Hk L (M) ' Hk dR(M), as follow: (i) Let i : Ωk L (M) ,→ Ω k dR(M) be the inclusion map . Prove: i induces a linear map i∗ : Hk L (M) → Hk dR(M). (ii) Prove i∗ is injective. Hint: Let dg be a normalized Haar measure on G, in other words, dg is the measure associated to a volume form α on G which is both left and right invariant, such that R G α = 1. [You may try to prove the existence of such dg if you want.] For each ω ∈ Ω k (M) define the averaging of ω with respect to G to be A(ω) = Z G τ ∗ g ω dg. Show that A is a linear map from Ωk (M) to Ωk L (M), which induces a linear map A∗ : Hk dR(M) → Hk L (M). Moreover, prove A∗ ◦ i∗ = Id. (iii) Prove: i∗ is surjective. Hint: It’s enough to prove [A(ω)] = [ω] for any [ω] ∈ Hk dR(M). First notice that the map A above can be rewritten as A(ω) = Z G τ ∗ω ∧ π ∗α, where τ : G × M → M is the action of G on M, and π : G × M → G is the projection, and one regards the differential form τ ∗ω ∧π ∗α as a top form on G (with M variables as parameters). Take a contractible neighborhood U of e in G, and a top form β on G which is supported in U and satisfies R G β = 1. Then there exists a differential form η on G so that α−β = dγ. (Why?) Let τU : U ×M → M be the restriction of τ . Then τ ∗ω ∧ π ∗β = τ ∗ U ω ∧ π ∗β. Finally prove τ ∗ U ω = π ∗ Mω + dη for some differential form η on U × M, where πM : U × M → M is the projection. (c) Prove: Hk dR(T n ) ' span{dxi1 ∧ · · · ∧ dxik | 1 ≤ i1 < · · · < ik ≤ n.}