PROBLEM SET 6: DE RHAM COHOMOLOGY
DUE: DEC. 26
(1) (a)
(b)
(c)
(d)
Find a closed 1-form on R2 \ {(0, 0)} that is not exact.
1 (M ) is infinitely dimensional.
Let M = R2 \ Z2 . Prove: HdR
k
n
Compute HdR (T ) via the Künneth formula.
k (Rn \ {p, q}).
Compute HdR
(2) (a) Let M = M1 ∪ M2 be the disjoint union of two smooth manifolds. Find the relation
k (M ) and H k (M ) and the relation between H k (M ) and H k (M ).
between HdR
i
i
c
c
dR
(b) What if M = ∪∞
i=1 Mi is the disjoint union of countably many smooth manifolds?
(3) Suppose f ∈ C ∞ (M ), X, Xi ∈ Γ∞ (T M ) and ω ∈ Ωk (M ). Prove:
(a) Lf X ω = f LX ω + df ∧ ιX ω.
(b) ι[X1 ,X2 ] ω = LX1 ιX2 ω − ιX2 LX1 ω.
(c) L[X1 ,X2 ] ω = LX1 LX2 ω − LX2 LX1 ω.
(d) If ϕ : N → M is a diffeomorphism, then ϕ∗ LX ω = Lϕ∗ X ϕ∗ ω.
P
(e) (LX ω)(X1 , · · · , Xk ) = LX (ω(X1 , · · · , Xk )) − ki=1 ω(X1 , · · · , LX Xi , · · · , Xk ).
(4) Let M be a smooth manifold, and I an open interval that contains 0. A smooth map
ρ : M ×I → M is called an isotopy if for each t, ρt = ρ(·, t) : M → M is a diffeomorphisms
with ρ0 = Id. Given an isotopy ρ, one can define a time-dependent vector field on M by
dρs (ρ−1
Xt (p) =
t (p)).
ds s=t
In other words, Xt (with t ∈ I) is a family of vector fields on M such that
dρt
(p) = Xt (ρt (p)).
dt
Prove: For any k-form ω ∈ Ωk (M ) and any t ∈ I,
d ∗
ρ ω = ρ∗t LXt ω.
dt t
(5) Some missing proofs in lecture 20:
(a) Prove Proposition 1.1.
(b) Prove ker(δk ) ⊂ Im(βk ) and Im(δk ) ⊂ ker(αk+1 ) in theorem 1.3.
(c) Prove that the map Ψ on cohomologies in the proof of theorem 2.6 is well-defined.
(d) Prove the Five Lemma (lemma 2.7).
(6) (a) Let f : S 1 → S 1 be the map f (θ) = eikθ . Find deg(f ).
(b) Let f : Tn → Tn be given by f (eiθ1 , · · · , eiθn ) = (eik1 θ1 , · · · , eikn θn ). Find deg(f ).
(c) Consider the map f : R2 → R2 , f (x, y) = (x2 − y 2 , 2xy). Find deg(f ),
1
2
PROBLEM SET 6: DE RHAM COHOMOLOGY DUE: DEC. 26
(7) (a) Prove: If M is a compact oriented manifold of dimension n > 1, then for any p ∈ M ,
n (M \ {p}) = 0. (Hint: Use Poincaré duality.)
HdR
k (T2 − {p}) for all k.
(b) Compute HdR
k (T2 #T2 ) for all k.
(c) Compute HdR
(d) In general, find the de Rham cohomology groups of T2 # · · · #T2 .
(8) Let G be a compact connected Lie group acting smoothly on M .
(a) For each k, define left-invariant k-forms on M , and then define the “k th left-invariant
de Rham cohomology group” HLk (M ).
k (G), as follow:
(b) Prove HLk (G) ' HdR
(i) Let i : ΩkL (M ) ,→ ΩkdR (M ) be the inclusion map . Prove: i induces a linear
k (M ).
map i∗ : HLk (M ) → HdR
(ii) Prove i∗ is injective.
Hint: Let dg be a normalized Haar measure on G, in other words, dg is the
measure associated toR a volume form α on G which is both left and right
invariant, such that G α = 1. [You may try to prove the existence of
such dg if you want.] For each ω ∈ Ωk (M ) define the averaging of ω with
respect to G to be
Z
A(ω) =
τg∗ ω dg.
G
Show that A is a linear map from Ωk (M ) to ΩkL (M ), which induces a
k (M ) → H k (M ). Moreover, prove A ◦ i = Id.
linear map A∗ : HdR
∗
∗
L
(iii) Prove: i∗ is surjective.
k (M ). First notice that
Hint: It’s enough to prove [A(ω)] = [ω] for any [ω] ∈ HdR
the map A above can be rewritten as
Z
A(ω) =
τ ∗ ω ∧ π ∗ α,
G
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 UR of e in G, and a top form β on G which is supported in
U and satisfies 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 .
∗ ω + dη for some
Then τ ∗ ω ∧ π ∗ β = τU∗ ω ∧ π ∗ β. Finally prove τU∗ ω = πM
differential form η on U × M , where πM : U × M → M is the projection.
k
(c) Prove: HdR (Tn ) ' span{dxi1 ∧ · · · ∧ dxik | 1 ≤ i1 < · · · < ik ≤ n.}.