Math 524
Homework 1
Due Wed, Jan 31.
Solve 4 of the problems below:
1. Let Sn⊂Rn+1 be the real n– dimensional sphere
{(x1, ..., xn+1)|
n+1
X
i=1
x2
i= 1}.
Fix the points N= (1,0, ..., 0) and S= (−1,0, ..., 0) ∈Sn, let UN=Sn\{N}
and US=Sn\ {S}. We define the atlas Aon Sngiven by two local charts
τNand τS, where τN:UN→Rnis the projection of vertex Non the
n–dimensional plane {(x1, ..., xn+1)|x1= 0} ⊂ Rn+1, and τSis the similar
projection of vertex S.
a) Prove that Agives Sna structure of a differential manifold.
b) Let ξbe a tangent vector field and ube a cotangent vector field on
Sn. Write ξand uin local coordinates and compute the transition maps for
ξand ugiven by the atlas A.
2. Let Mbe a C∞manifold and let pbe a positive integer. Prove that
VpT∗
Mis a C∞vector bundle.
3. Let M,M0be C∞manifolds. Prove that for two homotopically equiv-
alent maps fand g:M→M0, the pullback morphisms between de Rham
cohomology groups coincide: F∗=G∗. Follow these steps:
Let I= [0,1]. Let K:Ap+1(M×I)→ Ap(M) be a C∞map that
sends any p+ 1– form uon M×Ito a p–form K(u) on M, where in local
coordinates,
u(x, t) = X
|I|=p
uI(x, t)dt ∧dxI+X
|J|=p+1
˜uJ(x, t)dxJ,
K(u)(x) := X
|I|=p
(Z1
0
uI(x, t)dt)dxI.
a) Prove that Kis well defined.
b) Prove the following identity of functions from Ap+1(M×I) to Ap(M)
dK +Kd =i∗
1−i∗
0,
where for any t∈I, we define it:M→M×Iby it(x) = (x,t).
c) Use b) to prove that i∗
1=i∗
0:Hp
DR(M×I , R)→Hp
DR(M , R).
As a corollary, show that for any contractible C∞manifold M,Hp
DR(M , R) =
0 for p > 0 and H0
DR(M , R) = R.
4. Use exercises 1 and 3 above to prove that Hp
DR(Sn,R) = 0 for
p6= 0, n and H0
DR(Sn,R) = Hn
DR(Sn,R) = R.Hint: show that for p > 0,
Hp
DR(Sn,R)∼
=Hp−1
DR (Sn−1,R).
5. Let Mbe a complex manifold, Ap,q(M) be the space of (p, q)–forms.
Let Ωp(M)⊂ Ap,0(M) be the set of holomorphic forms ω(such that ¯
∂ω = 0).
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