Math 8250 HW #7
Due 11:15 AM Friday, March 29
1. What are the de Rham cohomology groups of CPn ? (Hint: What is the homotopy type of the
complement of a point? Feel free to use the fact that CP1 is diffeomorphic to S 2 . Please do
not just say “By the de Rham theorem the de Rham cohomology groups are the same as the
singular cohomology groups and we know the singular cohomology groups of CPn are . . .”.
You can use the de Rham theorem if you like, but then you need to prove that the singular
cohomology groups are what you say they are.)
2. Suppose M and N are closed, oriented manifolds. Let πM : M ×N → M and πN : M ×N → N
p
q
p+q
be the obvious projections. The cross-product × : HdR
(M ) ⊕ HdR
(N ) → HdR
(M × N ) is
defined by
∗
∗
[ω] × [η] = [πM
ω ∧ πN
η].
(a) Show that × is well-defined and that
]
]
α × β = πM
α ^ πN
β
p
q
for any α ∈ HdR
(M ) and β ∈ HdR
(N ). Here ^ is the cup product.
(b) Let ∆ : M → M × M be the diagonal map; i.e., ∆(p) = (p, p). Show that
α ^ β = ∆∗ (α × β).
3. Compute the de Rham cohomology groups of the Klein bottle.
4. Let ∆ be the Laplace-Beltrami operator.
(a) Show that if f is a smooth function on Rn , then
∆f = −
n
X
∂2f
k=1
∂x2i
.
(b) Let F~ be a vector field on R3 and let ω = F~ [ be the corresponding 1-form. Let ∆V be
the vector Laplacian ∆V = ∇(∇·) − ∇ × ∇×. Show that
(∆V F~ )[ = −∆ω.
1