Math 8250 HW #6
Due 11:15 AM Friday, March 22
1. (Proof of Proposition 3 from Day 21) Let M n be a compact, oriented, differentiable manifold,
let π : M × R → M be the projection, and for each a ∈ R let ia : M → M × R be given by
x 7→ (x, a). For each p ∈ {1, . . . , n + 1}, let hp : Ωp (M × R) → Ωp−1 (M × R) be the homotopy
operator defined in class. Show that
hp+1 dω + dhp ω = ω − π ∗ i∗0 ω
for any ω ∈ Ωp (M × R).
2. Use de Rham cohomology to show that the 2-sphere S 2 and the 2-torus T 2 = S 1 × S 1 are not
diffeomorphic.
3. For each part, determine whether the given statement is true or false. If the statement is
true, prove it. If the statement is false, give a counterexample.
(a) Every one-form on S 1 = {x ∈ R2 : |x| = 1} can be extended to a one-form on R2 . In
other words, if i : S 1 → R2 is the inclusion map, then for any ω ∈ Ω1 (S 1 ), there exists
η ∈ Ω1 (R2 ) so that i∗ η = ω.
(b) Every closed one-form on S 1 can be extended to a closed one-form on R2 .
4. Let M n be a smooth, compact, differentiable manifold and let Z ⊂ M be a closed, oriented,
p-dimensional submanifold.
(a) Suppose that ω1 , ω2 ∈ Ωp (M ) are cohomologous, meaning that they represent the same
de Rham cohomology class. Show that
Z
Z
ω1 =
ω2 .
Z
Z
(b) Show that integration over Z induces a linear map
Z
p
: HdR
(M ) → R.
Z
(c) Show that if Z is the (oriented) boundary of some
R compact, oriented, (p+1)-dimensional
submanifold of M , then the linear functional Z from part (b) is the zero map.
(d) Suppose the closed, oriented, p-dimensional submanifolds Z1 , Z2 ⊂ M are cobordant,
meaning that there exists a compact, oriented, (p + 1)-dimensional submanifold W ⊂ M
such that ∂W = Z1 t −Z2 (in other words, the oriented boundary of W is the disjoint
union of Z1 with its given Rorientation
R and Z2 with the opposite orientation). Show that
the two linear functionals Z1 and Z2 are equal.
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