Download MATH 215C 1. Let X be a manifold with boundary and f : X → Y a

MATH 215C
1. Let X be a manifold with boundary and f : X → Y a smooth map to
a smooth manifold Y . Suppose that ∂f is transverse to the closed map
g : Z → Y . Show that there exists a map f 0 : X → Y homotopic to f
such that ∂f 0 = ∂f and f 0 is transverse to g.
2. Suppose M is a compact connected orientable smooth manifold with finite
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fundamental group. show that HdR
(M ) = 0
3. 9.18, 10.4, 10.7 from Madsen and Tornehave.
4. Let f : M → N be a proper smooth map, where M and N are connected
oriented n-manifolds without boundary. For a regular value y ∈ N and
yi ∈ f −1 (y), let yi = +1 P
iff f∗ yi preserves orientation, and otherwise
yi = −1. Define degy (f ) = yi ∈f −1 (y) yi .
• Show that f ∗ : Hcn (N ) → Hcn (M ) is multiplication by degy (f ).
• Assume that M is a closed connected orientable manifold of dimension n , and f : M → Rn+1 is a smooth embedding. Use degree
theory to prove that Rn+1 − f (M ) has exactly two connected components and f (M ) is the set theoretic boundary of each.
5. Let N be a smooth manifold. Let j : N → N × N be the diagonal
embedding, and ∆ = j(N ).
• Prove that two smooth maps fi : Mi → N , i = 1, 2 are transverse if
and only if f1 × f2 : M1 × M2 → N × N is transverse to ∆.
• Prove that the normal bundle of ∆ in N × N is isomorphic to T N .
6. Let M be a smooth manifold and N ⊂ M be a smooth submanifold. Let
i : N → M denote the inclusion map.
• Prove that i∗ : Ωk (M ) → Ωk (N ) is surjective. Let Ωk (M, N ) =
Ker(i). Then (Ω∗ (M, N ), d) is a chain complex, and we let HdR (M, N )
denote its homology.
• Construct a natural long exact sequence
i∗
∂∗
k+1
k+1
k
k
. . . → HdR
(M ) −→ HdR
(N ) −→ HdR
(M, N ) → HdR
(M ) → . . .
• Construct a natural isomorphism
k
ψ : HdR
(M, N ) → H k (M, N, R).
• Let V → N be the normal bundle and r : V → N be the retraction.
Pick an inner product on V , and define a function V → [0, 1] by
s(v) = λ(k v k2 ), where λ : R → [0, 1] is a function with compact
support such that λ(t) = 1 for t < 1. Using the TNT, we can
think of V as an open neighborhood of N in M . Prove that ∂ ∗ [ω] =
[ds ∧ r∗ (ω)]
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