lecture36 - Lecture 36 The rst problem on todays homework...

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Lecture 36
The first problem on today’s homework will be to prove the inverse function
theorem for manifolds. Here we state the theorem and provide a sketch of the proof.
Let X, Y be n -dimensional manifolds, and let f : X → Y be a C ∞ map with
f ( p ) = p
1
.
Theorem 6.39. If df
p
: T
p
X → T
p
1
Y is bijective, then f maps a neighborhood V of
p diffeomorphically onto a neighborhood V
1
of p
1
.
Sketch of proof: Let φ : U → V be a parameterization of X at p , with φ ( q ) = p .
Similarly, let φ
1
: U
1
→ V
1
be a parameterization of Y at p
1
, with φ
1
( q
1
) = p
1
.
Show that we can assume that f : V → V
1
(Hint: if not, replace V by V ∩ f
− 1
( V
1
)).
Show that we have a diagram
�
⏐
⏐
f
V V
1
−−−→
�
⏐
⏐
(6.114)
φ φ
1
g
−−−→ U U
1
,
which defines g ,
g = φ
− 1
1
◦ f ◦ φ, (6.115)
g ( q ) = q
1
. (6.116)
So,
( dg )
q
= ( dφ
1
)
− 1
q
1
◦ df
p
◦ ( dφ )
q
. (6.117)
Note that all three of the linear maps on the r.h.s. are bijective, so ( dg )
q
is a bijection.
Use the Inverse Function Theorem for open sets in R n .
This ends our explanation of the first homework problem.
Last time we showed the following. Let X, Y be n -dimensional manifolds, and let
f : X → Y be a proper C ∞ map. We can define a topological invariant deg( f ) such
that for every ω ∈ Ω n
c
( Y ),
f
∗