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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
∗
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 Fall '04
 unknown
 Topology, Open set, Euclidean space, Hopf Theorem

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