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Math 598
Mar 20, 2005
1
Geometry and Topology II
Spring 2005, PSU
Lecture Notes 10
3.4 Proof of Sard’s Theorem
This section will be typeset later. In the meantime the reader is refered to
Milnor’s book on
Topology from DiFerentiable View Point
.
Exercise 1.
Show that Sard theorem immediately implies that if
f
:
M
→
N
is a smooth function and dim(
M
)
<
dim(
N
) then
f
(
M
) has measure zero.
Exercise 2.
Use Sard’s theorem to show that
S
n
is simply connected for
n
≥
2. (Hint: it is enough to show that every continuos map
f
:
S
1
→
S
2
is homotopic to a map
f
:
S
1
→
S
2
which is not onto. You also need to use
Wierstrauss’s approximation theorem.)
Exercise 3.
Let
M
n
be a compact manifold smoothly embedded in
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff
 Geometry, Topology

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