Math 343 Homework 5
Due 4pm Friday March 12, 2010.
I will be away at MSRI in Berkeley CA from March 7–19. We can set up times to
discuss this homework via Skype while I am away. The homework will be collected for me
at 4pm on the day it is due.
1
To be handed in for grading
1. Which of the following linear spaces intersect transversally? Explain your answers.
(a)
xy
plane and the
z
axis in
R
3
(b)
R
k
× {
0
}
and
{
0
} ×
R
l
in
R
n
. (Depends on
k,l,n
.)
(c)
V
× {
0
}
and the diagonal in
V
×
V
2.
Transversality and vector spaces.
(a) Suppose
V
and
W
are vector subspaces of
R
n
. The space
V
+
W
is deﬁned as
follows
V
+
W
=
{
~v
+
~w

~v
∈
V, ~w
∈
W
}
.
Check that
V
+
W
is a vector subspace of
R
n
.
(b) Check that
V
t
W
means just
V
+
W
=
R
n
.
3. If
U
⊂
R
k
and
V
⊂
H
k
are neighborhoods of 0, prove that there exists no diﬀeomor
phism of
V
with
U
.
4. Prove that if
f
:
X
→
Y
is a diﬀeomorphism of manifolds with boundary, then
∂f
maps
∂X
diﬀeomorphically onto
∂Y
. (Recall
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 Spring '10
 ElizabethDenne
 Vector Space, Brouwer Theorem, following linear spaces, solid hyperboloid x2

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