Serge Ballif
MATH 527 Homework 5
October 5, 2007
Problem 1.
Show that if
A
is a nonsingular
3
by
3
matrix having nonnegative entries, then
A
has a positive eigenvalue.
Consider the closed positive octant
P
of the sphere
S
2
. Since
A
has nonnegative
entries, matrix multiplication gives us a continuous map
f
A
:
P
→
P
with rule
of assignment
f
A
(
p
) =
Ap
k
Ap
k
. The space
P
is homeomorphic to the closed unit
disk in the plane
B
2
=
{
(
x,y
)

x
2
+
y
2
≤
1
}
. Hence, by the Brouwer ﬁxed point
theorem,
f
A
must have a ﬁxed point
p
(i.e.
f
A
(
p
) =
p
= 1
·
p
). Therefore, the
number
k
Ap
k
is an eigenvalue for the eigenvector
p
.
Problem 2.
Let
p
:
E
→
B
be a continuous and surjective map. Suppose
U
is an open set of
B
that is evenly covered
by
p
. Show that if
U
is connected, then the partition
p

1
(
U
)
into slices is unique.
Let
{
A
α
}
and
{
B
β
}
be partitions of
p

1
(
U
) into slices. Then each set
A
α
and
each set
B
β
is connected, since they are homeomorphic to
U
. Let
a
0
∈
A
∈ {
A
α
}
be an element such that
p
(
a
0
) =
u
∈
U
. Then there exists a set
B
∈ {
B
β
}
that
contains the point
a
0
.
Claim 1.
A
=
B
.
Proof.
Suppose
A
6
=
B
. Then without loss of generality we can assume that there
is an element
a
∈
A
that is not in
B
. Hence, each
a
in
A

B
is in some set
B
a
∈ {
B
β
}
. Then the set
A
∩
S
a
∈
A

B
B
a
together with the set
A
∩
B
form a
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 Fall '07
 ROTMAN,REGINA
 Math, Topology, Serge Ballif, constant map

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