Math 205C  Topology Midterm
Erin Pearse
1.
a)
State the definition of an
n
dimensional topological (differentiable) manifold.
An
n
dimensional topological manifold
is a topological space that is Haus
dorff, has a countable basis at every point, and is
locally Euclidean
.
That is,
every point has a neighbourhood which is homeomorphic to an open set of
R
n
.
An
n
dimensional differentiable manifold
M
is an ndimensional topolog
ical manifold with a
differentiable structure
.
That is, there is a collection of
coordinate charts
f
=
{
(
U
i
, ϕ
i
)
}
which cover
M
such that
i)
∀
x
∈
M,
∃
U
x
∈
f
s.t.
x
∈
U
x
and
U
x
∼
=
V
⊆
R
n
ii) For any two charts (
U, ϕ
) and (
V, ψ
) ,
U
∩
V
6
=
∅
=
⇒
ψ
◦
ϕ

1
and
ϕ
◦
ψ

1
are diffeomorphisms of
ϕ
(
U
∩
V
) and
ψ
(
U
∩
V
) in
R
n
.
iii)
U
is maximal in the sense that any chart (
U, ϕ
) which compatible with
f
in the sense of (ii) is included in
f
.
b)
Describe all 1dimensional manifolds.
If a 1dimensional manifold is compact, it is homeomorphic to
S
1
.
If a 1
dimensional manifold is not compact, it is homeomorphic to
R
1
. Anything not
falling into either category can readily be shown to be (i) not 1dimensional, or
(ii) not a topological manifold.
c)
Describe all closed orientable and nonorientable 2dimensional manifolds.
If a 2dimensional closed manifold is orientable, then it is a sphere, a torus,
or a connected sum of tori.
That is, it is an
n
genus torus (with a sphere
corresponding to genus 0).
If a 2dimensional closed manifold is nonorientable, then it is a Klein bottle,
projective plane, or connected sum of them.
d)
What are the fundamental groups of the manifolds in (b) and (c)?
π
1
(
S
1
)
∼
=
Z
and
S
1
has universal covering space
R
1
.
π
1
(
R
1
)
∼
=
{
1
}
and
R
1
is its own universal covering space.
π
1
(
S
2
)
∼
=
{
1
}
and
S
2
has universal covering space
R
2
.
π
1
(
T
2
)
∼
=
Z
×
Z
and
T
2
has universal covering space
R
2
.
π
1
(#
g
i
=1
T
2
)
∼
=
Z
2
g
and #
g
i
=1
T
2
has universal covering space
R
2
.
π
1
(
K
)
∼
=
F
(
a,b
)
{
aba

1
b
}
and
K
has universal covering space
R
2
.
π
1
(
R
P
2
)
∼
=
Z
2
and
R
P
2
has universal covering space
S
2
.
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2
Math 205C  Topology Midterm
Erin Pearse
2.
Explain why each of the following figures either is or is not a topological manifold.
a)
This object is clearly 1dimensional. Take
x
to be an inter
section point with open nbd
U
as shown. Assuming this space has the natural
(subspace) topology,
U
would have to be homeomorphic to an open set
V
⊂
R
1
.
Since
U
 {
x
}
consists of three connected components, and for
V
 {
f
(
x
)
}
con
sists of only two,
f
:
U
→
V
cannot be a homeomorphism. This is explained in
a little better detail in (v), below.
¥
b)
By the same argument as above, any open nbd of the in
tersection point
x
cannot be homemorphic to an open set of
R
1
.
¥
c)
Though not a differentiable manifold, this space is clearly
homeomorphic to (
a, b
)
⊂
R
1
and is thus a 1dimensional topological manifold.
¥
d)
This space is not Hausdorff, as any open set containing
x
also contains
y
. Since it is not Hausdorff, it fails to be a topological manifold.
¥
e)
Any point of this space has a nbd homeomorphic to an open
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 Spring '11
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 Logic, Topology, Metric space, Erin Pearse, Topology Midterm

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