1 Manifolds
A manifold is a space which looks like
R
n
at small scales (i.e. “locally”), but
which may be very diﬀerent from this at large scales (i.e. “globally”). In other
words, manifolds are made by gluing pieces of
R
n
together to make a more
complicated whole. We want to make this precise.
1.1 Topological manifolds
Deﬁnition 1.1.
A real, n-dimensional
topological manifold
is a Hausdorﬀ, sec-
ond countable topological space which is locally homeomorphic to
R
n
.
“Locally homeomorphic to
R
n
” simply means that each point
p
has an open
neighbourhood
U
for which we can ﬁnd a homeomorphism
ϕ
:
U
−→
V
to an
open subset
V
∈
R
n
. Such a homeomorphism
ϕ
is called a
coordinate chart
around
p
. A collection of charts which cover the manifold is called an
atlas
.
Remark 1.2.
Without the Hausdorﬀ assumption, we would have examples such
as the following: take the disjoint union
R
1

R
2
of two copies of the real line,
and form the quotient by the equivalence relation
R
1
\ {
0
} 
x
∼
ϕ
(
x
)
∈
R
2
\ {
0
}
,
(1)
where
ϕ
is the identiﬁcation
R
1
→
R
2
. The resulting quotient topological space is
locally homeomorphic to
R
but the points
[0
∈
R
1
]
,
[0
∈
R
2
]
cannot be separated
by open neighbourhoods.
Second countability is not as crucial, but will be necessary for the proof of
the Whitney embedding theorem, among other things.
We now give examples of topological manifolds. The simplest is, techni-
cally, the empty set. Then we have a countable set of points (with the discrete
topology), and
R
n
itself, but there are more:
Example 1.3
(Circle)
.
Deﬁne the circle
S
1
=
{
z
∈
C
:
|
z
|
= 1
}
. Then for
any ﬁxed point
z
∈
S
1
, write it as
z
=
e
2
πic
for a unique real number
0
≤
c <
1
,
and deﬁne the map
R
˜
ν
z


S
1
t
✤


e
2
πit
(2)
Let
I
c
= (
c
−
1
2
, c
+
1
2
)
, and note that
ν
z
= ˜
ν
z
|
I
c
is a homeomorphism from
I
c
to the neighbourhood of
z
given by
S
1
\{−
z
}
. Then
ϕ
z
=
ν
−
1
z
is a coordinate
chart near
z
.
By taking products of coordinate charts, we obtain charts for the Cartesian
product of manifolds. Hence the Cartesian product is a manifold.
Example 1.4
(n-torus)
.
S
1
× · · · ×
S
1
is a topological manifold (of dimension
given by the number
n
of factors), with charts
{
ϕ
z
1
× · · · ×
ϕ
z
n
:
z
i
∈
S
1
}
.
1

Example 1.5
(open subsets)
.
Any open subset
U
⊂
M
of a topological mani-
fold is also a topological manifold, where the charts are simply restrictions
ϕ
|
U
of charts
ϕ
for
M
. For instance, the real
n
×
n
matrices
Mat(
n,
R
)
form a vector
space isomorphic to
R
n
2
, and contain an open subset
GL
(
n,
R
) =
{
A
∈
Mat(
n,
R
) : det
A

= 0
}
,
(3)
known as the general linear group, which is a topological manifold.
Example 1.6
(Spheres)
.
The
n
-sphere is deﬁned as the subspace of unit vectors
in
R
n
+1
:
S
n
=
{
(
x
0
, . . . , x
n
)
∈
R
n
+1
:

x
2
i
= 1
}
.
Let
N
= (1
,
0
, . . . ,
0)
be the north pole and let
S
= (
−
1
,
0
, . . . ,
0)
be the south
pole in
S
n
. Then we may write
S
n
as the union
S
n
=
U
N
∪
U
S
, where
U
N
=
S
n
\{
S
}
and
U
S
=
S
n
\{
N
}
are equipped with coordinate charts
ϕ
N
, ϕ
S
into
R
n
, given by the “stereographic projections” from the points
S, N
respectively
ϕ
N
: (
x
0
, x
)
→
(1 +
x
0
)
−
1
x,
(4)
ϕ
S
: (
x
0
, x
)
→
(1
−
x
0
)
−
1
x.
(5)
Remark 1.7.
We have endowed the sphere
S
n
with a certain topology, but is
it possible for another topological manifold
˜
S
n
to be homotopy equivalent to
S
n
without
being homeomorphic to it? The answer is no, and this is known as the
topological Poincaré conjecture, and is usually stated as follows: any homotopy
n
-sphere is homeomorphic to the
n
-sphere. It was proven for
n >
4
by Smale,
for
n
= 4
by Freedman, and for
n
= 3
is equivalent to the smooth Poincaré
conjecture which was proved by Hamilton-Perelman. In dimensions
n
= 1
,
2
it
is a consequence of the classiﬁcation of topological 1- and 2-manifolds.
Example 1.8
(Projective spaces)
.
Let
K
=
R
or
C
. Then
K
P
n
is deﬁned to be
the space of lines through
{
0
}
in
K
n
+1
, and is called the projective space over
K
of dimension
n
.
More precisely, let
X
=
K
n
+1
\{
0
}
and deﬁne an equivalence relation on
X
via
x
∼
y
iﬀ
∃
λ
∈
K
∗
=
K
\{
0
}
such that
λx
=
y
, i.e.
x, y
lie on the same line
through the origin. Then
K
P
n
=
X/
∼
,
and it is equipped with the quotient topology.
The projection map
π
:
X
−→
K
P
n
is an
open
map, since if
U
⊂
X
is
open, then
tU
is also open
∀
t
∈
K
∗
, implying that
∪
t
∈
K
∗
tU
=
π
−
1
(
π
(
U
))
is
open, implying
π
(
U
)
is open. This immediately shows, by the way, that
K
P
n
is second countable.
To show
K
P
n
is Hausdorﬀ (which we must do, since Hausdorﬀ is preserved
by subspaces and products, but
not
quotients), we show that the graph of the
equivalence relation is closed in
X
×
X
(this, together with the openness of
π
,
gives us the Hausdorﬀ property for
K
P
n
). This graph is simply
Γ
∼
=
{
(
x, y
)
∈
X
×
X
:
x
∼
y
}
,
2

and we notice that
Γ
∼
is actually the common zero set of the following contin-
uous functions
f
ij
(
x, y
) = (
x
i
y
j
−
x
j
y
i
)
i

=
j.
An atlas for
K
P
n
is given by the open sets
U
i
=
π
(
˜
U
i
)
, where
˜
U
i
=
{
(
x
0
, . . . , x
n
)
∈
X
:
x
i

= 0
}
,
and these are equipped with charts to
K
n
given by
ϕ
i
([
x
0
, . . . , x
n
]) =
x
−
1
i
(
x
0
, . . . , x
i
−
1
, x
i
+1
, . . . , x
n
)
,
(6)
which are indeed invertible by
(
y
1
, . . . , y
n
)
→
(
y
1
, . . . , y
i
,
1
, y
i
+1
, . . . , y
n
)
.
Sometimes one ﬁnds it useful to simply use the “coordinates”
(
x
0
, . . . , x
n
)
for
K
P
n
, with the understanding that the
x
i
are well-deﬁned only up to overall
rescaling. This is called using “projective coordinates” and in this case a point
in
K
P
n
is denoted by
[
x
0
:
· · ·
:
x
n
]
.
Example 1.9
(Connected sum)
.
Let
p
∈
M
and
q
∈
N
be points in topological
manifolds and let
(
U, ϕ
)
and
(
V, ψ
)
be charts around
p, q
such that
ϕ
(
p
) = 0
and
ψ
(
q
) = 0
.
Choose

small enough so that
B
(0
,
2

)
⊂
ϕ
(
U
)
and
B
(0
,
2

)
⊂
ϕ
(
V
)
, and
deﬁne the map of annuli
B
(0
,
2

)
\
B
(0
, 
)
φ


B
(0
,
2

)
\
B
(0
, 
)
x
✤


2

2
|
x
|
2
x
(7)
This is a homeomorphism of the annulus to itself, exchanging the boundaries.
Now we deﬁne a new topological manifold, called the
connected sum
M
#
N
, as
the quotient
X/
∼
, where
X
= (
M
\
ϕ
−
1
(
B
(0
, 
)))

(
N
\
ψ
−
1
(
B
(0
, 
)))
,
and we deﬁne an identiﬁcation
x
∼
ψ
−
1
φϕ
(
x
)
for
x
∈
ϕ
−
1
(
B
(0
,
2

))
. If
A
M
and
A
N
are atlases for
M, N
respectively, then a new atlas for the connect sum is
simply
A
M
|
M
\
ϕ
−
1
(
B
(0
,
))
∪ A
N
|
N
\
ψ
−
1
(
B
(0
,
))
Remark 1.10.
The homeomorphism type of the connected sum of connected
manifolds
M, N
is independent of the choices of
p, q
and
ϕ, ψ
, except that it
may depend on the two possible orientations of the gluing map
ψ
−
1
φϕ
. To
prove this, one must appeal to the so-called
annulus theorem
.
Remark 1.11.
By iterated connect sum of
S
2
with
T
2
and
R
P
2
, we can obtain
all compact 2-dimensional manifolds.
3

Example 1.12.
Let
F
be a topological space. A ﬁber bundle with ﬁber
F
is a
triple
(
E, p, B
)
, where
E, B
are topological spaces called the “total space” and
“base”, respectively, and
p
:
E
−→
B
is a continuous surjective map called the
“projection map”, such that, for each point
b
∈
B
, there is a neighbourhood
U
of
b
and a homeomorphism
Φ :
p
−
1
U
−→
U
×
F,
such that
p
U
◦
Φ =
p
, where
p
U
:
U
×
F
−→
U
is the usual projection. The
submanifold
p
−
1
(
b
)
∼
=
F
is called the “ﬁber over
b
”.
When
B, F
are topological manifolds, then clearly
E
becomes one as well.
We will often encounter such manifolds.
Example 1.13
(General gluing construction)
.
To construct a topological mani-
fold “from scratch”, we glue open subsets of
R
n
together using homeomorphisms,
as follows.
Begin with a countable collection of open subsets of
R
n
:
A
=
{
U
i
}
. Then
for each
i
, we choose ﬁnitely many open subsets
U
ij
⊂
U
i
and gluing maps
U
ij
ϕ
ij


U
ji
,
(8)
which we require to satisfy
ϕ
ij
ϕ
ji
= Id
U
ji
, and such that
ϕ
ij
(
U
ij
∩
U
ik
) =
U
ji
∩
U
jk
for all
k
, and most important of all,
ϕ
ij
must be
homeomorphisms
.
Next, we want the pairwise gluings to be consistent (transitive) and so we
require that
ϕ
ki
ϕ
jk
ϕ
ij
= Id
U
ij
∩
U
jk
for all
i, j, k
. This will ensure that the
equivalence relation in (10) is well-deﬁned.
Second countability of the glued manifold is guaranteed since we started with
a countable collection of opens, but the Hausdorﬀ property is not necessarily
satisﬁed without a further assumption: we require that the graph of
ϕ
ij
, namely
{
(
x, ϕ
ij
(
x
)) :
x
∈
U
ij
}
(9)
is a closed subset of
U
i
×
U
j
.
The ﬁnal glued topological manifold is then
M
=

U
i
∼
,
(10)
for the equivalence relation
x
∼
ϕ
ij
(
x
)
for
x
∈
U
ij
, for all
i, j
. This space has a
distinguished atlas
A
, whose charts are simply the inclusions of the
U
i
in
R
n
.
4