1
Differential Topology/Geometry
1.1
Manifolds and Differentiable Structures
n
Manifold.
Let
X
be a topological space that is Hausdorff with a countable basis.
Then
X
is a
topological manifold
if for all
x
∈
X
, there exists a set
U
3
x
such that there is a homeomophism
ϕ
:
U
→
D
n
⊂
n
. We write
X
n
for an
n
manifold. The sets and functions
U
and
φ
are called a
local chart
.
Atlas.
Let
X
be a topological manifold.
A collection of charts
{
U
α
, ϕ
α
}
that covers
X
is called an
atlas
. The atlas
{
U
α
, ϕ
α
}
is a
smooth atlas
if for all
α, β
, the map
ϕ
β
◦
ϕ

1
α
is smooth (i.e. in
C
∞
). Two atlases are
equivalent
if their union is still a smooth atlas. A
smooth manifold
is an
equivalence class of atlases. The maps
ϕ
β
◦
ϕ

1
α
are called the
transition maps
.
Examples of Manifolds.
0.
n
with
U
α
open balls.
1.
S
n
with an atlas of 2
n
+ 2 charts that are simply the hemispheres.
S
n
could have an atlas of only 2
charts, namely the stereographic projections from the North and South poles.
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 Spring '06
 DAVIDCOOK
 Math, Logic, Geometry, Topology, Manifold, Topological manifold, smooth atlas, U Dn Rn, Dn Rn

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