Graphics Reference
In-Depth Information
3
2
p
p
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
1
1
Y
:
0
,
Æ
S
and
Y
:
-
p
,
Æ
S
1
2
2
defined by
()
=
()
=
(
)
YY
u
u
cos
u
,
sin
u
1
2
are a simpler set of local C
•
parameterizations. There are lots of different local C
•
parameterizations for
S
1
. On the other hand, there are also lots of local parameteri-
zations that are not C
•
. For example the map
(
)
Æ
1
j :
-
11
,
S
+
defined by
(
)
2
()
=-
(
)
(
]
j u
11
- +
u
,
1
+
u
,
u
Œ-
10
,
,
(
)
2
[
=--
11
(
u
)
,
1
-
u
,
u
Œ
01
,
)
,
is parameterization of a neighborhood of the point (0,1) that is continuous (C
0
) but
not C
1
because the derivative does not exist at u = 0.
The next theorem states an important property of differentiable manifolds,
although its proof is much too involved to present here. The main consequence for
us is that simplicial homology groups are defined for such manifolds and we can use
what we know about pseudomanifolds and homology manifolds.
8.3.3. Theorem.
Every C
r
manifold
M
, r ≥ 1, admits a triangulation that is infinite
in general, but if
M
is compact, then
M
has a finite triangulation that makes it into a
pseudomanifold if it is connected. Every closed compact connected C
r
manifold, r ≥
1, is a homology manifold.
Proof.
See [Munk61].
Manifolds are defined in terms of parameterizations and that is the most common
way they are presented, but there is another way, namely, they can sometimes be
defined as the set of zeros of a function. For example, the sphere
S
2
is the set of zeros
of the polynomial
(
)
=++-
2
2
2
px yz
,,
x
y
z
1
.
Let f :
R
n
Æ
R
m
. Define the set of zeros of f , V(f) , by
Definition.
{
}
()
=
-1
()
=Œ
n
()
=
Vf
f
0pR p
f
0
.
In practice, f is usually a polynomial, and in that case V(f) is also called an (affine
algebraic)
variety
. Algebraic geometry is that field in mathematics which tries to
analyze the topological structure of V(f) in terms of algebraic invariants associated to
