Graphics Reference
In-Depth Information
(
)
=-
(
)
rx x
,
,...,
x
x x
,
,...,
x
12
n
+
1
12
n
+
1
has degree -1.
Proof.
It is not hard to prove the theorem by simple computations using an appro-
priate triangulation (K,j) of
S
n
and the simplicial map of K corresponding to r, but
the details are rather messy and we do not repeat them here. See [AgoM76]. Intu-
itively, think of
S
n
as a cell complex with two n-cells that consist of two hemispheres
(the parts with the x-coordinates nonnegative or nonpositive). Then a generator of
H
n
(
S
n
) can be represented by a cycle that consists of the sum of these two cells ori-
ented appropriately. The reflection will then map each of these oriented n-cells into
the other, but with the opposite orientation.
Let n ≥ 1. The antipodal map f :
S
n
Æ
S
n
, f(
p
) =-
p
, has degree
7.5.1.3. Theorem.
(-1)
n+1
.
Consider the reflections r
i
:
S
n
Æ
S
n
, 1 £ i £ n+1, defined by
Proof.
(
)
=
(
)
rx x
,
,...,
x
x
,...,
x
,
-
x x
,
,...,
x
.
i
12
n
+
1
1
i
-
1
i
i
+
1
n
+
1
Clearly, f = r
1
°
r
2
°
...
°
r
n+1
. Therefore, the theorem follows easily from Theorems
7.5.1.1(4) and 7.5.1.2.
Actually, the case where n is odd can be proved directly without appealing to
Theorem 7.5.1.2. In terms of coordinates,
(
)
=-
(
)
fx x
,
,...,
x
x
,
-
x
,...,
-
x
.
12
n
+
1
1
2
n
+
1
If n = 1, then f is just a rotation through 180 degrees and is homotopic to the
identity. The map
1
¥
[
Æ
1
h:
S
01
,
S
defined by
(
)
=
(
)
--
(
)
+
(
)
(
)
hx x t
,
,
cos
p
t
x
,
x
sin
p
t x
,
-
x
12
1
2
2
1
is one such homotopy. In the case of an arbitrary odd n, we have an even number of
coordinates and we can again define a homotopy between f and the identity by using
a map like h for each pair of coordinates x
2i-1
and x
2i
, i = 1,2, ...(n + 1)/2.
Using the properties of the degree of a map, we can easily deduce some
well-known theorems. See also Section 8.5.
S
n
is not a retract of
D
n+1
.
7.5.1.4. Theorem.
Proof.
Suppose that we have a retraction r :
D
n+1
Æ
S
n
. Let i :
S
n
Æ
D
n+1
be the natural
inclusion map. The maps
