Algebraic Topology - Computer Graphics and Geometric Modeling - page 445

æÆæææ r

n n +1 n

S D S

lead to maps

() ææ ( ) ææ ()

i r

n n + 1 n

* *

H S H D H S .

n n n

This would imply that the degree of the identity map r ° i : S n Æ S n is zero because
H n ( D n+1 ) = 0. Since the identity map has degree 1 (Theorem 7.5.1.1(1)), the retraction
r cannot exist.

More generally, if W is an orientable pseudomanifold whose boundary ∂ W is non-
empty and connected, then ∂ W is not a retract of W . One can use the same argument,
but one would have to prove that H n ( W ) = 0 first, which involves facts about homol-
ogy groups that we have not proved. Intuitively, the fact is clear however because there
is no nonzero n-cycle since the boundary of the sum of all the n-simplices in a trian-
gulation is nonzero.

7.5.1.5. Theorem. (The Brouwer Fixed Point Theorem) Every continuous map

f: D n Æ D n has a fixed point.

Proof. If f has no fixed points, then the map

n n -1

r : DS Æ

defined by

n-1 n-1

() = () r p point of q S where the ray from f p through meets p S

is a retraction of D n onto S n-1 , which is impossible by Theorem 7.5.1.4.

Finally,

Definition. Let M n be a closed orientable pseudomanifold. A homeomorphism
h: M n Æ M n is said to be orientation preserving if it has degree +1 and orientation
reversing if it has degree -1.

Note that by Theorem 7.5.1.1(5), the degree of h is ±1. More generally,

Definition. Let M n and W n be closed oriented pseudomanifolds and let f : M n Æ W n
be a continuous map. As indicated earlier, we can interpret the orientations as corre-
sponding to a choice of generators m M ΠH n ( M n ) and m W ΠH n ( W n ). Define the degree
of f , denoted by deg f, to be the unique integer defined by the property that

* (

) = ( )

f m deg f m .

M W

If f is a homeomorphism, then f is said to be orientation preserving if it has degree +1
and orientation reversing if it has degree -1.

The degree of a map between pseudomanifolds satisfies properties similar to those
stated in Theorem 7.5.1.1. We leave their statements and proofs as exercises for the
reader.