Algebraic Topology - Computer Graphics and Geometric Modeling - page 443

Proof. One can find more details for the argument sketched above in [SeiT80]. Part
(1) actually follows from part (2) because a simply connected space cannot have any
nontrivial covering space.

7.5.1 The Degree of a Map and Applications

This section describes some applications related to orientable pseudomanifolds. We
shall yet again assume that polyhedra have well-defined homology groups.

Definition. Let M n be a closed orientable pseudomanifold. If f : M n Æ M n is a con-
tinuous map, define the degree of f , denoted by deg f, to be the unique integer defined
by the property that

* () = (

) fa deg fa

for all a Œ H n M n (ª Z ).

7.5.1.1. Theorem. Let M n be a closed orientable pseudomanifold and let f, g : M n Æ
M n be continuous maps.

(1) If f is the identity map, then deg f = 1.
(2) If f is a constant map, then deg f = 0.
(3) If f and g are homotopic, then deg f = deg g.
(4) deg (f ° g) = (deg f )(deg g).
(5) If f is a homeomorphism, then deg f =±1.

Proof. Easy.

To better understand the degree of a map f : S n Æ S n for arbitrary
n ≥ 1, note that by Theorems 7.4.1.16, 7.4.3.4, and 7.4.3.5 there is a commutative
diagram

() ææ ()

n n

p S p S

n n

f

*

m Ø Ø m ,

() ææ ()

n n

H S H S

n n

f

*

with the Hurewicz homomorphisms m being isomorphisms for all such n. Homology
classes are more algebraic and less geometric than elements of homotopy groups. By
looking at what happens in the diagram at the homotopy group level it is more intu-
itively obvious that the degree of f roughly states how many times S n is wrapped

around itself by f.

Let n ≥ 1. The reflection r : S n Æ S n defined by

7.5.1.2. Theorem.