Simplicial Structures in Topology

124 III Homology of Polyhedra
1-simplexes:
2-simplexes:
{0,y0},...,{0,yn−1}
{y0,∞},...,{yn−1,∞}
{y0,y1},...,{yn−1,y0}
{0,y0,y1},...,{0,y1,y2},...,{0,yn−1,y0}
{∞,y0,y1},...,{∞,y1,y2},...,{∞,yn−1,y0} ,
whose geometric realization is also homeomorphic to S 2 . Note that
f (0)=0 , f (∞)=∞ and f (xs)=ys.
Let z be a generator of H2(L;Z) (the sum of all oriented 2-simplexes of L) andz ′
be the corresponding generator of K (a subdivision of L); we can easily see that
H2( f )(z ′ )=nz, which means that gr ( f )=n.
Since the degree of a map is invariant up to homotopy, we conclude that
gr (P)=n. Assuming that P is not a surjection, there should exist a p ∈ S 2 such
that P(S 2 ) ⊂ S 2 \p ≈ R 2 . We could then interpret P as a map from S 2 to R 2 and,
therefore, homotopic to the constant map from S 2 to 0 ∈ R 2 (if X is a topological
space and h: X → R 2 is a continuous function, then h is homotopic to the constant
map to 0 through the homotopy ht(x)=tx). But a constant map has degree 0,
contradicting the fact that P has degree n > 0. �
Exercises
1. Let A: S 1 → S 1 be the antipodal function. Prove that gr A = 1.
2. Prove that there is no retraction of the unit disk D n ⊂ R n onto its boundary S n−1 .
3. A polyhedron |K| has the fixed point property if every map f : |K|→|K| has at
least one fixed point. Prove that the fixed point property is invariant up to homeomorphism.
4. Let |K| be a polyhedron with the fixed point property. Prove that if |L| is a retraction
of |K| then also |L| has the fixed point property.
5. Consider the space
X = {(x0,x1,x2) ∈ R 3 |(∀i = 0,1,2)xi ≥ 0}
with the topology induced by R 3 and let f : X → X be a given continuous function.
Prove that it is possible to find a unit vector�v ∈ X and a nonnegative real number λ
such that f (�v)=λ�v.