Simplicial Structures in Topology

122 III Homology of Polyhedra
(III.3.6) Corollary. If a map f : S 2 → S 2 has no fixed point, there exists p ∈ S 2 such
that f (p)=−p.
Proof. Suppose that f has no fixed point; by the preceeding lemma gr ( f )=−1. In
particular, the antipodal function
A: S 2 → S 2 , p ↦→−p
has degree −1. Since gr (Af)=gr (A)gr ( f )=1, it follows that
Λ(Af)=1 +(−1) 2 gr (Af)=2
and so, there is p ∈ S 2 such that Af(p)=p; hence, f (p)=−p. �
(III.3.7) Corollary. There is no map f : S 2n → S 2n with n ≥ 1, such that the vectors
p and f (p) are perpendicular for every p ∈ S 2n .
Proof. Suppose there is such a function. Then
and we may therefore define a map
(∀p ∈ S 2n ) ||(1 −t) f (p)+tp|| �= 0
F : S 2n × I → S 2n , (p,t) ↦→
(1 − t) f (p)+tp
||(1 −t) f (p)+tp|| .
This map is a homotopy between function f and the identity function 1 S 2n. It follows
that
Λ( f )=Λ(1 S 2n)=1 +(−1) 2n = 2
and so (∃p ∈ S 2n ) f (p)=p, against the hypothesis on f . �
(III.3.8) Remark. A tangent vector field on a sphere S n is a set of vectors of R n+1
tangent to S n , one at each point p ∈ S n , and such that the length and direction of
the vector at p vary continuously with p. Corollary (III.3.7) states that no sphere of
even dimension has a nonvanishing tangent vector field. Odd-dimensional spheres
have such fields; for instance,
S 2n−1 → S 2n−1 , (x1,...,x2n) ↦→ (−x2,x1,...,−x2n,x2n−1) .
The important Fundamental Theorem of Algebra may be easily proved as a
consequence of the Lefschetz Fixed Point Theorem:
(III.3.9) Corollary. A polynomial
f (z)=z n + a1z n−1 + ...+ an−1z + an ∈ C[z]
(with n ≥ 1) has a complex root. Consequently, the equation f (z) =0 has n solutions,
not necessarily distinct.