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