Simplicial Structures in Topology

III.3 Some Applications 123
Proof. We consider that polynomial as a map P: R 2 → R 2 To prove that P is
surjective and that, as a consequence, there exists an α ∈ C corresponding to 0 ∈ R 2
such that
α n + a1α n−1 + ...+ an−1α + an = 0 .
We take the compactification S 2 of R 2 obtained by adding a point ∞ to R 2 and we
extend P to a map P: S 2 → S 2 by setting P(∞)=∞. Themap
H : S 2 × I → S 2 , (p,t) ↦→ z n +(1−t) � a1z n−1 �
+ ...+ an , ∞ ↦→ ∞
is a homotopy between maps P and
f : S 2 → S 2 , z ↦→ z n .
We wish to prove that gr ( f )=n. To this end, we construct two simplicial complexes
K and L homeomorphic to S 2 , and consider a simplicial approximation of f .Letus
take the complex numbers
xs = e 2πis
n 2 , s = 0,1,...,n 2 − 1
and construct the simplicial complex K,formedby
vertices:
{0},{x0},...,{x n 2 −1 },{∞} ;
1-simplexes:
2-simplexes:
{0,x0},...,{0,x n 2 −1 }
{x0,∞},...,{x n 2 −1 ,∞}
{x0,x1},...,{x n 2 −1 ,x0}
{0,x0,x1},...,{0,x1,x2},...,{0,x n 2 −1 ,x0}
{∞,x0,x1},...,{∞,x1,x2},...,{∞,x n 2 −1 ,x0} ,
whose geometric realization is homeomorphic to S 2 .
We now take the complex numbers
yt = e 2πit
n , t = 0,1,...,n − 1
and consider the simplicial complex L,formedby
vertices:
{0},{y0},...,{yn−1},{∞} ;