Simplicial Structures in Topology

VI.3 Homotopy Groups 225
The following definition is important to Sect. VI.4.
(VI.3.14) Definition. A path-connected space Y is n-simple if, for a given y0 ∈ Y ,
the action of π1(Y,y0) on πn(Y,y0) is trivial; in other words, if φ n λ is the identity
isomorphism, for every loop with base at y0. This happens, for instance, if Y is
simply connected, thatistosay,ifπ1(Y,y0) ∼ = 0. Therefore, the n-sphere Sn is
n-simple. In other words, Y is n-simple if, for every pair of points y0 and y1 of
Y, πn(Y,y0) ∼ = πn(Y,y1), and this isomorphism does not depend on the choice of the
path from y0 to y1.
VI.3.2 On the Homotopy Groups of Spheres
We know that spheres have a very simple triangulation; in fact, the n-dimensional
sphere is homeomorphic to the polyhedron | •
σn+1|. This gives us an easy way to
prove that the fundamental group of S1 is isomorphic to Z and that π1(S2 ,e0) ∼ = 0
(see Sect. VI.1). We now prove that the “lower groups” of the spheres are trivial,
that is to say,
(VI.3.15) Theorem. For every n ≥ 2 and every r with 1 ≤ r ≤ n − 1,
πr(S n ,e0) ∼ = 0.
Proof. Let us choose [ f ] ∈ πr(Sn ,e0) arbitrarily; we must prove that the map f is
homotopic to the constant map ce0 .LetK =(X,Φ) and L =(Y,Ψ) be two simplicial
complexes such that |K| ∼ = Sr and |L| ∼ = Sn . By the Simplicial Approximation
Theorem, there exists a simplicial function g: K (t) → L, of a suitable barycentric
subdivision of K such that |g|∼ fF,whereF : |K (t) |→|K| is the homeomorphism
defined in Sect. III.1. Let us suppose that |K| and |K (t) | are identified with Sr ;in
addition, let us identify L with Sn and let us assume that |g| and f are maps from
Sr to Sn ; hence, notwithstanding the homeomorphisms, we have that |g| ∼ f .Yet,
since dimK (t) = r and r < n, the simplicial function g cannot be surjective and so
|g|: S r −→ S n
is not surjective. Let p be a point of S n that does not belong to the image of |g|;
hence,
|g|: S r −→ S n � {p} ∼ = R n
(the homeomorphism φ : S n � {p} →R n is a stereographic projection). Now, let
c: S r → R n be the constant map at the point φ(e0); from the identification S n �
{p}≡R n , we conclude that |g| and c are homotopic, with the homotopy given by
the map
H : S r × I −→ R n
(∀(x,t) ∈ S r × I) H(x,t)=(1 −t)|g|(x)+tc.
Wemaythensaythat f is homotopic to a constant function. �