Simplicial Structures in Topology

232 VI Homotopy Groups
8. Prove that the function p: R → S 1 defined by
(∀t ∈ R) p(t)=e 2πit
is a fibration with fiber Z over every point of S 1 ; with this result and the previous
exercise, prove that πn(S 1 ) ∼ = 0, for every n ≥ 2.
9. Prove that, for every n ≥ 1 and for every (Y,y0) ∈ Top ∗, the function
defined by
is a group homomorphism.
VI.4 Obstruction Theory
Σ∗ : πn(Y,y0) −→ πn+1(ΣY,[y0])
(∀[ f ] ∈ πn(Y,y0)) Σ∗([ f ]) = [Σ f ]
In this last section, we put together the homotopy groups and the cohomology with
coefficients in a homotopy group to study the map extension problem. More precisely,
let |K| be a polyhedron, |L| a subpolyhedron of |K|, andWatopological space. We intend to study under what conditions a map f : |L|→W can be extended
to a map g: |K|→W, in other words, when it is possible to find f : |K|→W such
that the diagram
|L|
���
��
ι ��
f ���
��
���
|K| ��
W
f
commutes. Here, ι is the inclusion map. We answer this question in the case where
W is n-simple, with 1 ≤ n ≤ dimK − 1 (see Definition (VI.3.14)). Note that the
homotopy groups of W do not depend on the choice of a base point; then, we forgo
the base point and just write πn(W) for such groups.
Let us suppose that K =(X,Φ) and L =(Y,Ψ); sinceLisasubcomplex of K, it
follows that Y ⊂ X and Ψ ⊂ Φ. We start by giving an orientation to K (and consequently
also to L) so that we are able to compute their homology and cohomology;
let Kn be the n-dimensional subcomplex of K (in other words, the union of all simplexes
whose dimension is less than or equal to n). Suppose that we have extended
f to a map f n : |Kn ∪ L| →W. We note that, for every (n + 1)-simplex σ of K, the
simplicial complex •
σ is a subcomplex of Kn ∪ L. Letfn σ be the restriction of f n to
| •
σ|;since| •
σ| ∼ = Sn , we may regard f n σ as a map from Sn to W; then this map defines
an element [ f n σ ] ∈ πn(W). By linearity, we define the homomorphism