Simplicial Structures in Topology

VI.3 Homotopy Groups 221
VI.3.1 Action of the Fundamental Group on the Higher
Homotopy Groups
Like the fundamental group, the higher homotopy groups depend on the choice of a
base point. In fact, in Sect. VI.1 we have proved that a path λ : I → Y between two
points y0 and y1 of Y defines an isomorphism
as follows: for every [ f ] ∈ π1(Y,y0),
φ λ : π1(Y,y0) −→ π1(Y,y1)
φ λ ([ f ]) :=[λ −1 ∗ ( f ∗ λ)].
When λ is a closed path at y0, that is to say, a loop of Y with base at y0,wehavean
isomorphism from π1(Y,y0) onto itself and this defines an action
given by the conjugation:
φ : π1(Y,y0) × π1(Y,y0) −→ π1(Y,y0)
ν
−1
(∀[ f ],[g] ∈ π1(Y,y0)) φ([ f ],[g]) = [ f ] × [g] ν
× [ f ].
For the higher homotopy groups, we avail ourselves of the Homotopy Extension
Property for polyhedra. Let λ beapathfromy0to y1 in Y . We identify Sn with the
polyhedron | •
σn+1| and suppose that the base point e0 of Sn is identified with a vertex
of | •
σn+1|. By the Homotopy Extension Property, applied to the pair of polyhedra
(| •
σn+1|,e0), the function f and the path λ give rise to a homotopy (not necessarily
unique)
F : | •
σn+1|×I → Y
through the diagram
e0 ×{0}
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σn+1|×{0}
| •
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e0 × I
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λ
σn+1|×I
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f
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Y
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