Simplicial Structures in Topology

VI.3 Homotopy Groups 213
We now set the rule that, for every σ 2
i
, F j
i
(σ 2
i
∂2(σ 2
i )=F 0
i (σ 2
i ) − F 1 (σ 2
i )+F 2
i (σ 2
i ).
2
) is the jth-face of σi ; in this manner,
Similarly to what we have done before for k = 0,1,2, let λ k
i (0) (respectively, λ k
i (1))
be a path of 1-simplexes of |K| that joins ∗ to the first (respectively, second) vertex
of Fk (σ 2
i ); we now associate the loop μi, defined by the composition of loops
λ 0
i (0).F 0
i (σ 2
i ).(λ 0
ν
−1
i (1)) × λ 1
i (0).F 1
i (σ 2
i ).(λ 1
ν
−1
i (1)) × λ 2
i (0).F 2
i (σ 2
i ).(λ 2
i (1)) −1 ,
with every 2-simplex σ 2
i
(see Sect. VI.1 on p. 195). On the other hand,
[ f ]+[π(|K|),π(|K|)] = [ ν
×i (μi) mi ]+[π(|K|),π(|K|)]
and, since every μi is homotopic to the constant loop because |σ i 2 | is contractible,
[ f ] ∈ [π(|K|),π(|K|)]. �
VI.3 Homotopy Groups
The fundamental group π(Y,y0) of a based space (Y,y0) ∈ Top ∗, henceforth denoted
by π1(Y,y0), is the first of a series
{πn(Y,y0)|n ≥ 1}
of groups associated with (Y,y0). All these groups, called homotopy groups of Y
(with base point y0) are homotopy invariants; in addition, we shall prove that, for
every n ≥ 2, all groups πn(Y,y0) are Abelian, even if π1(Y,y0) may not be Abelian.
As we have seen before, the principal tool in constructing the fundamental group
of a based space is the comultiplication
ν : S 1 → S 1 ∨ S 1 , e 2πti �
(e0,e
↦→
2π2ti ) 0 ≤ t ≤ 1 2
(e2π(2t−1)i ,e0) 1 2 ≤ t ≤ 1.
The comultiplication ν, that we now indicate with ν1, has a very simple geometric
interpretation: it is essentially the quotient map obtained by the identification of the
points (1,0) and (−1,0) of the unit circle S 1 . We may pursue a similar idea for
defining a comultiplication in a unit sphere S n ⊂ R n+1 ,forn ≥ 2: let S n−1 be the
intersection of S n with the hyperplane zn+1 = 0andlet
qn : S n −→ S n /S n−1
be the quotient map; then, the comultiplication
νn : S n −→ S n ∨ S n