Simplicial Structures in Topology

VI.1 Fundamental Group 201
We now define
φ(gij) :=[αi{ai,a j}α −1
j ].
Notice that φ is independent from the choice of αi and α j: a path of 1-simplexes
contained in L is contractible because L is contractible; hence, if α ′ i and α′ j were
new paths of 1-simplexes from a0 to ai and a j, respectively, we would have αi ∼ α ′ i
and α j ∼ α ′ j for they are contractible.
Suppose {ai,a j,ak} to be a 2-simplex of K;then
φ(gij)φ(g jk)φ(gki)=[αi{ai,aj}α −1
j ][α j{a j,ak}α −1
k ][αk{ak,ai}α −1
i ]
=[αiα −1
i ]=1
and so φ may be extended to a group homomorphism.
Now, for every generator gij of Gp(S;R),
ψφ(gij)=ψ([αi{ai,aj}α −1
j ]) = gij
as the paths αi and α j consist of 1-simplexes of L (see the definition of φ). Therefore,
ψφ = 1 Gp(S;R).
On the other hand, for every 1-simplex {ai,aj} of K
φψ([αi{ai,a j}α −1
j ]) = [αi{ai,aj}α −1
j ]
(here αi and α j are paths of L taken accordingly to the definition of φ). Therefore,
for a closed path of 1-simplexes
we may write
α = {a0,ai}{ai,a j}...{ak,a0},
φψ(α)=φψ([α0{a0,ai}α −1
i ])...φψ([αk{ak,a0}α −1
0 ])
=[α0{a0,ai}α −1
i ]...[αk{ak,a0}α −1
0 ]=α,
and so φψ = 1 π(|K|,a0). �
We now compute the fundamental groups of some spaces by way of examples.
(VI.1.8) Example (π(S 1 ) ∼ = Z). We triangulate the circle with the simplicial complex
K of vertices a0,a1,a2 and 1-simplexes
{a0,a1},{a1,a2} and {a2,a0}.
Here, we have the spanning tree |L| given by the two simplexes {a0,a1} and
{a1,a2}. Hence, by Theorem (VI.1.7), π(S 1 ,a0) has only one generator.
(VI.1.9) Example (π(S 2 )=0). It follows directly from Theorem (VI.1.7), when
we triangulate S 2 as the boundary of a tetrahedron.