Simplicial Structures in Topology

VI.1 Fundamental Group 205
We take a disk D 2 with center c and divide its boundary ∂D 2 into n + 1partsby
means of the (distinct) points b0,...,bn. LetD =(Y,Ψ) be the simplicial complex
with vertices
Y = {c,b0,...,bn}
and simplexes
Ψ = {c,b0,...,bn;b0b1,b1b2,...,bnb0,cb0,cb1,...,cbn,
cb0b1,cb1b2,...,cbnb0}
(we omit the curly brackets for the simplexes). The one-dimensional simplicial complex
∂D formed by the simplexes
is a subcomplex of D; note that
{b0,...,bn;b0b1,b1b2,...,bnb0}
|D| ∼ = D 2 and |∂ D| ∼ = ∂D 2 ∼ = S 1 .
The construction is nothing but a cone (with vertex c) on a simplicial subdivision of
S 1 . We now consider the simplicial function
f : ∂D → K , (∀i = 0,1,...,n) f (bi)=ai
and an adjunction space X defined by the pushout
S 1 ∼ = |∂ D|
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D 2 ∼ = |D|
With these conditions we have the following:
(VI.1.15) Theorem. The fundamental group π(X,a0) of the space X, obtained
by adjoining a 2-cell to |K| by means of the path α, is obtained from the group
π(|K|,a0) together with the relation defined by the homotopy class of the same
path α.
Proof. We consider the barycentric subdivision of the triangulation of D 2 just described,
in other words, before gluing D 2 to K, we refine the triangulation of D 2 by
adding 2(n + 1) new vertices
| f |
| f |
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|K|
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X
c0,c1,...,cn; d0,d1,...,dn