Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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102 III Homology of Polyhedra<br />
where αi is the sum of the coefficients ζr of the elements zr whose first coord<strong>in</strong>ate<br />
equals xi; now, we can easily see that ∑ m i=0 αi = 1. After mak<strong>in</strong>g similar remarks<br />
for |pr2|(u), we conclude that<br />
ψφ = 1 |K×L| .<br />
S<strong>in</strong>ce |K × L| is compact, φ is a homeomorphism. �<br />
(III.1.2) Example. The diagram <strong>in</strong> Fig. III.1 illustrates the case <strong>in</strong> which K is<br />
(0, 1)<br />
(0, 0)<br />
(2, 1)<br />
(2, 0)<br />
(1, 1)<br />
(1, 0)<br />
Fig. III.1<br />
a 2-simplex with vertices {0,1,2} and L is the 1-simplex with vertices {0,1}.<br />
The result<strong>in</strong>g prism consists of the three 3-simplexes {(0,0),(1,0),(2,0),(2,1)},<br />
{(0,0),(1,0), (1,1),(2,1)} and {(0,0),(0,1),(1,1),(2,1)}, respectively.<br />
Let K =(X, Φ) be a simplicial complex; we construct an <strong>in</strong>f<strong>in</strong>ite sequence of<br />
simplicial complexes {K (r) |r ≥ 0} as follows:<br />
1. K (0) = K<br />
2. K (1) =(X (1) ,Φ (1) ) is the simplicial complex def<strong>in</strong>ed by:<br />
(i) X (1) = Φ<br />
(ii) Φ (1) is the set of all nonempty subsets of Φ such that<br />
3. K (r) is def<strong>in</strong>ed iteratively:<br />
{σi0 ,...,σ<strong>in</strong> }∈Φ(1) ⇔ σi0 ⊂ ...⊂ σ<strong>in</strong> ;<br />
K (r) =(K (r−1) ) (1) .<br />
The simplicial complex K (r) is the rth- barycentric subdivision of K.<br />
(III.1.3) Remark. The def<strong>in</strong>ition we gave of the first barycentric subdivision K (1)<br />
of a simplicial complex is, perhaps, a little convoluted, and so we give here another,<br />
which is equivalent but refers to the geometric realization |K|. Foreveryn-simplex<br />
σ = {x0,...,xn} of K,wedef<strong>in</strong>ethebarycenter of σ to be the po<strong>in</strong>t<br />
b(σ)=<br />
n<br />
∑<br />
i=0<br />
1<br />
n + 1 xi ;