Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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II.2 Abstract <strong>Simplicial</strong> Complexes 51<br />
d(| f |(p),| f |(q)) ≤ � 2(n + 1)d(p,q).<br />
Case 3: Let us assume that s(p) ⊂ s(q). Rewrite the <strong>in</strong>dices of the elements of s(p)<br />
and s(q) <strong>in</strong> such a way that, xi = yi for every i = 0,...,n. Similar to the previous<br />
case, we consider the elements<br />
�<br />
xi = yi, 0 ≤ i ≤ n<br />
zi =<br />
y j, n + 1 ≤ j ≤ m<br />
and the real numbers<br />
�<br />
−αi + βi, 0 ≤ i ≤ n<br />
γi =<br />
β j, n + 1 ≤ j ≤ m.<br />
If −αi +βi ≥ 0foreveryi = 0,...,n,thens(p)=s(q) and p = q, because ∑ n i=0 αi =<br />
1. Hence, there exists a number 0 ≤ i ≤ n such that −αi + βi < 0. At this po<strong>in</strong>t, we<br />
argue as <strong>in</strong> the previous case. If s(q) ⊂ s(p), we use an analogous procedure. �<br />
In particular, the follow<strong>in</strong>g result holds true:<br />
(II.2.8) Theorem. Any piecewise l<strong>in</strong>ear function (the simplicial realization of a<br />
simplicial function)<br />
�<br />
is cont<strong>in</strong>uous.<br />
F : |K|→|L| , F<br />
�<br />
n<br />
∑ αixi<br />
i=0<br />
=<br />
n<br />
∑<br />
i=0<br />
αiF(xi)<br />
Hence | | is a functor.<br />
We now <strong>in</strong>vestigate some of the properties of the geometric realization of a simplicial<br />
complex. Recall that it is possible to characterize a convex set X of an<br />
Euclidean space as follows: for every p,q ∈ X, the segment [p,q], with end-po<strong>in</strong>ts p<br />
and q, is conta<strong>in</strong>ed <strong>in</strong> X. As we are go<strong>in</strong>g to see <strong>in</strong> the next theorem, this convexity<br />
property is valid for the geometric realization of the complex σ (called geometric<br />
simplex), for every simplex σ of a simplicial complex K.<br />
(II.2.9) Theorem. Let K =(X,Φ) be a simplicial complex. The follow<strong>in</strong>g results<br />
hold true:<br />
(i) The geometric realization σ of any simplex σ ∈ Φ is convex.<br />
(ii) For every two simplexes σ,τ ∈ Φ we have<br />
(iii) For every σ ∈ Φ, σ is compact .<br />
|σ|∩|τ| = |σ ∩ τ|.<br />
Proof. (i) Assume that σ = {x0,...,xn} and let p,q be arbitrary po<strong>in</strong>ts of |σ|; suppose<br />
that p = ∑ n i=0 αixi and q = ∑ n i=0 βixi. The segment [p,q] is the set of all po<strong>in</strong>ts<br />
r = tp+(1 −t)q,foreveryt ∈ [0,1]. Then