Topology in Molecular Biology

9 Euler, Dehn–Sommerville Characteristics, and Their Applications 165

Fig. 9.2.Barycentric subdivision

Example 4.LetK be an (n−1)-dim simplicial complex on the vertex set
{v 0 ,...,vm− 1 },andσn−^1 the standard simplex on the vertices{α 0 ,...,αn− 1 }.
Then there exists a canonical simplicial map

φ:K′↪→σn−^1

determined byφ(ω)=αk,1≤k≤n, whereω=(vi 1 ,...,vik) is a simplex
ofK.

The mapφfrom the previous example belongs to a very special class of
simplicial maps, the so-calledbranched combinatorial coverings.
A mapp:K 1 →K 2 between simplicial complexesK 1 andK 2 is called a
branched combinatorial coveringif:

1. For any relatively open simplex
   ◦
   τ ∈K 2 the preimagep−^1 (
   ◦
   τ) is a finite
   non-empty disjoint union of relatively open simplices

◦

ωi(τ);

2. The mapp:

◦

ωi(τ)→

◦
τis a homeomorphism for alli.
Two simplicial complexesK 1 andK 2 are said to becombinatorially equiv-
alentif there exists a simplicial complexKisomorphic to a subdivision of
each of them. The combinatorial neighbourhood of a simplexτ ∈Kis the
subcomplex consisting of all simplices, together with their boundaries having
the simplexτas a face. A simplicial complexKis called ann-dim piecewise
linear (PL-) manifoldif after application of a sequence of barycentric subdi-
visions the combinatorial neighbourhood of each simplex becomes a complex
combinatorially equivalent toσn.

9.3 Euler Characteristic and Dehn–Sommerville Characteristics

Characteristics

Thef-vector of an (n−1)-dim simplicial complexKis given by

f(K)=(f 0 ,f 1 ,...,fn− 1 ),

wherefiis the number ofi-dim simplices ofKn−^1.
TheEuler characteristicofKn−^1 is

χ(Kn−^1 )=f 0 −f 1 +···+(−1)n−^1 fn− 1.