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:
- 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);
- 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.