164 V.M. Buchstaber
α 0 α 0 α 1
0-dim 1-dim
Fig. 9.1.SimplexA pointx∈σncan be written inbarycentric coordinatesasx=∑nj=0xjαj,∑nj=0xj=1,xj≥ 0.Afaceofσnis the simplex determined by a subset of verticesα 0 ,...,αn.
The empty subset of vertices determined the empty face.
Example 1.A face of dim (n−1) is given byσjn−^1 =(α 0 ,...,α̂j,...,αn).
A finitesimplicial complexKis a finite collection of simplices satisfying
the following two properties:
- Each face of a simplex from the collection belongs to the collection
- The intersection of any two simplices from the collection is a face of each
Example 2.The boundary of ann-dim simplexσnis the union∪jσnj−^1 of
its (n−1)-dim facets, together with all their faces. This is an (n−1)-dim
simplicial complex, thestandardsimplicial subdivision of the sphereSn−^1.
Amap of simplicesσn 1 →σm 2 is a map from the vertices ofσ 1 nto the
vertices ofσ 2 mextended linear to the whole ofσn 1 .Asimplicial mapf:K 1 →
K 2 of complexes is a map whose restriction to each simplex is a map of
simplices. Therefore, a simplicial map is determined by the imagesf(αj)=βk,
where{αj}and{βk}are the sets of vertices ofK 1 andK 2.
Example 3.LetKbe any simplicial complex on the vertex set{v 0 ,...,vm− 1 },
andσm−^1 the standard simplex on the vertices{α 0 ,...,αm− 1 }. Then there
exists a canonical simplicial map (inclusion)
f:K↪→σm−^1determined byf(vj)=αj.
Given two simplicial complexesK 1 andK 2 , we say thatK 2 is a subdivision
ofK 1 if each simplex ofK 1 is a union of finitely many simplices ofK 2 and
the simplices ofK 2 are contained linearly in the simplices ofK 1.
The barycentric subdivisionK′provides a standard way to subdivide any
simplicial complexK.
The barycentric subdivision may be defined inductively: To subdivide an
n-simplexσnwe first barycentrically subdivide the faces ofσn, then introduce
yet another vertexα∈σnin the centre ofσn, and add new simplices of the
form (β 0 ,...,βk,α).