Last updates: 14 June 2012
Let be a
A simplicial complex with vertex set is a collection
of finite subsets of such that
- (a) If then
- (b) If and
Let be a simplicial complex.
The simplicial complex is partially ordered by inclusion.
A simplex is an element of .
A vertex is a simplex with one element.
- A face of a simplex
is a subset of .
A chamber is a maximal simplex.
Let be a simplicial complex
with vertex set .
- The exterior algebra of
with the relations
- The homology of is the homology of the complex
with boundary map
The standard -simplex is
More generally, let be a real vector space and let
be linearly independent vectors in
A geometric realization of is a topological space
whose structure is completely controlled by the simplicial complex
where each -simplex in
corresponds to a standard -simplex in
. It is a bit challenging to make precise sense of the
"completely controlled by" in sufficient generality so it is better to ignore this
problem and use simplicial complexes for examples but avoid simplicial
complexes in general theory (see also the discussion in [Hatcher, p.107]).
Another historical solution is to use simplicial sets (see
[Gelfand-Manin Ch. 1 Sec. 2.1.2]).
Notes and References
These notes are part of attempt to sort out the zoo of definitions of homology and
cohomology of a space (and the various corresponding definitions of a "space").
The definition of homology of a
simplicial complex is one of the traditional starting points in algebraic topology as
found, for example in [Mu, Ch. 1 \S 5]
N. Bourbaki, Algebra I,
Chapter I, Section 9 No. 4, Springer-Verlag, Berlin 1989.
General Topology, Chapter IV, Section 1, Springer-Verlag, Berlin 1989.
J.R. Munkres, Elements of algebraic topology,
Addison-Wesley Publishing Company, Menlo Park, CA, 1984. ix+454 pp. ISBN: 0-201-04586-9