Geometric complex

A set of simplices in Euclidean or in Hilbert spaces satisfying certain conditions. A finite geometric complex is a finite set of closed simplices in a Euclidean space, any two simplices either having no points in common or intersecting along a common face. Two geometric complexes are considered to be isomorphic if it is possible to establish a one-to-one correspondence between their vertices which induces a one-to-one correspondence between all their simplices. Any number of simplices of a geometric complex constitutes a subcomplex of it. Any geometric complex is isomorphic to a subcomplex of some simplex of sufficiently high dimension. The dimension of a finite geometric complex is the largest dimension of its constituent simplices.

An infinite geometric complex is defined, up to isomorphism, as a set of simplices of some, not necessarily countably-dimensional, Hilbert space; the vertices of the simplices are the ends of the vectors of some orthonormal basis. Any geometric complex defines a topological space, constituted by all the points of its simplices and known as the polyhedron of the geometric complex. The dimension of an infinite geometric complex is the least upper bound of the dimensions of its simplices. For an infinite geometric complex, the topology of the polyhedron induced by its imbedding into the ambient Hilbert space is not the only topology compatible with the ordinary topology on all its simplices; the weak topology may serve as an example.

Comments

In topology and differential geometry, these complexes occur most frequently as realizations of abstract simplicial complexes (cf. Simplicial complex). In fact, the phrase "simplicial complex" occasionally serves as a substitute for "geometric complex" . A surrounding Euclidean or Hilbert space occurs only implicitly as the set of all real-valued functions on the set of vertices. Specific examples are triangulations (cf. Triangulation) of polyhedra and nerves of open coverings (cf. Nerve of a family of sets). Note that a geometric complex is also a cellular complex (cf. Cell complex).

References

[a1]	J.R. Munkres, "Elements of algebraic topology" , Addison-Wesley (1984)
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. I, Sect. 6
[a3]	B. Grünbaum, "Convex polytopes" , Interscience (1967)
[a4]	B. Grünbaum, "Polytopes, graphs and complexes" Bull. Amer. Math. Soc. , 76 (1970) pp. 1131–1201