Subdivision

of a geometric simplicial complex

A geometric simplicial complex such that the underlying space coincides with the underlying space and such that each simplex of is contained in some simplex of . In practice, the transition to a subdivision is carried out by decomposing the simplices in into smaller simplices such that the decomposition of each simplex is matched to the decomposition of its faces. In particular, each vertex of is a vertex of . The transition to a subdivision is usually employed to demonstrate invariance of the combinatorially defined characteristics of polyhedra (cf. Polyhedron, abstract; for example, the Euler characteristic or the homology groups, cf. Homology group), and also to obtain triangulations (cf. Triangulation) with the necessary properties (for example, sufficiently small triangulations). A stellar subdivision of a complex with centre at a point is obtained as follows. The closed simplices of that do not contain remain unaltered. Each closed simplex containing is split up into cones with their vertices at over those faces of that do not contain . For any two triangulations and of the same polyhedron there exists a triangulation of obtained not only from but also from by means of a sequence of stellar subdivisions. The concept of a stellar subdivision may be formalized in the language of abstract simplicial complexes (simplicial schemes). Any stellar subdivision of a closed subcomplex can be extended to a stellar subdivision of the entire complex. The derived complex of a complex is obtained as the result of a sequence of stellar subdivisions with centres in all open simplices of in the order of decreasing dimensions. For an arbitrary closed subcomplex of a complex , the subcomplex is complete in the following sense: From the fact that all the vertices of a certain simplex lie in it follows that . If one takes as the centres of the derived complex the barycentres of the simplices, one gets the barycentric subdivision. If the diameter of each simplex of an -dimensional complex does not exceed , the diameters of the simplices in its barycentric subdivision are bounded by . The diameters of the simplices in the -fold barycentric subdivision of are bounded by , and so they can be made arbitrarily small by selecting sufficiently large.

References

[1]	P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)
[2]	P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1965)

References

[a1]	C.R.F. Maunder, "Algebraic topology" , Cambridge Univ. Press (1980)
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Sects. 4.4; 5.4