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Polyhedral chain

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A linear expression in a region , where are -dimensional simplices lying in . By an -dimensional simplex (cf. Simplex (abstract)) in one means an ordered set of points in whose convex hull lies in . The boundary of a polyhedral chain is defined in the usual way. The concept of a polyhedral chain occupies a position intermediate between those of a simplicial chain of a triangulation of and a singular chain in , but differs from the latter in the linearity of the simplices.

References

[1] P.S. Aleksandrov, "Introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian)


Comments

The points making up a simplex are required to be in general position, i.e. they are not all contained in some -dimensional affine subspace of .

References

[a1] L.C. Glaser, "Geometrical combinatorial topology" , 1–2 , v. Nostrand (1970)
[a2] C.R.F. Maunder, "Algebraic topology" , v. Nostrand (1972)
How to Cite This Entry:
Polyhedral chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polyhedral_chain&oldid=31889
This article was adapted from an original article by S.V. Matveev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article