Polyhedral chain

A linear expression $\sum_{i=1}^md_it_i^r$ in a region $U\subset\mathbf R^n$, where $t_i^r$ are $r$-dimensional simplices lying in $U$. By an $r$-dimensional simplex (cf. Simplex (abstract)) in $U$ one means an ordered set of $r+1$ points in $U$ whose convex hull lies in $U$. 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 $U$ and a singular chain in $U$, but differs from the latter in the linearity of the simplices.

#### References[edit]

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

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#### Comments[edit]

The $r+1$ points making up a simplex are required to be in general position, i.e. they are not all contained in some $(r-1)$-dimensional affine subspace of $\mathbf R^n$.

#### References[edit]

[a1] |  L.C. Glaser, "Geometrical combinatorial topology" , 1–2 , v. Nostrand (1970)

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[a2] |  C.R.F. Maunder, "Algebraic topology" , v. Nostrand (1972)