View source for Polyhedral chain

{{TEX|done}}
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)|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Introduction to homological dimension theory and general combinatorial topology" , Moscow  (1975)  (In Russian)</TD></TR></table>



====Comments====
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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.C. Glaser,   "Geometrical combinatorial topology" , '''1–2''' , v. Nostrand  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.R.F. Maunder,   "Algebraic topology" , v. Nostrand  (1972)</TD></TR></table>