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Difference between revisions of "Polyhedral chain"

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A linear expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736201.png" /> in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736202.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736203.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736204.png" />-dimensional simplices lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736205.png" />. By an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736206.png" />-dimensional simplex (cf. [[Simplex (abstract)|Simplex (abstract)]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736207.png" /> one means an ordered set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736208.png" /> points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p0736209.png" /> whose convex hull lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p07362010.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p07362011.png" /> and a singular chain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p07362012.png" />, but differs from the latter in the linearity of the simplices.
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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====
 
====References====
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====Comments====
 
====Comments====
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p07362013.png" /> points making up a simplex are required to be in general position, i.e. they are not all contained in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p07362014.png" />-dimensional affine subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073620/p07362015.png" />.
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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====
 
====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>
 
<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>

Latest revision as of 15:34, 20 April 2014

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

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


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

[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