Difference between revisions of "Chain"
From Encyclopedia of Mathematics
(TeX) |
(section headings) |
||
| Line 1: | Line 1: | ||
{{TEX|done}} | {{TEX|done}} | ||
| + | |||
| + | ===In ordered sets=== | ||
The same as a [[Totally ordered set|totally ordered set]]. | The same as a [[Totally ordered set|totally ordered set]]. | ||
| + | ===In algebraic topology=== | ||
A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$. | A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)</TD></TR></table> | ||
Revision as of 22:30, 23 November 2014
In ordered sets
The same as a totally ordered set.
In algebraic topology
A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$.
References
| [1] | N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966) |
| [2] | P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) |
How to Cite This Entry:
Chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain&oldid=34920
Chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain&oldid=34920
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article