Chain Complex

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/ChainComplex.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ChainComplex.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Commentary and figure by
Todd Rowland
Wolfram Research, Inc.
Dec 11, 2001

A basis for the chains for the simplicial homology of the circle

<<Graphics`Arrow`
<<Graphics`Colors`

Simplicial homology is defined for simplical complexes.  The circle is  decomposed into simplices in the diagram below.  The  corresponding chain complex is given by these 'chains', i.e., integral combinations of the one dimenionsal pieces A and B in (Subscript[C, 0])and integral combinations of C and D in (Subscript[C, 1]).   The boundary of C is B-A and the boundary of D is A-B.  Hence C+D has no boundary, and is called a cycle.

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