Chain Homology
For every p, the kernel of partial_p:C_p->C_(p-1) is called the group of cycles,
| Z_p={c in C_p:partial(c)=0}. |
(1)
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The letter Z is short for the German word for cycle, "Zyklus." The image partial(C_(p+1)) is contained in the group of cycles because partial degreespartial=0, and is called the group of boundaries,
| B_p={c in C_p:( exists b in C_(p+1):partial(b)=c)}. |
(2)
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The quotients H_p=Z_p/B_p are the homology groups of the chain.
Given a short exact sequence of chain complexes
| 0->A_*->B_*->C_*->0, |
(3)
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there is a long exact sequence in homology.
| ...->H_p(A)->H_p(B)->H_p(C)-->^deltaH_(p-1)(A)->.... |
(4)
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In particular, a cycle a in A_p with partiala=0, is mapped to a cycle b in B_p. Similarly, a boundary partiala^' in A_p gets mapped to a boundary partialb^' in B_p. Consequently, the map between homologies H_p(A)->H_p(B) is well-defined. The only map which is not that obvious is delta, called the connecting homomorphism, which is well-defined by the snake lemma.
Proofs of this nature are (with a modicum of humor) referred to as diagram chasing.
See also
Chain Complex, Chain Equivalence, Chain Homomorphism, Chain Homotopy, Homology, Snake LemmaThis entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd. "Chain Homology." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ChainHomology.html