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Zig-Zag Lemma

A diagram lemma which states that every short exact sequence of chain complexes and chain homomorphisms

0-->C-->^phiD-->^psiE-->0

gives rise to a long exact sequence in homology

...-->H_p(C)-->^(Phi_*)H_p(D)-->^(Psi_*)H_p(E)-->^(partial_*)H_(p-1)(C)-->^(Phi_*)H_(p-1)(D)-->...,

where the map partial_* is the chain homomorphism induced by the boundary operator of the chain complex D.

The name of this lemma is due to its proof, which consists of diagram chasing along a staircase-like path.

See also

Chain complex, Chain Homomorphism, Diagram Chasing, Diagram Lemma, Exact Sequence, Homology

This entry contributed by Margherita Barile

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References

Davis, J. F. and Kirk, P. Lecture Notes in Algebraic Topology. Providence, RI: Amer. Math. Soc., p. 18, 2001.Munkres, J. R. "The Zig-Zag Lemma." §24 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 136-142, 1993.

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Zig-Zag Lemma

Cite this as:

Barile, Margherita. "Zig-Zag Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Zig-ZagLemma.html

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