A diagram lemma which states that every short exact sequence of chain complexes and chain
homomorphisms
gives rise to a long exact sequence in homology
where the map 
is the chain homomorphism induced by the boundary
operator of the chain complex 
.
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.Referenced
on Wolfram|Alpha
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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