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Five Lemma


FiveLemma

A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds:

1. If f_0 is surjective, and f_1 and f_3 are injective, then f_2 is injective;

2. If f_4 is injective, and f_1 and f_3 are surjective, then f_2 is surjective.

If f_0,f_1,f_3 and f_4 are bijective, the hypotheses of (1) and (2) are satisfied simultaneously, and the conclusion is that f_2 is bijective. This statement is known as the Steenrod five lemma.

FiveLemma2

If A_0, B_0, A_4, and B_4 are the zero group, then f_0 and f_4 are zero maps, and thus are trivially injective and surjective. In this particular case the diagram reduces to that shown above. It follows from (1), respectively (2), that f_2 is injective (or surjective) if f_1 and f_3 are. This weaker statement is sometimes referred to as the "short five lemma."


See also

Commutative Diagram, Diagram Lemma, Exact Sequence, Four Lemma

This entry contributed by Margherita Barile

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References

Eilenberg, S. and Steenrod, N. Foundations of Algebraic Topology. Princeton, NJ: Princeton University Press, p. 16, 1952.Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, pp. 346-347, 1995.Lang, S. Algebra, rev. 3rd ed. New York: Springer Verlag, p. 169, 2002.Mac Lane, S. Homology. Berlin: Springer-Verlag, p. 14, 1967.Mitchell, B. Theory of Categories. New York: Academic Press, pp. 35-36, 1965.Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub., p. 140, 1993.Rotman, J. J. An Introduction to Algebraic Topology. New York: Springer-Verlag, pp. 98-99, 1988.Spanier, E. H. Algebraic Topology. New York: McGraw-Hill, pp. 185-186, 1966.

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Five Lemma

Cite this as:

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

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