The five lemma states: given a commutative diagram with two exact rows of five terms each, if four of the five vertical maps are isomorphisms, then the fifth is also an isomorphism. More precisely, if the first and third vertical maps are surjective and the second and fourth are injective, then the middle map is injective; the dual statement gives surjectivity. The five lemma is the standard tool for proving that a map between homology groups is an isomorphism, by embedding it in a map of long exact sequences where the surrounding terms are known.
The five lemma is the most frequently used diagram lemma in homological algebra and algebraic topology. Consider a commutative diagram with two exact rows of five terms, connected by five vertical maps:
A_1 -> A_2 -> A_3 -> A_4 -> A_5
|alpha |beta |gamma |delta |epsilon
B_1 -> B_2 -> B_3 -> B_4 -> B_5
The five lemma states: if alpha, beta, delta, and epsilon are isomorphisms, then gamma is an isomorphism. The proof is a diagram chase in two parts (injectivity and surjectivity of gamma), each using only two of the four assumed isomorphisms.
Injectivity of gamma: Suppose gamma(a_3) = 0 in B_3. We want to show a_3 = 0. Map a_3 to A_4: the element d_3(a_3) in A_4 maps to d_3'(gamma(a_3)) = d_3'(0) = 0 in B_4 by commutativity. Since delta is injective, d_3(a_3) = 0. By exactness of the top row at A_3, a_3 = d_2(a_2) for some a_2 in A_2. Now beta(a_2) maps to gamma(d_2(a_2)) = gamma(a_3) = 0 in B_3, so d_2'(beta(a_2)) = 0 in B_3. By exactness of the bottom row at B_2, beta(a_2) = d_1'(b_1) for some b_1 in B_1. Since alpha is surjective, b_1 = alpha(a_1) for some a_1. Then beta(a_2) = d_1'(alpha(a_1)) = beta(d_1(a_1)) by commutativity. Since beta is injective, a_2 = d_1(a_1). Therefore a_3 = d_2(a_2) = d_2(d_1(a_1)) = 0 by exactness (im(d_1) subset ker(d_2)).
Surjectivity of gamma proceeds dually: start with b_3 in B_3, use surjectivity of delta to control d_3'(b_3), use exactness and surjectivity of alpha to adjust, and arrive at a preimage in A_3.
The five lemma is used in algebraic topology whenever a map between spaces induces a map of long exact sequences. For instance, to show that a map f : X -> Y inducing isomorphisms on the homology of subspaces A subset X and B subset Y also induces isomorphisms on relative homology H_*(X, A) -> H_*(Y, B): the long exact sequences of the pairs (X, A) and (Y, B) are connected by f_*, and the five lemma (applied term by term) upgrades the known isomorphisms on H_*(A) and H_*(X) to isomorphisms on H_*(X, A).
The short five lemma (the special case where A_1 = A_5 = B_1 = B_5 = 0) is the most common version in practice. It states: in a morphism of short exact sequences, if the "end" maps are isomorphisms, the "middle" map is too. This is used to show that homology is an invariant of the chain homotopy type, to prove the uniqueness of homology theories satisfying the Eilenberg-Steenrod axioms, and to compare different definitions of cohomology. The five lemma, together with the snake lemma, forms the core technical foundation of homological algebra.