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The Five Lemma

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Exact Sequences in Homological Algebra Group Homomorphisms +1 more→→The Kunneth Formula

Core Idea

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.

Explainer

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

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.

Practice Questions 4 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Introduction to Exponents → Order of Operations → Integer Order of Operations → Variable Expressions → The Distributive Property → Variables and Expressions Review → Introduction to Polynomials → Adding and Subtracting Polynomials → Multiplying Polynomials → Factorial → Permutations → Combinations → Counting Principles: Addition and Multiplication Rules → Introduction to Graph Theory → Propositional Logic Foundations → Logical Equivalences → Set Operations: Union, Intersection, and Complement → Proof by Cases → Proving by Cases and Exhaustion → Vacuous Truth and Trivial Cases → Proof by Cases (Proof by Exhaustion) → Mathematical Induction → Binary Operations and Algebraic Structures → Group Definition and Examples → Basic Properties of Groups → Group Homomorphisms → Group Isomorphisms → Cayley's Theorem → Cosets and Lagrange's Theorem → Normal Subgroups → Quotient Groups → First Isomorphism Theorem for Groups → First Isomorphism Theorem for Rings → First Isomorphism Theorem for Groups → Third Isomorphism Theorem for Groups → First Isomorphism Theorem for Rings → Integral Domains → Principal Ideal Domains → Unique Factorization Domains → Polynomial Rings → Field Extensions → Algebraic and Transcendental Elements → Splitting Fields → Finite Fields → Galois Groups → Fundamental Theorem of Galois Theory → Second Isomorphism Theorem for Groups → Direct Products of Groups → Classification of Finite Abelian Groups → Chain Complexes and the Boundary Operator → Exact Sequences in Homological Algebra → The Snake Lemma → The Five Lemma

Longest path: 105 steps · 462 total prerequisite topics

Prerequisites (3)

Exact Sequences in Homological Algebrahard Group Homomorphismshard The Snake Lemmasoft

Leads To (1)

The Kunneth Formulasoft