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The Mayer-Vietoris Sequence

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Relative Homology and the Long Exact Sequence of a Pair The Excision Theorem +1 more→→Homology of Spheres Jordan Curve Theorem (Homological Proof)+1 more

Core Idea

The Mayer-Vietoris sequence is the homological analogue of the inclusion-exclusion principle: it computes the homology of a union X = A union B from the homology groups of A, B, and their intersection A intersect B. The long exact sequence ... -> H_n(A intersect B) -> H_n(A) + H_n(B) -> H_n(X) -> H_{n-1}(A intersect B) -> ... systematically relates these groups, and the connecting homomorphism captures how the topology of the intersection constrains the topology of the whole. It is the primary computational tool for singular homology.

Explainer

The Mayer-Vietoris sequence is the most frequently used computational tool in homology. Suppose X = A union B, where A and B are open subsets (or more generally, the interiors of A and B cover X). Then there is a long exact sequence: ... -> H_n(A intersect B) -(i_*,-j_*)-> H_n(A) direct sum H_n(B) -(k_*+l_*)-> H_n(X) -partial-> H_{n-1}(A intersect B) -> ... The first map sends a class [c] in H_n(A intersect B) to the pair (i_*[c], -j_*[c]) of its images in A and B (with a sign chosen for exactness). The second map adds the images of classes in A and B to produce a class in X. The connecting homomorphism partial links the homology of X in one dimension to the homology of the intersection in one dimension lower.

The derivation of Mayer-Vietoris from excision and the long exact sequence of a pair is instructive. Start with the pair (X, A). Its long exact sequence involves H_n(X, A). Excision (with Z = X \ B, noting cl(X \ B) subset int(A) when {int(A), int(B)} covers X) gives H_n(X, A) = H_n(B, A intersect B). Now the long exact sequence of the pair (B, A intersect B) involves H_n(A intersect B) and H_n(B). Splicing these two long exact sequences together and rearranging yields the Mayer-Vietoris sequence. Understanding this derivation shows that Mayer-Vietoris is not a separate axiom but a consequence of the more fundamental excision property.

The Mayer-Vietoris sequence is most powerful when A, B, or A intersect B have simple homology. The classic example is the computation of H_*(Sⁿ) by covering the sphere with two contractible hemispheres whose intersection is S^n-1. Since contractible spaces have trivial higher homology, the Mayer-Vietoris sequence collapses to isomorphisms H_k(Sⁿ) = H_{k-1}(S^n-1) for k >= 2, giving the homology of all spheres by induction. For surfaces, one typically decomposes into pieces that retract to graphs or circles, and the Mayer-Vietoris sequence assembles the answer from these simple building blocks.

The Mayer-Vietoris sequence also exists for cohomology (with arrows reversed) and for reduced homology (which simplifies the low-dimensional terms). There are relative versions for pairs and versions for arbitrary coverings (the Mayer-Vietoris spectral sequence). The Euler characteristic satisfies the clean formula chi(A union B) = chi(A) + chi(B) - chi(A intersect B), which follows immediately from the exactness of the Mayer-Vietoris sequence and the additivity of the Euler characteristic on exact sequences. This "inclusion-exclusion for topology" is one of the most elegant consequences of the sequence and reinforces its role as the homological upgrade of a familiar combinatorial principle.

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 → Relative Homology and the Long Exact Sequence of a Pair → The Excision Theorem → The Mayer-Vietoris Sequence

Longest path: 107 steps · 475 total prerequisite topics

Prerequisites (3)

Relative Homology and the Long Exact Sequence of a Pairhard The Excision Theoremhard Exact Sequences in Homological Algebrasoft

Leads To (3)

Homology of Sphereshard Jordan Curve Theorem (Homological Proof)hard The Kunneth Formulasoft