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The Excision Theorem

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12topics build on this

474prerequisites beneath it

Relative Homology and the Long Exact Sequence of a Pair Singular Homology Groups→→Cellular Homology Jordan Curve Theorem (Homological Proof)+1 more

Core Idea

The excision theorem states that if Z is a subspace of A whose closure is contained in the interior of A, then the inclusion (X \ Z, A \ Z) -> (X, A) induces isomorphisms H_n(X \ Z, A \ Z) = H_n(X, A) for all n. In other words, we can "cut out" (excise) the subspace Z from both X and A without changing the relative homology. This theorem is what gives homology its local-to-global computational power: relative homology depends only on the behavior near the boundary of A in X, not on what happens deep inside A or far from A.

Explainer

The excision theorem is one of the Eilenberg-Steenrod axioms for homology theories, and it is the property that gives homology its remarkable computational power. The precise statement: if (X, A) is a pair and Z is a subset of A with cl(Z) contained in int(A), then the inclusion map (X \ Z, A \ Z) hookrightarrow (X, A) induces isomorphisms H_n(X \ Z, A \ Z) -> H_n(X, A) for all n. Equivalently (and often more useful in practice): if X = A union B with int(A) and int(B) covering X, then the inclusion (B, A intersect B) hookrightarrow (X, A) induces isomorphisms H_n(B, A intersect B) -> H_n(X, A).

The intuition behind excision is that relative homology H_n(X, A) measures the topology of X in the neighborhood of the "boundary" between A and its complement X \ A. What happens deep inside A is invisible (it is already quotiented out), and what happens far from A in X contributes nothing to the relative chains (which must have boundaries in A). Therefore, cutting out a piece Z that is buried deep inside A has no effect on the relative homology. The formal proof uses the technique of barycentric subdivision: singular chains can be subdivided into smaller and smaller pieces until every singular simplex maps either entirely into A or entirely into B = X \ Z, allowing chains to be decomposed into local contributions. This subdivision process does not change homology (subdivided chains are homologous to the originals), and once chains are local, the excision isomorphism follows.

The most important consequence of excision is the identification H_n(X, A) = reduced H_n(X/A) for good pairs (pairs where A is a neighborhood deformation retract in X). The proof: let X/A be the quotient space obtained by collapsing A to a point p. Then H_n(X/A, p) = H_n(X/A) for n > 0 (since a point has trivial higher homology). Excision (in the equivalent formulation) identifies H_n(X, A) with H_n(X/A, A/A) = H_n(X/A, p). This identification makes relative homology geometric: H_n(X, A) measures the holes in X that survive when we crush A to a point.

Excision is the engine behind the Mayer-Vietoris sequence. Given X = A union B with open cover, the long exact sequence of the pair (X, A) involves H_n(X, A). Excision identifies H_n(X, A) with H_n(B, A intersect B) (excise Z = X \ B). Substituting this into the long exact sequence and rearranging gives the Mayer-Vietoris sequence: ... -> H_n(A intersect B) -> H_n(A) direct sum H_n(B) -> H_n(X) -> H_{n-1}(A intersect B) -> ... This sequence allows computation of H_n(X) from the homology of the pieces A, B, and their intersection. Similarly, excision underlies cellular homology: the relative group H_n(Xⁿ, X^n-1) for a CW complex is computed by excising the (n-2)-skeleton, reducing to a wedge of spheres and giving H_n(Xⁿ, X^n-1) = Z^{number of n-cells}. Without excision, neither of these fundamental computational tools would exist.

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

Longest path: 106 steps · 474 total prerequisite topics

Prerequisites (2)

Relative Homology and the Long Exact Sequence of a Pairhard Singular Homology Groupshard

Leads To (3)

Cellular Homologyhard Jordan Curve Theorem (Homological Proof)soft The Mayer-Vietoris Sequencehard