Relative Homology and Excision

Given a topological pair $(X,A)$ with $A\subseteq X$, relative homology is the homology of the quotient chain complex $S_{\bullet }(X)/S_{\bullet }(A)$, where $S_{\bullet }$ denotes singular chains. It records the homology of $X$ while declaring chains in $A$ to be "free" — geometrically it captures cycles relative to a boundary allowed to wander inside $A$. Its importance is structural: the relative groups sit in a long exact sequence linking $H_{\bullet }(A)$, $H_{\bullet }(X)$, and $H_{\bullet }(X,A)$, converting hard absolute computations into bookkeeping. The excision theorem is the deep geometric input — it is what makes singular homology computable and what equates relative homology with the reduced homology of the quotient $X/A$ for good pairs. The surprising fact: although a single singular simplex can be large and overlap a cut, subdivision (barycentric, iterated) shrinks chains until excision becomes a purely algebraic chain-homotopy statement.

The Core Insight

Relative homology is the homology of a quotient complex; excision says that quotient complex is insensitive to the interior of what is quotiented out. The bridge between these two facts is the small simplices theorem: any singular chain is homologous to one built from arbitrarily small simplices, each fitting inside a member of a fixed open cover. Once chains are small enough, removing a set $U$ buried inside $A$ changes nothing because no surviving simplex needs $U$.

Formal Statement

The Relative Chain Complex

Let $(X,A)$ be a pair, $A\subseteq X$. Let $S_n(X)$ be the free abelian group on singular $n$-simplices $\sigma :{\Delta }^n\to X$. The inclusion $A\hookrightarrow X$ induces an injection $S_n(A)\hookrightarrow S_n(X)$. The relative chain group is the quotient

The boundary $\partial :S_n(X)\to S_{n-1}(X)$ satisfies $\partial S_n(A)\subseteq S_{n-1}(A)$, so it descends to $\bar{\partial }:S_n(X,A)\to S_{n-1}(X,A)$ with ${\bar{\partial }}^2=0$. The relative homology groups are

Relative Cycles and Boundaries — Geometric Form

A relative $n$-cycle is $c\in S_n(X)$ with $\partial c\in S_{n-1}(A)$. A relative $n$-boundary is $c=\partial d+e$ with $d\in S_{n+1}(X)$, $e\in S_n(A)$. Then

Long Exact Sequence of a Pair

For any pair $(X,A)$ there is a natural long exact sequence

Excision

Let $Z\subseteq A\subseteq X$ with $\overline{Z}\subseteq \mathrm{int}(A)$. Then the inclusion $(X\setminus Z,A\setminus Z)\hookrightarrow (X,A)$ induces an isomorphism

Relative Homology and Quotients (Good Pairs)

If $(X,A)$ is a good pair (i.e., $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$), then the quotient map $q:(X,A)\to (X/A,A/A)$ induces an isomorphism

Why It Holds / Derivation

Excision via Barycentric Subdivision

Fix the cover $\mathcal{U}=\{\mathrm{int}A,\mathrm{int}B\}$ of $X$ with $X=\mathrm{int}A\cup \mathrm{int}B$. Let $S_n^{\mathcal{U}}(X)\subseteq S_n(X)$ be the subgroup generated by simplices whose image lies in $A$ or in $B$ ("$\mathcal{U}$-small" simplices).

Samuel Eilenberg & Norman Steenrod | 1913–1998 / 1910–1971

Eilenberg (Polish-American, a founder of category theory with Mac Lane) and Steenrod (American, of Steenrod squares fame) isolated the formal properties that any reasonable homology theory must satisfy. In their 1952 Foundations of Algebraic Topology they listed seven Eilenberg–Steenrod axioms — homotopy invariance, exactness, excision, dimension, additivity — and proved these axioms uniquely determine homology on finite CW complexes. Excision was elevated from a computational trick into a defining axiom, recasting decades of ad hoc methods (Čech, Vietoris, singular) into one structural framework.

Building Intuition

Think of $H_n(X,A)$ as "homology of $X$ where you're allowed to dump boundaries into $A$." A relative cycle is a chain whose edge is permitted to terminate inside $A$ — like a strip of fabric whose frayed ends are tucked under the rug $A$. Nontrivial classes are cavities of $X$ that can only be closed off using $A$.

Excision says: the cavity doesn't care about junk buried deep inside the rug. If $Z$ is well inside $A$ (closure inside the interior), then carving $Z$ out of both $X$ and $A$ leaves the relative cavities untouched — there were never any cycles that needed the buried region $Z$ to register their class.

The mechanism is subdivision: a giant simplex straddling the cut $Z$ can always be chopped into tiny ones, each entirely on one side. Tiny simplices never need to cross into $Z$.

Excision: cutting out a deep subset $Z\subset\operatorname{int}A$ leaves relative homology unchanged

Worked Examples

Spheres via the Pair $(D^n,S^{n-1})$

Local Homology of a Manifold

Common Misconceptions

The Closure Condition Is Not Optional

Excision requires $\overline{Z}\subseteq \mathrm{int}(A)$, not merely $Z\subseteq A$. If $Z$ touches the boundary of $A$ — i.e., if $\mathrm{int}A\cup \mathrm{int}(X\setminus Z)\ne X$ — the subdivision argument fails because there is no open cover to make simplices small relative to. People routinely "excise the boundary" and get false isomorphisms. Always verify the topological openness/closure conditions, or use the equivalent statement $X=\mathrm{int}A\cup \mathrm{int}B$.

Excising Up to the Boundary Fails

Take $X=[0,1]$, $A=[0,1]$, and try $Z=\{1\}$. Here $\overline{Z}=\{1\}\not\subset \mathrm{int}A=(0,1)$. Then $H_n(X,A)=0$ (since $X=A$), but $H_n(X\setminus Z,A\setminus Z)=H_n([0,1),[0,1))=0$ too — vacuously fine here. The real failure appears with $X=S^1$, $A=$ upper closed semicircle, $Z=$ an arc touching an endpoint of $A$: then $\overline{Z}\not\subset \mathrm{int}A$ and the inclusion-induced map on relative homology is not an isomorphism. The slogan "you can always delete subsets of $A$" is false: you may only delete what is buried in the interior.

$H_n(X,A)\ne \widetilde H_n(X/A)$ for Bad Pairs

Theorem [6] needs the good pair hypothesis (a neighborhood deformation retract). For pathological subspaces — e.g., $A=$ the closed topologist's-sine-curve segment, or $A$ a non-neighborhood-retract Cantor-like set — the natural map $H_n(X,A)\to {\tilde{H}}_n(X/A)$ can fail to be an isomorphism. Do not blindly compute relative homology by collapsing $A$.

Connections

Relative homology is the prototype of a relative invariant, paralleling Relative Cohomology and the Long Exact Sequence of a Pair. Excision is one of the Eilenberg–Steenrod Axioms, and together with homotopy invariance and exactness it pins down ordinary homology on CW complexes. The triple $(X,A,B)$ generalizes the pair, producing the Long Exact Sequence of a Triple. Excision underlies the Mayer–Vietoris Sequence (via the pair $(X,B)$ with $X=\mathrm{int}A\cup \mathrm{int}B$) and the Suspension Isomorphism. The good-pair theorem links to CW Complexes and cellular homology, where $H_n(X^n,X^{n-1})$ is free on $n$-cells. Local homology 14 feeds directly into Orientation of Manifolds and Poincaré Duality. The excision argument's subdivision machinery reappears in Singular vs Simplicial Homology comparison theorems and in Bredon Equivariant Homology.

Real-World Applications

1. Brouwer fixed-point & invariance of domain. Local homology of ${\mathbb{R}}^n$ (Example 2) proves ${\mathbb{R}}^m\ncong {\mathbb{R}}^n$ for $m\ne n$ and that open sets of ${\mathbb{R}}^n$ map to open sets — foundational in topology of Euclidean spaces.
2. Degree theory. The fundamental class via $H_n(M,M\setminus \{x\})\cong \mathbb{Z}$ defines the local degree of a map, used in nonlinear PDE existence proofs and bifurcation theory.
3. Morse theory. Relative homology $H_{\bullet }(M^{\le b},M^{\le a})$ of sublevel sets, computed by excising away from critical points, yields the Morse inequalities used in geodesics and gauge theory.
4. Persistent homology / TDA. Relative and excisive computations let topological data analysis localize features in point-cloud data; software (Ripser, GUDHI) exploits subdivision-style reductions.
5. Knot and link invariants. Relative homology of knot complements rel boundary tori underlies computations of Alexander modules.
6. Configuration spaces in robotics. Excision computes homology of motion-planning complexes by cutting out obstacle neighborhoods.
7. Condensed-matter topology. Local homology and orientation classes formalize bulk–boundary correspondence and topological invariants of band structures.

Historical Development

• 1895–1904 — Poincaré introduces homology informally via Analysis Situs, using relative cycles bounded by submanifolds in disguise.
• 1920s — Veblen, Alexander, and Lefschetz develop combinatorial (simplicial) homology; relative chains appear in Lefschetz's duality work.
• 1930s — Čech and Vietoris construct homology for general spaces; excision-type arguments are implicit in covering refinements.
• 1935–1942 — Eilenberg systematizes singular homology, replacing combinatorial approximations with continuous singular simplices.
• 1944 — Eilenberg's singular homology paper establishes barycentric subdivision and the small simplices theorem rigorously, making excision a theorem.
• 1945 — Eilenberg & Mac Lane introduce categorical/functorial language, framing relative homology as a functor on the category of pairs.
• 1952 — Eilenberg & Steenrod publish Foundations of Algebraic Topology, axiomatizing homology with excision as a core axiom and proving uniqueness on finite complexes.
• 1960s — Generalized (extraordinary) homology theories (K-theory, cobordism) by Atiyah, Hirzebruch, Thom satisfy all axioms except dimension, confirming excision's universality.
• 1970s–80s — Bredon and others extend excision to equivariant and sheaf-theoretic settings; excision becomes a guiding principle in motivic and étale cohomology (Voevodsky's later work).

Subtleties & Limits

Excision holds cleanly for ordinary singular homology, but its generalizations are delicate. In algebraic geometry, the analog — excision for $K$-theory and motivic cohomology — fails for general schemes; pinning down when "excision holds" (e.g., for noncommutative rings, the excision problem in algebraic K-theory solved partially by Suslin–Wodzicki via $H$-unitality, and refined by Cortiñas, Land–Tamme) remains an active research arc. Similarly, for Borel–Moore homology and intersection homology, excision must be restricted or modified, and the correct "stratified excision" theorems for singular spaces are subtle. In condensed mathematics (Clausen–Scholze), formulating excision for analytic/condensed sheaves to recover classical results is ongoing. The general meta-question — for which (co)homology-like functors does excision hold, and what structure (descent, H-unitality, pro-cdh) forces it? — is not fully resolved.

See Also