Relative Homology and Excision

⚡ Key Takeaways

Relative homology $H_n(X,A)$ measures the homology of a space $X$ "rel" a subspace $A$ , vanishing on cycles that live in $A$ . Excision asserts that homology classes do not "see" pieces you cut out from the interior of $A$ : $H_n(X,A)\cong H_n(X\setminus U, A\setminus U)$ when $\overline U\subset \operatorname{int}A$ . Together they are the engine behind the long exact sequence of a pair, the suspension isomorphism, and essentially every computation in singular homology. Introduced in the 1930s–40s (Eilenberg, Steenrod) and axiomatized in their 1952 Foundations of Algebraic Topology.

💡 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 $U$ .

DefinitionThe 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

1

\quad S_n(X,A)\;=\;S_n(X)\big/ S_n(A).

Derivation hint: $S_n(A)$ is a subgroup because every simplex into $A$ is a simplex into $X$ ; the quotient is free on simplices whose image is not contained in $A$ . 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

2

\quad H_n(X,A)\;=\;\ker\bar\partial_n\big/\operatorname{im}\bar\partial_{n+1}.

Derivation hint: well-defined since $\partial^2=0$ on $S_\bullet(X)$ passes to the quotient; relative cycles are chains $c$ with $\partial c\in S_{n-1}(A)$ .

DefinitionRelative 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

3

\quad H_n(X,A)=\frac{\{c\in S_n(X): \partial c\in S_{n-1}(A)\}}{\{\partial d+e : d\in S_{n+1}(X),\,e\in S_n(A)\}}.

Derivation hint: lift cosets in $S_n(X,A)$ back to representatives in $S_n(X)$ ; the kernel/image conditions translate exactly to "boundary lands in $A$ " and "differs from a genuine boundary by a chain in $A$ ."

TheoremLong Exact Sequence of a Pair

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

4

\quad \cdots\to H_n(A)\xrightarrow{i_*} H_n(X)\xrightarrow{j_*} H_n(X,A)\xrightarrow{\partial_*} H_{n-1}(A)\to\cdots

where $i:A\hookrightarrow X$ , $j$ is the quotient $S_\bullet(X)\to S_\bullet(X,A)$ , and $\partial_*[c]=[\partial c]$ is the connecting homomorphism. Derivation hint: apply the zig-zag/snake lemma to the short exact sequence of chain complexes $0\to S_\bullet(A)\to S_\bullet(X)\to S_\bullet(X,A)\to 0$ .

TheoremExcision

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

5

\quad H_n(X\setminus Z,\,A\setminus Z)\;\xrightarrow{\;\cong\;}\;H_n(X,A)\qquad \text{for all } n.

Equivalently, if $X=\operatorname{int}(A)\cup\operatorname{int}(B)$ then $H_n(B,\,A\cap B)\cong H_n(X,A)$ . Derivation hint: set $B=X\setminus Z$ ; the closure condition $\overline Z\subset\operatorname{int}A$ is exactly $\operatorname{int}A\cup\operatorname{int}B=X$ .

TheoremRelative 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

6

\quad H_n(X,A)\;\cong\;H_n(X/A,\,A/A)\;\cong\;\widetilde H_n(X/A),

where $\widetilde H_\bullet$ is reduced homology. Derivation hint: excise $A$ minus a collar; the deformation retraction collapses the collar, identifying the pair with the based quotient.

ProofExcision via Barycentric Subdivision

Fix the cover $\mathcal U=\{\operatorname{int}A,\operatorname{int}B\}$ of $X$ with $X=\operatorname{int}A\cup\operatorname{int}B$ . 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).

Step 1 — Barycentric subdivision operator. Define $\mathrm{Sd}:S_n(X)\to S_n(X)$ by linearly subdividing each simplex about its barycenter, extended affinely then pushed forward. One checks the chain-map identity

7

\quad \partial\circ\mathrm{Sd}=\mathrm{Sd}\circ\partial.

Derivation hint: prove first on the universal model $\Delta^n$ (identity map) by induction on $n$ using the cone formula $\mathrm{Sd}(c)=b\!\cdot\!\mathrm{Sd}(\partial c)$ where $b$ is the barycenter; push forward by naturality.

Step 2 — Chain homotopy to the identity. Construct $T:S_n(X)\to S_{n+1}(X)$ (the prism/cone operator) with

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\quad \partial T + T\partial \;=\; \mathrm{id}-\mathrm{Sd}.

Derivation hint: build $T$ on $\Delta^n$ by coning the homotopy $T(c)=b\!\cdot\!\big(c-\mathrm{Sd}(c)-T(\partial c)\big)$ inductively; naturality transports it to any $X$ .

Step 3 — Iterate to shrink. By the Lebesgue number lemma applied to $\Delta^n$ , sufficiently many iterations $\mathrm{Sd}^m\sigma$ make every piece $\mathcal U$ -small. The iterated homotopy

9

\quad D_m=\sum_{i=0}^{m-1} T\,\mathrm{Sd}^{i}\quad\text{gives}\quad \partial D_m+D_m\partial=\mathrm{id}-\mathrm{Sd}^m,

so each chain is homologous to a $\mathcal U$ -small chain. Hence $H_n^{\mathcal U}(X)\to H_n(X)$ is an isomorphism (the small simplices theorem). Derivation hint: sum the telescoping homotopies $T\,\mathrm{Sd}^i$ ; the diameter of subdivided simplices shrinks by factor $\le n/(n+1)$ each pass.

Step 4 — Algebraic excision. With small chains, the inclusion

S_\bullet(B)\big/S_\bullet(A\cap B)\;\xrightarrow{\;\cong\;}\;S_\bullet^{\mathcal U}(X)\big/S_\bullet(A)

is an isomorphism of complexes: every $\mathcal U$ -small simplex not in $A$ lies in $B$ , and those in $A$ die in both quotients. Combining with Step 3 (which gives $H_n^{\mathcal U}(X,A)\cong H_n(X,A)$ ) yields

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\quad H_n(B,A\cap B)\;\cong\;H_n(X,A).

Setting $B=X\setminus Z$ gives the stated form. $\blacksquare$ Derivation hint: the only subtlety is checking that the subdivision homotopy descends to the relative complex; it does because $T$ and $\mathrm{Sd}$ are natural and preserve $S_\bullet(A)$ .

∎

mathematician · 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.

🧠 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 $Z$ .

Figure

Excision: cutting out a deep subset

Z\subset\operatorname{int}A

leaves relative homology unchanged

ExampleSpheres via the Pair

(D^n,S^{n-1})

Given: Compute $H_k(D^n,S^{n-1})$ and use it to find $\widetilde H_k(S^n)$ .

Step 1. $D^n$ is contractible, so $\widetilde H_k(D^n)=0$ for all $k$ . The long exact sequence [4] in reduced form gives

11

\quad \cdots\to \widetilde H_k(D^n)\to H_k(D^n,S^{n-1})\xrightarrow{\partial_*}\widetilde H_{k-1}(S^{n-1})\to \widetilde H_{k-1}(D^n)\to\cdots

Derivation hint: replace absolute by reduced groups; $H_k(D^n,S^{n-1})$ is unchanged since reduced/unreduced agree relatively.

Step 2. With $\widetilde H_k(D^n)=\widetilde H_{k-1}(D^n)=0$ , the connecting map is an isomorphism:

12

\quad H_k(D^n,S^{n-1})\;\cong\;\widetilde H_{k-1}(S^{n-1}).

Derivation hint: a map sandwiched between two zeros in an exact sequence is an isomorphism.

Step 3. Since $(D^n,S^{n-1})$ is a good pair and $D^n/S^{n-1}\cong S^n$ , theorem [6] gives $H_k(D^n,S^{n-1})\cong\widetilde H_k(S^n)$ . Combining with [12]:

13

\quad \widetilde H_k(S^n)\;\cong\;\widetilde H_{k-1}(S^{n-1}).

Boxed answer: inducting from $\widetilde H_k(S^0)=\mathbb Z$ for $k=0$ else $0$ :

\boxed{\;\widetilde H_k(S^n)=\begin{cases}\mathbb Z,& k=n\\ 0,&k\ne n\end{cases}\;}

ExampleLocal Homology of a Manifold

Given: Let $M$ be an $n$ -manifold, $x\in M$ . Compute the local homology $H_k(M,M\setminus\{x\})$ .

Step 1. Choose a chart $U\cong\mathbb R^n$ around $x$ . Apply excision with $Z=M\setminus U$ (closed) and $A=M\setminus\{x\}$ . Check $\overline Z=Z\subset M\setminus\{x\}=\operatorname{int}A$ (since $A$ is open). Then

14

\quad H_k(M,M\setminus\{x\})\;\cong\;H_k(U,U\setminus\{x\})\;\cong\;H_k(\mathbb R^n,\mathbb R^n\setminus\{0\}).

Derivation hint: excise everything outside the chart; locality of homology.

Step 2. $\mathbb R^n\setminus\{0\}$ deformation retracts to $S^{n-1}$ , and $\mathbb R^n$ is contractible. The reduced long exact sequence gives, as in [12],

15

\quad H_k(\mathbb R^n,\mathbb R^n\setminus\{0\})\cong \widetilde H_{k-1}(S^{n-1})\cong\begin{cases}\mathbb Z,&k=n\\0,&\text{else.}\end{cases}

Derivation hint: same sandwich argument; $\widetilde H_{k-1}(S^{n-1})=\mathbb Z$ iff $k-1=n-1$ .

Boxed answer:

\boxed{\;H_k(M,M\setminus\{x\})=\begin{cases}\mathbb Z,&k=n\\0,&k\ne n\end{cases}\;}

This $\mathbb Z$ in degree $n$ is the seed of orientation and fundamental classes; a choice of generator is a local orientation at $x$ .

⚠️ The Closure Condition Is Not Optional

Excision requires $\overline Z\subseteq\operatorname{int}(A)$ , not merely $Z\subseteq A$ . If $Z$ touches the boundary of $A$ — i.e., if $\operatorname{int}A\cup\operatorname{int}(X\setminus Z)\ne X$ — 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 $X=\operatorname{int}A\cup\operatorname{int}B$ .

CounterexampleExcising Up to the Boundary Fails

Take $X=[0,1]$ , $A=[0,1]$ , and try $Z=\{1\}$ . Here $\overline Z=\{1\}\not\subset\operatorname{int}A=(0,1)$ . 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 $\overline Z\not\subset\operatorname{int}A$ 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 $H_n(X,A)\to\widetilde H_n(X/A)$ can fail to be an isomorphism. Do not blindly compute relative homology by collapsing $A$ .

connection

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 $X=\operatorname{int}A\cup\operatorname{int}B$ ) 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.

🔬 Where this appears

⏳ History

- 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).

open-problem

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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also