Mayer-Vietoris Sequence

⚡ Key Takeaways

The Mayer–Vietoris sequence is a long exact sequence relating the (co)homology of a space $X$ to the (co)homology of two subspaces $A,B$ whose interiors cover $X$ and of their intersection $A\cap B$ . In words: if you can cut a space into two overlapping pieces, you can compute its homology from the homology of the pieces and their overlap, with a connecting map measuring the "twisting" along the overlap. It applies in singular, simplicial, cellular, sheaf, de Rham, and generalized (co)homology theories. Named after Walther Mayer and Leopold Vietoris, who developed it around 1929–1930.

💡 The Core Insight

A cycle on $X$ can be subdivided so that each piece lies in $A$ or in $B$ . The obstruction to writing a cycle on $X$ as the sum of a cycle on $A$ and a cycle on $B$ is recorded in $A\cap B$ : when you split a chain $c = c_A + c_B$ , even if $\partial c = 0$ the individual pieces satisfy $\partial c_A = -\partial c_B$ , a cycle living in $A\cap B$ . The connecting map sends $[c]$ to this overlap cycle. Exactness packages this single observation into a machine that recovers the global from the local.

DefinitionExcisive couple / Mayer–Vietoris triad

Let $X$ be a topological space and let $A,B \subseteq X$ . The pair $(A,B)$ is called an excisive couple (and $(X;A,B)$ a Mayer–Vietoris triad) if

1

\quad X = \operatorname{int}(A) \cup \operatorname{int}(B),

i.e. the interiors of $A$ and $B$ cover $X$ . Here:

$X$ is the ambient space;
$A, B$ are the two covering subspaces;
$A\cap B$ is their intersection;
$\operatorname{int}(\cdot)$ denotes topological interior in $X$ .

This condition guarantees that the inclusion $(A, A\cap B) \hookrightarrow (X,B)$ is an excision, which is the technical input that makes the sequence exact.

TheoremMayer–Vietoris, homology

Let $(X;A,B)$ be a Mayer–Vietoris triad. Denote the inclusions

i:A\cap B\hookrightarrow A,\quad j:A\cap B\hookrightarrow B,\quad k:A\hookrightarrow X,\quad l:B\hookrightarrow X.

Then there is a long exact sequence of homology groups (with coefficients in any abelian group $G$ )

2

\quad \cdots \to H_n(A\cap B)\xrightarrow{\ \varphi\ } H_n(A)\oplus H_n(B)\xrightarrow{\ \psi\ } H_n(X)\xrightarrow{\ \partial\ } H_{n-1}(A\cap B)\to\cdots

where the maps are

3

\quad \varphi(x) = (i_*x,\ j_*x),\qquad \psi(a,b) = k_*a - l_*b,

and $\partial$ is the connecting homomorphism. The signs ensure $\psi\circ\varphi = 0$ .

DefinitionReduced and cohomological versions

For the reduced homology $\tilde H_n$ , the same sequence holds provided $A\cap B \neq \varnothing$ , terminating

4

\quad \cdots \to \tilde H_0(A\cap B)\to \tilde H_0(A)\oplus\tilde H_0(B)\to \tilde H_0(X)\to 0.

In cohomology, the arrows reverse and the connecting map raises degree:

5

\quad \cdots \to H^n(X)\xrightarrow{(k^*,l^*)} H^n(A)\oplus H^n(B)\xrightarrow{i^*-j^*} H^n(A\cap B)\xrightarrow{\ \delta\ } H^{n+1}(X)\to\cdots

ProofThe short exact sequence of chains

Step 1 — Small chains. Let $\mathcal{U}=\{A,B\}$ and let $C_n^{\mathcal U}(X)$ denote the subgroup of $C_n(X)$ generated by singular simplices whose image lies entirely in $A$ or entirely in $B$ . Concretely

6

\quad C_n^{\mathcal U}(X) = C_n(A) + C_n(B)\ \subseteq\ C_n(X).

Derivation hint: A chain is " $\mathcal U$ -small" iff each generating simplex factors through one of the cover elements; the sum of subcomplexes is the span of such simplices.

Step 2 — Small Simplices Theorem. Because $X=\operatorname{int}A\cup\operatorname{int}B$ , barycentric subdivision shows the inclusion $C_\bullet^{\mathcal U}(X)\hookrightarrow C_\bullet(X)$ is a chain-homotopy equivalence, hence

7

\quad H_n\big(C_\bullet^{\mathcal U}(X)\big)\ \cong\ H_n(X).

Derivation hint: Iterated barycentric subdivision $\mathrm{Sd}^m$ eventually pushes every simplex into the open cover by the Lebesgue number lemma; the subdivision operator is chain homotopic to the identity.

Step 3 — The short exact sequence. Form the sequence of chain complexes

8

\quad 0 \to C_\bullet(A\cap B)\xrightarrow{\ \alpha\ } C_\bullet(A)\oplus C_\bullet(B)\xrightarrow{\ \beta\ } C_\bullet^{\mathcal U}(X) \to 0,

with $\alpha(c) = (c,c)$ and $\beta(a,b) = a - b$ . Derivation hint: $\alpha$ is injective (a chain in the intersection mapping to zero in both must vanish); $\beta$ is surjective by definition of $C^{\mathcal U}$ ; exactness in the middle is the statement that $a-b=0$ forces $a=b\in C_\bullet(A\cap B)$ .

Step 4 — Snake lemma. A short exact sequence of chain complexes yields a long exact sequence in homology. Substituting $H_n(C^{\mathcal U})\cong H_n(X)$ from Step 2 gives [2]. The connecting homomorphism is constructed as follows: given $[z]\in H_n(X)$ , write $z = a - b$ with $a\in C_n(A)$ , $b\in C_n(B)$ ; then $\partial a = \partial b \in C_{n-1}(A\cap B)$ is a cycle, and

9

\quad \partial[z] = [\partial a]\ =\ [\partial b]\ \in\ H_{n-1}(A\cap B).

Derivation hint: This is the standard zig-zag of the snake lemma. Since $\partial(a-b)=\partial z=0$ we get $\partial a=\partial b$ ; that common boundary is a cycle in the intersection, well-defined up to boundaries.

∎

⚠️ Common mistake

The covering condition [1] is essential and is not satisfied merely by $X = A\cup B$ as sets. If $A$ and $B$ are closed and meet only along a "thin" set whose interiors do not cover, the naive sequence can fail. For CW complexes one typically thickens subcomplexes to open neighborhoods (which deformation retract) so that the interior condition holds; the maps remain those of the subcomplexes up to homotopy equivalence.

mathematician · Walther Mayer & Leopold Vietoris | 1887–1948 / 1891–2002

Walther Mayer, an Austrian mathematician, served as Albert Einstein's assistant ("Einstein's calculator") at Princeton and worked extensively on combinatorial topology and tensor calculus. Leopold Vietoris, also Austrian, is famous both for his topological contributions — the Vietoris topology, Vietoris–Rips complexes used today in topological data analysis, and the Vietoris homology theory — and for his extraordinary longevity, dying at age 110. Around 1929–1930 Mayer, prompted by lectures and discussions with Vietoris, formulated the exact-sequence relation between the homology of a complex and that of two subcomplexes. Vietoris had developed the relevant homology machinery for general (non-triangulable) spaces. The synthesis of their work into the sequence that bears both names became one of the cornerstones of the homological-algebra revolution of the 1940s–50s.

🧠 Intuition

Think of $X$ as a country, $A$ and $B$ as two overlapping maps that together cover it, and homology classes as "loops/holes that cannot be filled." If you have a global loop on $X$ , you can cut it where the two maps overlap. The pieces on map $A$ and map $B$ might each be open arcs (boundaries), but where you cut — in the overlap $A\cap B$ — you create endpoints. The connecting map $\partial$ remembers those endpoints as a cycle in the overlap. Exactness says: a global hole exists precisely when the overlap forces an obstruction that cannot be canceled by holes already present in $A$ or $B$ . The sphere $S^n$ is the cleanest example: both hemispheres are contractible (no holes), yet their overlap is a homotopy $S^{n-1}$ , and $\partial$ transports the $(n-1)$ -dimensional hole of the equator up to the $n$ -dimensional hole of the whole sphere.

Figure

Mayer–Vietoris decomposition of the circle

S^1

into two arcs with two-component overlap

ExampleHomology of the

n

-sphere by induction

Given: $S^n$ for $n\ge 1$ . Cover by $A = S^n\setminus\{\text{south pole}\}$ and $B = S^n\setminus\{\text{north pole}\}$ .

Steps: 1. Each of $A,B$ is homeomorphic to $\mathbb R^n$ , hence contractible: $\tilde H_*(A)=\tilde H_*(B)=0$ . 2. $A\cap B = S^n\setminus\{2\text{ poles}\}$ deformation retracts onto the equator $S^{n-1}$ . 3. The reduced sequence [4] in degree $n$ gives, since the $\oplus$ term vanishes,

10

\quad 0 \to \tilde H_n(S^n)\xrightarrow{\partial} \tilde H_{n-1}(S^{n-1})\to 0,

Derivation hint: Between two zero groups (the $A\oplus B$ terms vanish in every positive degree), $\partial$ is an isomorphism by exactness. 4. So $\partial$ is an isomorphism: $\tilde H_n(S^n)\cong \tilde H_{n-1}(S^{n-1})$ . 5. Base case $S^0$ (two points): $\tilde H_0(S^0)=\mathbb Z$ , all else $0$ . Induct.

Boxed answer:

\boxed{\ \tilde H_k(S^n)=\begin{cases}\mathbb Z & k=n\\ 0 & k\neq n\end{cases}\ }

ExampleHomology of the torus

T^2 = S^1\times S^1

Given: Decompose $T^2$ as $A$ = a neighborhood of one circle's worth (a "fat" cylinder) and $B$ = the complementary cylinder, so $A\cup B = T^2$ , with $A\cap B$ = two disjoint annuli.

Steps: 1. $A \simeq S^1$ , $B\simeq S^1$ , so $H_*(A)=H_*(B)=H_*(S^1)$ . 2. $A\cap B \simeq S^1\sqcup S^1$ , so $H_0=\mathbb Z^2,\ H_1=\mathbb Z^2$ . 3. Write the sequence in low degrees ( $G=\mathbb Z$ ):

11

\quad 0\to H_2(T^2)\to H_1(A\cap B)\xrightarrow{\varphi} H_1(A)\oplus H_1(B)\xrightarrow{\psi} H_1(T^2)\to \tilde H_0(A\cap B)\to 0.

Derivation hint: $H_2(A)=H_2(B)=0$ and $H_2(A\cap B)=0$ force $H_2(T^2)\hookrightarrow H_1(A\cap B)$ ; the rest is the tail of [2]. 4. The map $\varphi:\mathbb Z^2\to\mathbb Z^2$ sends each of the two boundary circles to the same generator (both annuli wrap the longitude), so $\varphi$ has rank 1; thus $\ker\varphi\cong\mathbb Z$ and $\operatorname{coker}\varphi\cong\mathbb Z$ . 5. Therefore $H_2(T^2)=\ker\varphi=\mathbb Z$ ; counting ranks gives $H_1(T^2)=\mathbb Z^2$ .

Boxed answer:

\boxed{\ H_0(T^2)=\mathbb Z,\quad H_1(T^2)=\mathbb Z^2,\quad H_2(T^2)=\mathbb Z\ }

(Consistent with $T^2$ having one connected component, two independent loops, one enclosed void; Euler characteristic $1-2+1=0$ .)

⚠️ Common mistake

The connecting map is not natural in an obvious "first guess" way. Students often expect $\partial$ to be a forgetful or restriction map. It is neither — it is defined through the splitting of a chain (choices $a,b$ with $z=a-b$ ) and the snake-lemma diagram chase. It is, however, natural with respect to maps of triads, and this naturality is what allows Mayer–Vietoris to fit into long exact ladders and spectral-sequence comparisons.

⚠️ Common mistake

Direct sum, not direct product, and watch the sign. The middle term is $H_n(A)\oplus H_n(B)$ with the map $\psi(a,b)=k_*a - l_*b$ . Dropping the minus sign breaks $\psi\circ\varphi=0$ and destroys exactness. Some texts instead put the sign in $\varphi$ ; either convention works, but you must be consistent.

CounterexampleFailure without the interior-cover condition

Take $X=\mathbb R$ , $A=(-\infty,0]$ , $B=[0,\infty)$ . These are closed, $X=A\cup B$ , and $A\cap B=\{0\}$ . Here $\operatorname{int}A=(-\infty,0)$ and $\operatorname{int}B=(0,\infty)$ , whose union is $\mathbb R\setminus\{0\}\neq\mathbb R$ , so [1] fails at the single point $0$ . In this benign case all spaces are contractible and the sequence still works by accident. But replace with a pathological pair where the overlap is not a deformation retract of a neighborhood (e.g. the topologist's sine curve glued to an arc), and the excision step Step 2 collapses: $C^{\mathcal U}\hookrightarrow C(X)$ need not be a homotopy equivalence, and the resulting "sequence" need not be exact. The lesson: closedness is not enough — you genuinely need open interiors to cover, or subspaces that admit collared/neighborhood-retract structure.

connection

Mayer–Vietoris is the abelian, all-degrees shadow of Seifert–van Kampen Theorem, which computes $\pi_1$ from a two-piece cover (a pushout of groups rather than an exact sequence). Both are instances of descent / gluing. The sequence is the special case of the Spectral Sequence of an open cover (the Mayer–Vietoris / Čech–to–derived-functor spectral sequence) when the cover has only two elements. In sheaf theory it is precisely the long exact sequence associated to the short exact sequence $0\to \mathcal F\to j_{A*}\mathcal F\oplus j_{B*}\mathcal F\to j_{(A\cap B)*}\mathcal F\to 0$ , linking it to Sheaf Cohomology and Čech Cohomology. In differential geometry the de Rham version connects it to de Rham Cohomology via partitions of unity, and the dual statement for compactly supported forms gives Poincaré Duality. It is also a corollary of the Eilenberg–Steenrod Axioms (excision + exactness of pairs), hence holds for any Generalized Homology Theory such as K-theory and cobordism.

🔬 Where this appears

⏳ History

- 1895–1904: Henri Poincaré founds combinatorial topology and defines Betti numbers and torsion in Analysis Situs, providing the homological objects that the sequence will later relate.

- 1925–1926: Emmy Noether reframes homology in terms of groups and homomorphisms rather than numbers, making exact-sequence reasoning conceivable.

- 1929: Leopold Vietoris develops a homology theory for general metric (non-triangulable) spaces, introducing the limit/nerve techniques that underlie the gluing philosophy.

- 1929–1930: Walther Mayer, in correspondence and lectures with Vietoris in Vienna, formulates the exact relationship between the homology of a complex and of two subcomplexes — the prototype of the sequence.

- 1930s: The relation is used in special cases (spheres, projective spaces) but stated in cumbersome combinatorial language, without the modern notion of "exact sequence."

- 1941–1942: Witold Hurewicz, and independently others, sharpen the language of exact sequences; the term exact sequence enters usage.

- 1945: Samuel Eilenberg and Norman Steenrod axiomatize homology theory (Eilenberg–Steenrod axioms), from which Mayer–Vietoris follows formally from excision and the exact sequence of a pair.

- 1952: Eilenberg–Steenrod's Foundations of Algebraic Topology presents Mayer–Vietoris in its now-standard categorical form, valid for any homology theory.

- 1990s–2000s: Carlsson, Ghrist, de Silva, Edelsbrunner and others revive Vietoris's gluing ideas in persistent homology and TDA, returning the sequence to computational practice.

open-problem

The two-piece sequence generalizes to arbitrary open covers via the Mayer–Vietoris (Čech) spectral sequence, but extracting closed-form homology from many overlapping pieces is genuinely hard: the spectral sequence may have nonzero higher differentials and nontrivial extension problems, both of which can obstruct mechanical computation. In persistent (co)homology, a fully functorial, stable Mayer–Vietoris formalism for persistence modules — controlling how barcodes glue with interleaving/stability guarantees — is an active research area; the relevant "exactness up to interleaving distance" statements are subtle and not yet a closed theory. For generalized homology theories with products (e.g. complex cobordism, $K$ -theory) the multiplicative structure of the connecting map and its interaction with the cup/external product (the Mayer–Vietoris analogue of the cohomology ring's behavior under gluing) requires care; clean gluing formulas for ring structures are theory-dependent. Finally, the non-excisive case — pairs not satisfying [1], common in singular or stratified spaces — generally requires intersection homology or homotopy-coherent descent rather than the naive sequence.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also