Mayer–Vietoris sequence

For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:[9]

${\displaystyle \cdots \to H_{n+1}(X)\,\xrightarrow {\partial _{*}} \,H_{n}(A\cap B)\,\xrightarrow {\left({\begin{smallmatrix}i_{*}\\j_{*}\end{smallmatrix}}\right)} \,H_{n}(A)\oplus H_{n}(B)\,\xrightarrow {k_{*}-l_{*}} \,H_{n}(X)\,\xrightarrow {\partial _{*}} \,H_{n-1}(A\cap B)\to \cdots }$

${\displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdots \to H_{0}(A)\oplus H_{0}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{0}(X)\to 0.}$

Here i : A∩B ↪ A, j : A∩B ↪ B, k : A ↪ X, and l : B ↪ X are inclusion maps and ${\displaystyle \oplus }$ denotes the direct sum of abelian groups.

Illustration of the boundary map ∂_∗ on the torus where the 1-cycle x = u + v is the sum of two 1-chains whose boundary lies in the intersection of A and B.

The boundary maps ∂_∗ lowering the dimension may be defined as follows.[10] An element in H_n(X) is the homology class of an n-cycle x which, by barycentric subdivision for example, can be written as the sum of two n-chains u and v whose images lie wholly in A and B, respectively. Thus ∂x = ∂(u + v) = 0 so that ∂u = −∂v. This implies that the images of both these boundary (n − 1)-cycles are contained in the intersection A∩B. Then ∂_∗([x]) can be defined to be the class of ∂u in H_n−1(A∩B). Choosing another decomposition x = u′ + v′ does not affect [∂u], since ∂u + ∂v = ∂x = ∂u′ + ∂v′, which implies ∂u − ∂u′ = ∂(v′ − v), and therefore ∂u and ∂u′ lie in the same homology class; nor does choosing a different representative x′, since then ∂x′ = ∂x = 0. Notice that the maps in the Mayer–Vietoris sequence depend on choosing an order for A and B. In particular, the boundary map changes sign if A and B are swapped.

For reduced homology there is also a Mayer–Vietoris sequence, under the assumption that A and B have non-empty intersection.[11] The sequence is identical for positive dimensions and ends as:

${\displaystyle \cdots \to {\tilde {H}}_{0}(A\cap B)\,{\xrightarrow {(i_{*},j_{*})}}\,{\tilde {H}}_{0}(A)\oplus {\tilde {H}}_{0}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,{\tilde {H}}_{0}(X)\to 0.}$

There is an analogy between the Mayer–Vietoris sequence (especially for homology groups of dimension 1) and the Seifert–van Kampen theorem.[10][12] Whenever ${\displaystyle A\cap B}$ is path-connected, the reduced Mayer–Vietoris sequence yields the isomorphism

${\displaystyle H_{1}(X)\cong (H_{1}(A)\oplus H_{1}(B))/{\text{Ker}}(k_{*}-l_{*})}$

where, by exactness,

${\displaystyle {\text{Ker}}(k_{*}-l_{*})\cong {\text{Im}}(i_{*},j_{*}).}$

This is precisely the abelianized statement of the Seifert–van Kampen theorem. Compare with the fact that ${\displaystyle H_{1}(X)}$ is the abelianization of the fundamental group ${\displaystyle \pi _{1}(X)}$ when ${\displaystyle X}$ is path-connected.[13]

The decomposition for X = S²

To completely compute the homology of the k-sphere X = S^k, let A and B be two hemispheres of X with intersection homotopy equivalent to a (k − 1)-dimensional equatorial sphere. Since the k-dimensional hemispheres are homeomorphic to k-discs, which are contractible, the homology groups for A and B are trivial. The Mayer–Vietoris sequence for reduced homology groups then yields

${\displaystyle \cdots \longrightarrow 0\longrightarrow {\tilde {H}}_{n}\!\left(S^{k}\right)\,{\xrightarrow {\overset {}{\partial _{*}}}}\,{\tilde {H}}_{n-1}\!\left(S^{k-1}\right)\longrightarrow 0\longrightarrow \cdots }$

Exactness immediately implies that the map ∂_* is an isomorphism. Using the reduced homology of the 0-sphere (two points) as a base case, it follows[14]

${\displaystyle {\tilde {H}}_{n}\!\left(S^{k}\right)\cong \delta _{kn}\,\mathbb {Z} ={\begin{cases}\mathbb {Z} &{\mbox{if }}n=k\\0&{\mbox{if }}n\neq k\end{cases}}}$

where δ is the Kronecker delta. Such a complete understanding of the homology groups for spheres is in stark contrast with current knowledge of homotopy groups of spheres, especially for the case n > k about which little is known.[15]

The Klein bottle (fundamental polygon with appropriate edge identifications) decomposed as two Möbius strips A (in blue) and B (in red).

A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the Klein bottle X. One uses the decomposition of X as the union of two Möbius strips A and B glued along their boundary circle (see illustration on the right). Then A, B and their intersection A∩B are homotopy equivalent to circles, so the nontrivial part of the sequence yields[16]

${\displaystyle 0\rightarrow {\tilde {H}}_{2}(X)\rightarrow \mathbb {Z} \ {\xrightarrow {\overset {}{\alpha }}}\ \mathbb {Z} \oplus \mathbb {Z} \rightarrow \,{\tilde {H}}_{1}(X)\rightarrow 0}$

and the trivial part implies vanishing homology for dimensions greater than 2. The central map α sends 1 to (2, −2) since the boundary circle of a Möbius band wraps twice around the core circle. In particular α is injective so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, −1) as a basis for Z², it follows

${\displaystyle {\tilde {H}}_{n}\left(X\right)\cong \delta _{1n}\,(\mathbb {Z} \oplus \mathbb {Z} _{2})={\begin{cases}\mathbb {Z} \oplus \mathbb {Z} _{2}&{\mbox{if }}n=1\\0&{\mbox{if }}n\neq 1\end{cases}}}$

This decomposition of the wedge sum X of two 2-spheres K and L yields all the homology groups of X.

Let X be the wedge sum of two spaces K and L, and suppose furthermore that the identified basepoint is a deformation retract of open neighborhoods U ⊆ K and V ⊆ L. Letting A = K ∪ V and B = U ∪ L it follows that A ∪ B = X and A ∩ B = U ∪ V, which is contractible by construction. The reduced version of the sequence then yields (by exactness)[17]

${\displaystyle {\tilde {H}}_{n}(K\vee L)\cong {\tilde {H}}_{n}(K)\oplus {\tilde {H}}_{n}(L)}$

for all dimensions n. The illustration on the right shows X as the sum of two 2-spheres K and L. For this specific case, using the result from above for 2-spheres, one has

${\displaystyle {\tilde {H}}_{n}\left(S^{2}\vee S^{2}\right)\cong \delta _{2n}\,(\mathbb {Z} \oplus \mathbb {Z} )=\left\{{\begin{matrix}\mathbb {Z} \oplus \mathbb {Z} &{\mbox{if }}n=2\\0&{\mbox{if }}n\neq 2\end{matrix}}\right.}$

This decomposition of the suspension X of the 0-sphere Y yields all the homology groups of X.

If X is the suspension SY of a space Y, let A and B be the complements in X of the top and bottom 'vertices' of the double cone, respectively. Then X is the union A∪B, with A and B contractible. Also, the intersection A∩B is homotopy equivalent to Y. Hence the Mayer–Vietoris sequence yields, for all n,[18]

${\displaystyle {\tilde {H}}_{n}(SY)\cong {\tilde {H}}_{n-1}(Y)}$

The illustration on the right shows the 1-sphere X as the suspension of the 0-sphere Y. Noting in general that the k-sphere is the suspension of the (k − 1)-sphere, it is easy to derive the homology groups of the k-sphere by induction, as above.

A relative form of the Mayer–Vietoris sequence also exists. If Y ⊂ X and is the union of the interiors of C ⊂ A and D ⊂ B, then the exact sequence is:[19]

${\displaystyle \cdots \to H_{n}(A\cap B,C\cap D)\,{\xrightarrow {(i_{*},j_{*})}}\,H_{n}(A,C)\oplus H_{n}(B,D)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{n}(X,Y)\,{\xrightarrow {\partial _{*}}}\,H_{n-1}(A\cap B,C\cap D)\to \cdots }$

The homology groups are natural in the sense that if ${\displaystyle f:X_{1}\to X_{2}}$ is a continuous map, then there is a canonical pushforward map of homology groups ${\displaystyle f_{*}:H_{k}(X_{1})\to H_{k}(X_{2})}$ such that the composition of pushforwards is the pushforward of a composition: that is, ${\displaystyle (g\circ h)_{*}=g_{*}\circ h_{*}.}$ The Mayer–Vietoris sequence is also natural in the sense that if

${\displaystyle {\begin{matrix}X_{1}=A_{1}\cup B_{1}\\X_{2}=A_{2}\cup B_{2}\end{matrix}}\qquad {\text{and}}\qquad {\begin{matrix}f(A_{1})\subset A_{2}\\f(B_{1})\subset B_{2}\end{matrix}}}$

then the connecting morphism of the Mayer–Vietoris sequence, ${\displaystyle \partial _{*},}$ commutes with ${\displaystyle f_{*}}$.[20] That is, the following diagram commutes[21] (the horizontal maps are the usual ones):

${\displaystyle {\begin{matrix}\cdots &H_{n+1}(X_{1})&\longrightarrow &H_{n}(A_{1}\cap B_{1})&\longrightarrow &H_{n}(A_{1})\oplus H_{n}(B_{1})&\longrightarrow &H_{n}(X_{1})&\longrightarrow &H_{n-1}(A_{1}\cap B_{1})&\longrightarrow &\cdots \\&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }\\\cdots &H_{n+1}(X_{2})&\longrightarrow &H_{n}(A_{2}\cap B_{2})&\longrightarrow &H_{n}(A_{2})\oplus H_{n}(B_{2})&\longrightarrow &H_{n}(X_{2})&\longrightarrow &H_{n-1}(A_{2}\cap B_{2})&\longrightarrow &\cdots \\\end{matrix}}}$

The Mayer–Vietoris long exact sequence for singular cohomology groups with coefficient group G is dual to the homological version. It is the following:[22]

${\displaystyle \cdots \to H^{n}(X;G)\to H^{n}(A;G)\oplus H^{n}(B;G)\to H^{n}(A\cap B;G)\to H^{n+1}(X;G)\to \cdots }$

where the dimension preserving maps are restriction maps induced from inclusions, and the (co-)boundary maps are defined in a similar fashion to the homological version. There is also a relative formulation.

As an important special case when G is the group of real numbers R and the underlying topological space has the additional structure of a smooth manifold, the Mayer–Vietoris sequence for de Rham cohomology is

${\displaystyle \cdots \to H^{n}(X)\,{\xrightarrow {\rho }}\,H^{n}(U)\oplus H^{n}(V)\,{\xrightarrow {\Delta }}\,H^{n}(U\cap V)\,{\xrightarrow {d^{*}}}\,H^{n+1}(X)\to \cdots }$

where {U, V} is an open cover of X, ρ denotes the restriction map, and Δ is the difference. The map ${\displaystyle d^{*}}$ is defined similarly as the map ${\displaystyle \partial _{*}}$ from above. It can be briefly described as follows. For a cohomology class [ω] represented by closed form ω in U∩V, express ω as a difference of forms ${\displaystyle \omega _{U}-\omega _{V}}$ via a partition of unity subordinate to the open cover {U, V}, for example. The exterior derivative dω_U and dω_V agree on U∩V and therefore together define an n + 1 form σ on X. One then has d^∗([ω]) = [σ].

For de Rham cohomology with compact supports, there exists a "flipped" version of the above sequence:

${\displaystyle \cdots \to H_{c}^{n}(U\cap V)\,\xrightarrow {\delta } \,H_{c}^{n}(U)\oplus H_{c}^{n}(V)\,\xrightarrow {\Sigma } \,H_{c}^{n}(X)\,\xrightarrow {d^{*}} \,H_{c}^{n+1}(U\cap V)\to \cdots }$

where ${\displaystyle U}$,${\displaystyle V}$,${\displaystyle X}$ are as above, ${\displaystyle \delta }$ is the signed inclusion map ${\displaystyle \delta$ :\omega \mapsto (i_{*}^{U}\omega ,-i_{*}^{V}\omega )} where ${\displaystyle i^{U}}$ extends a form with compact support to a form on ${\displaystyle U}$ by zero, and ${\displaystyle \Sigma }$ is the sum.[23]

Consider the long exact sequence associated to the short exact sequences of chain groups (constituent groups of chain complexes)

${\displaystyle 0\to C_{n}(A\cap B)\,{\xrightarrow {\alpha }}\,C_{n}(A)\oplus C_{n}(B)\,{\xrightarrow {\beta }}\,C_{n}(A+B)\to 0}$

where α(x) = (x, −x), β(x, y) = x + y, and C_n(A + B) is the chain group consisting of sums of chains in A and chains in B.[9] It is a fact that the singular n-simplices of X whose images are contained in either A or B generate all of the homology group H_n(X).[24] In other words, H_n(A + B) is isomorphic to H_n(X). This gives the Mayer–Vietoris sequence for singular homology.

The same computation applied to the short exact sequences of vector spaces of differential forms

${\displaystyle 0\to \Omega ^{n}(X)\to \Omega ^{n}(U)\oplus \Omega ^{n}(V)\to \Omega ^{n}(U\cap V)\to 0}$

yields the Mayer–Vietoris sequence for de Rham cohomology.[25]

From a formal point of view, the Mayer–Vietoris sequence can be derived from the Eilenberg–Steenrod axioms for homology theories using the long exact sequence in homology.[26]

The derivation of the Mayer–Vietoris sequence from the Eilenberg–Steenrod axioms does not require the dimension axiom,[27] so in addition to existing in ordinary cohomology theories, it holds in extraordinary cohomology theories (such as topological K-theory and cobordism).

From the point of view of sheaf cohomology, the Mayer–Vietoris sequence is related to Čech cohomology. Specifically, it arises from the degeneration of the spectral sequence that relates Čech cohomology to sheaf cohomology (sometimes called the Mayer–Vietoris spectral sequence) in the case where the open cover used to compute the Čech cohomology consists of two open sets.[28] This spectral sequence exists in arbitrary topoi.[29]