# MayerâVietoris sequence Information

https://en.wikipedia.org/wiki/MayerâVietoris_sequence

In mathematics, particularly algebraic topology and homology theory, the MayerâVietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

The MayerâVietoris sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds for those theories satisfying the EilenbergâSteenrod axioms, and it has variations for both reduced and relative (co)homology. Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the MayerâVietoris sequence in the hope of obtaining partial information. Many spaces encountered in topology are constructed by piecing together very simple patches. Carefully choosing the two covering subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the MayerâVietoris sequence is analogous to the Seifertâvan Kampen theorem for the fundamental group, and a precise relation exists for homology of dimension one.

## Background, motivation, and history

Leopold Vietoris on his 110th birthday

Like the fundamental group or the higher homotopy groups of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of linear algebra, many other important (co)homology theories, especially singular (co)homology, are not computable directly from their definition for nontrivial spaces. For singular (co)homology, the singular (co)chains and (co)cycles groups are often too big to handle directly. More subtle and indirect approaches become necessary. The MayerâVietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection.

The most natural and convenient way to express the relation involves the algebraic concept of exact sequences: sequences of objects (in this case groups) and morphisms (in this case group homomorphisms) between them such that the image of one morphism equals the kernel of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are topological manifolds, simplicial complexes, or CW complexes, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability.

Mayer was introduced to topology by his colleague Vietoris when attending his lectures in 1926 and 1927 at a local university in Vienna. [1] He was told about the conjectured result and a way to its solution, and solved the question for the Betti numbers in 1929. [2] He applied his results to the torus considered as the union of two cylinders. [3] [4] Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence. [5] The concept of an exact sequence only appeared in print in the 1952 book Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod [6] where the results of Mayer and Vietoris were expressed in the modern form. [7]

## Basic versions for singular homology

Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The MayerâVietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection Aâ©B. [8] There is an unreduced and a reduced version.

### Unreduced version

For unreduced homology, the Mayerâ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 : Aâ©B âȘ A, j : Aâ©B âȘ B, k : A âȘ X, and l : B âȘ X are inclusion maps and ${\displaystyle \oplus }$ denotes the direct sum of abelian groups.

### Boundary map

Illustration of the boundary map â 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 â lowering the dimension may be defined as follows. [10] An element in Hn(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 âx = â(u + v) = 0 so that âu = ââv. This implies that the images of both these boundary (n â 1)-cycles are contained in the intersection Aâ©B. Then â([x]) can be defined to be the class of âu in Hn−1(Aâ©B). Choosing another decomposition x = uâČ + vâČ does not affect [âu], since âu + âv = âx = âuâČ + âvâČ, which implies âu − âuâČ = â(vâČv), and therefore âu and âuâČ lie in the same homology class; nor does choosing a different representative xâČ, since then âxâČ = âx = 0. Notice that the maps in the MayerâVietoris sequence depend on choosing an order for A and B. In particular, the boundary map changes sign if A and B are swapped.

### Reduced version

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

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

### Analogy with the Seifertâvan Kampen theorem

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

## Basic applications

### k-sphere

The decomposition for X = S2

To completely compute the homology of the k-sphere X = Sk, 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 Mayerâ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 â* 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 ÎŽ 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]

### Klein bottle

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

A slightly more difficult application of the Mayerâ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 MĂ¶bius strips A and B glued along their boundary circle (see illustration on the right). Then A, B and their intersection Aâ©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 Î± sends 1 to (2, −2) since the boundary circle of a MĂ¶bius band wraps twice around the core circle. In particular Î± is injective so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, −1) as a basis for Z2, it follows

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

### Wedge sums

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 â K and V â L. Letting A = K âȘ V and B = U âȘ L it follows that A âȘ B = X and A â© B = U âȘ 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

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

### Suspensions

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 AâȘB, with A and B contractible. Also, the intersection Aâ©B is homotopy equivalent to Y. Hence the Mayerâ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.

## Further discussion

### Relative form

A relative form of the MayerâVietoris sequence also exists. If Y â X and is the union of C â A and D â B, then the exact sequence is: [19]

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

### Naturality

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 Mayerâ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 MayerâVietoris sequence, ${\displaystyle \partial _{*},}$ commutes with ${\displaystyle f_{*}}$. [20] That is, the following diagram commutes [21] (the horizontal maps are the usual ones):

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

### Cohomological versions

The Mayerâ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 Mayerâ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, Ï denotes the restriction map, and Î 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 [Ï] represented by closed form Ï in Uâ©V, express Ï 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 dÏU and dÏV agree on Uâ©V and therefore together define an n + 1 form Ï on X. One then has d([Ï]) = [Ï].

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]

### Derivation

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 Î±(x) = (x, âx), ÎČ(x, y) = x + y, and Cn(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 Hn(X). [24] In other words, Hn(A + B) is isomorphic to Hn(X). This gives the Mayerâ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 MayerâVietoris sequence for de Rham cohomology. [25]

From a formal point of view, the MayerâVietoris sequence can be derived from the EilenbergâSteenrod axioms for homology theories using the long exact sequence in homology. [26]

### Other homology theories

The derivation of the MayerâVietoris sequence from the Eilenbergâ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).

### Sheaf cohomology

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

## Notes

1. ^
2. ^
3. ^ DieudonnĂ© 1989, p. 39
4. ^ Mayer 1929, p. 41
5. ^
6. ^ Corry 2004, p. 345
7. ^ Eilenberg & Steenrod 1952, Theorem 15.3
8. ^
9. ^ a b Hatcher 2002, p. 149
10. ^ a b Hatcher 2002, p. 150
11. ^ Spanier 1966, p. 187
12. ^ Massey 1984, p. 240
13. ^ Hatcher 2002, Theorem 2A.1, p. 166
14. ^ Hatcher 2002, Example 2.46, p. 150
15. ^ Hatcher 2002, p. 384
16. ^ Hatcher 2002, p. 151
17. ^ Hatcher 2002, Exercise 31 on page 158
18. ^ Hatcher 2002, Exercise 32 on page 158
19. ^ Hatcher 2002, p. 152
20. ^ Massey 1984, p. 208
21. ^ Eilenberg & Steenrod 1952, Theorem 15.4
22. ^ Hatcher 2002, p. 203
23. ^ Bott, Raoul. Differential forms in algebraic topology. Tu, Loring W. New York. ISBN  978-0-387-90613-3. OCLC  7597142.
24. ^ Hatcher 2002, Proposition 2.21, p. 119
25. ^ Bott & Tu 1982, Â§I.2
26. ^ Hatcher 2002, p. 162
27. ^ KĆno & Tamaki 2006, pp. 25â26
28. ^ Dimca 2004, pp. 35â36
29. ^ Verdier 1972 (SGA 4.V.3)

## References

• KĆno, Akira; Tamaki, Dai (2006) [2002], Generalized cohomology, Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs, vol. 230 (Translated from the 2002 Japanese edition by Tamaki ed.), Providence, RI: American Mathematical Society, ISBN  978-0-8218-3514-2, MR  2225848