Editing Mayer–Vietoris sequence (section)

==Basic versions for singular homology==
Let ''X'' be a [[topological space]] and ''A'', ''B'' be two subspaces whose [[Interior (topology)|interiors]] cover ''X''. (The interiors of ''A'' and ''B'' need not be disjoint.) The Mayer–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 (set theory)|intersection]] ''A''∩''B''.<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=§15}}</ref> There is an unreduced and a reduced version.

===Unreduced version===
For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:<ref name="Hatcher149">{{harvnb|Hatcher|2002|p=149}}</ref>

:<math>\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 </math>
:<math> 
\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.
</math>

Here <math> i : A\cap B \hookrightarrow A, j : A\cap B \hookrightarrow B, k : A \hookrightarrow X</math> , and <math> l : B \hookrightarrow X</math>  are [[inclusion map]]s and <math>\oplus</math> denotes the [[direct sum of abelian groups]].

===Boundary map===
[[Image:Mayer Vietoris sequence boundary map on torus.png|thumb|280px|right|Illustration of the boundary map ∂<sub>&lowast;</sub> 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 ∂<sub>&lowast;</sub> lowering the dimension may be defined as follows.<ref name="Hatcher 2002 150">{{harvnb|Hatcher|2002|p=150}}</ref> An element in ''H<sub>n</sub>''(''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'') = ∂''u'' + ∂''v''. Since ''x'' is a cycle, ∂x = 0, so ∂''u'' = −∂''v''. This implies that the images of both these boundary (''n'' − 1)-cycles are contained in the intersection ''A''∩''B''. Then ∂<sub>&lowast;</sub>([''x'']) can be defined to be the class of ∂''u'' in ''H''<sub>''n''&minus;1</sub>(''A''∩''B''). Choosing another decomposition ''x'' = ''u′'' + ''v′'' does not affect [∂''u''], since ∂''u'' + ∂''v'' = ∂''x'' = ∂''u′'' + ∂''v′'', which implies ∂''u'' &minus; ∂''u′'' = ∂(''v′'' &minus; ''v''), and therefore ∂''u'' and ∂''u′'' lie in the same homology class; nor does choosing a different representative ''x′'', since then ''x′'' - ''x'' = ∂''φ'' for some ''φ'' in ''H''<sub>''n''+1</sub>(''X''). 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.

===Reduced version===
For [[reduced homology]] there is also a Mayer–Vietoris sequence, under the assumption that ''A'' and ''B'' have [[non-empty]] intersection.<ref>{{harvnb|Spanier|1966|p=187}}</ref> The sequence is identical for positive dimensions and ends as:

:<math>\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.</math>

===Analogy with the Seifert–van Kampen theorem===
There is an analogy between the Mayer–Vietoris sequence (especially for homology groups of dimension 1) and the [[Seifert–van Kampen theorem]].<ref name="Hatcher 2002 150"/><ref>{{harvnb|Massey|1984|p=240}}</ref> Whenever <math>A\cap B</math> is [[path-connected]], the reduced Mayer–Vietoris sequence yields the isomorphism

:<math>H_1(X) \cong (H_1(A)\oplus H_1(B))/\text{Ker} (k_* - l_*)</math>

where, by exactness,

:<math>\text{Ker} (k_* - l_*) \cong \text{Im} (i_*, j_*).</math>

This is precisely the [[Commutator subgroup#Abelianization|abelianized]] statement of the Seifert–van Kampen theorem. Compare with the fact that <math>H_1(X)</math> is the abelianization of the [[fundamental group]] <math>\pi_1(X)</math> when <math>X</math> is path-connected.<ref>{{harvnb|Hatcher|2002|loc=Theorem 2A.1, p. 166}}</ref>