Editing Mayer–Vietoris sequence (section)

==Further discussion==

===Relative form===
A [[relative homology|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:<ref>{{harvnb|Hatcher|2002|p=152}}</ref>

:<math>\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</math>

===Naturality===
The homology groups are [[Natural (category theory)|natural]] in the sense that if <math>f:X_1 \to X_2</math> is a [[Continuous function (topology)|continuous]] map, then there is a canonical [[pushforward (homology)|pushforward]] map of homology groups <math>f_*: H_k(X_1) \to H_k(X_2)</math> such that the composition of pushforwards is the pushforward of a composition: that is, <math>(g\circ h)_* = g_*\circ h_*.</math> The Mayer–Vietoris sequence is also natural in the sense that if

:<math>\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}</math>,

then the connecting morphism of the Mayer–Vietoris sequence, <math>\partial_*,</math> commutes with <math>f_*</math>.<ref>{{harvnb|Massey|1984|p=208}}</ref> That is, the following diagram [[Commutative diagram|commutes]]<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.4}}</ref>  (the horizontal maps are the usual ones):

:<math>\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}</math>

===Cohomological versions===

The Mayer–Vietoris long exact sequence for [[singular cohomology]] groups with coefficient [[group (mathematics)|group]] ''G'' is [[Duality (mathematics)|dual]] to the homological version. It is the following:<ref>{{harvnb|Hatcher|2002|p=203}}</ref>

:<math>\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</math>

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 number]]s '''R''' and the underlying topological space has the additional structure of a [[smooth manifold]], the Mayer–Vietoris sequence for [[de Rham cohomology]] is

:<math>\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</math>

where {{math|{{mset|''U'', ''V''}}}} is an [[open cover]] of {{mvar|X, ρ}} denotes the restriction map, and {{math|Δ}} is the difference. The map <math>d^*</math> is defined similarly as the map <math>\partial_*</math> from above. It can be briefly described as follows. For a cohomology class {{math|[''ω'']}} represented by [[closed and exact differential forms|closed form]] {{mvar|ω}} in {{math|''U''∩''V''}}, express {{mvar|ω}} as a difference of forms <math>\omega_U - \omega_V</math> via a [[partition of unity]] subordinate to the open cover {{math|{{mset|''U'', ''V''}}}}, for example. The exterior derivative {{mvar|dω<sub>U</sub>}} and {{mvar|dω<sub>V</sub>}} agree on {{math|''U''∩''V''}} and therefore together define an {{math|''n'' + 1}} form {{mvar|σ}} on {{mvar|X}}. One then has {{math|1=''d''<sup>&lowast;</sup>([''ω'']) = [''σ'']}}.

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

:<math>\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</math>

where <math>U</math>,<math>V</math>,<math>X</math> are as above, <math>\delta</math> is the signed inclusion map <math>\delta : \omega \mapsto (i^U_*\omega,-i^V_*\omega)</math> where <math>i^U</math> extends a form with compact support to a form on <math>U</math> by zero, and <math>\Sigma</math> is the sum.<ref>{{Cite book|last=Bott, Raoul|url=https://www.worldcat.org/oclc/7597142|title=Differential forms in algebraic topology|others=Tu, Loring W.|date=16 May 1995 |isbn=978-0-387-90613-3|location=New York|oclc=7597142}}</ref>

===Derivation===
Consider the [[Homological algebra#Functoriality|long exact sequence associated to]] the [[short exact sequence]]s of chain groups (constituent groups of [[chain complex]]es)

:<math>0 \to C_n(A\cap B)\,\xrightarrow{\alpha}\,C_n(A) \oplus C_n(B)\,\xrightarrow{\beta}\,C_n(A+B) \to 0</math>,

where α(''x'') = (''x'', −''x''), β(''x'', ''y'') = ''x'' + ''y'', and ''C''<sub>''n''</sub>(''A'' + ''B'') is the chain group consisting of sums of chains in ''A'' and chains in ''B''.<ref name="Hatcher149"/> 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''<sub>''n''</sub>(''X'').<ref>{{harvnb|Hatcher|2002|loc=Proposition 2.21,  p. 119}}</ref> In other words, ''H''<sub>''n''</sub>(''A'' + ''B'') is isomorphic to ''H''<sub>''n''</sub>(''X''). This gives the Mayer–Vietoris sequence for singular homology.

The same computation applied to the short exact sequences of vector spaces of [[differential form]]s

:<math>0\to\Omega^{n}(X)\to\Omega^{n}(U)\oplus\Omega^{n}(V)\to\Omega^{n}(U\cap V)\to 0 </math>

yields the Mayer–Vietoris sequence for de Rham cohomology.<ref>{{harvnb|Bott|Tu|1982|loc=§I.2}}</ref>

From a formal point of view, the Mayer–Vietoris sequence can be derived from the [[Eilenberg–Steenrod axioms]] for [[homology theory|homology theories]] using the [[long exact sequence in homology]].<ref>{{harvnb|Hatcher|2002|p=162}}</ref>

===Other homology theories===
The derivation of the Mayer–Vietoris sequence from the Eilenberg–Steenrod axioms does not require the [[dimension axiom]],<ref>{{harvnb|Kōno|Tamaki|2006|pp=25–26}}</ref> so in addition to existing in [[List of cohomology theories#Ordinary homology theories|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 Mayer–Vietoris sequence is related to [[Čech cohomology]]. Specifically, it arises from the [[Spectral sequence|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.<ref>{{harvnb|Dimca|2004|pp=35–36}}</ref> This spectral sequence exists in arbitrary [[Topos|topoi]].<ref>{{harvnb|Verdier|1972}} (SGA 4.V.3)</ref>