Editing Mayer–Vietoris sequence (section)

===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>