Editing Mayer–Vietoris sequence (section)

===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.