Editing Mayer–Vietoris sequence (section)

===Suspensions===
[[Image:0-Sphere Suspension - Mayer-Vietoris Cover.svg|right|500px|thumb|This decomposition of the suspension ''X'' of the 0-sphere ''Y'' yields all the homology groups of ''X''.]]
If ''X'' is the [[Suspension (topology)|suspension]] ''SY'' of a space ''Y'', let ''A'' and ''B'' be the [[Complement (set theory)|complements]] in ''X'' of the top and bottom 'vertices' of the double cone, respectively. Then ''X'' is the union ''A''∪''B'', with ''A'' and ''B'' contractible. Also, the intersection ''A''∩''B'' is homotopy equivalent to ''Y''. Hence the Mayer–Vietoris sequence yields, for all ''n'',<ref>{{harvnb|Hatcher|2002|loc=Exercise 32 on page 158}}</ref>
:<math>\tilde{H}_n(SY)\cong \tilde{H}_{n-1}(Y).</math>

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'' &minus; 1)-sphere, it is easy to derive the homology groups of the ''k''-sphere by induction, [[#k-sphere|as above]].
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