Editing Mayer–Vietoris sequence (section)

===Wedge sums===
[[Image:WedgeSumSpheres.png|right|300px|thumb|This decomposition of the wedge sum ''X'' of two 2-spheres ''K'' and ''L'' yields all the homology groups of ''X''.]]
Let ''X'' be the [[wedge sum]] of two spaces ''K'' and ''L'', and suppose furthermore that the identified [[Pointed space|basepoint]] is a [[deformation retract]] of [[Neighbourhood (mathematics)|open neighborhoods]] ''U'' ⊆ ''K'' and ''V'' ⊆ ''L''. Letting ''A'' = ''K'' ∪ ''V'' and ''B'' = ''U'' ∪ ''L'' it follows that ''A'' ∪ ''B'' = ''X'' and ''A'' ∩ ''B'' = ''U'' ∪ ''V'', which is [[contractible]] by construction. The reduced version of the sequence then yields (by exactness)<ref>{{harvnb|Hatcher|2002|loc=Exercise 31 on page 158}}</ref>
:<math>\tilde{H}_n(K\vee L)\cong \tilde{H}_n(K)\oplus\tilde{H}_n(L)</math>
for all dimensions ''n''. The illustration on the right shows ''X'' as the sum of two 2-spheres ''K'' and ''L''. For this specific case, using the result [[#k-sphere|from above]] for 2-spheres, one has
:<math>\tilde{H}_n\left(S^2\vee S^2\right)\cong\delta_{2n}\,(\mathbb{Z}\oplus\mathbb{Z})=\left\{\begin{matrix} 
\mathbb{Z}\oplus\mathbb{Z} & \mbox{if } n=2,   \\ 
0 & \mbox{if } n \ne 2.   \end{matrix}\right.</math>
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