Editing Mayer–Vietoris sequence (section)

===''k''-sphere===
[[Image:SphereCoverStriped.png|thumb|250px|right|The decomposition for ''X'' = ''S''<sup>2</sup>]]
To completely compute the homology of the [[n-sphere|''k''-sphere]] ''X'' = ''S''<sup>''k''</sup>, let ''A'' and ''B'' be two hemispheres of ''X'' with intersection [[homotopy equivalent]] to a (''k'' &minus; 1)-dimensional equatorial sphere. Since the ''k''-dimensional hemispheres are [[homeomorphic]] to ''k''-discs, which are [[contractible]], the homology groups for ''A'' and ''B'' are [[Trivial group|trivial]]. The Mayer–Vietoris sequence for [[reduced homology]] groups then yields

:<math>
\cdots \longrightarrow 0 \longrightarrow 
\tilde{H}_{n}\!\left(S^k\right)\, \xrightarrow{\overset{}{\partial_*}}\,\tilde{H}_{n-1}\!\left(S^{k-1}\right) 
\longrightarrow 0 \longrightarrow \cdots
</math>

Exactness immediately implies that the map ∂<sub>*</sub> is an isomorphism. Using the [[reduced homology]] of the [[0-sphere]] (two points) as a [[Mathematical induction|base case]], it follows<ref>{{harvnb|Hatcher|2002|loc=Example 2.46,  p. 150}}</ref>

:<math>\tilde{H}_n\!\left(S^k\right)\cong\delta_{kn}\,\mathbb{Z}=
\begin{cases} 
\mathbb{Z} & \mbox{if } n=k,   \\ 
0 & \mbox{if } n \ne k, 
\end{cases}</math>

where δ is the [[Kronecker delta]]. Such a complete understanding of the homology groups for spheres is in stark contrast with current knowledge of [[homotopy groups of spheres]], especially for the case ''n'' > ''k'' about which little is known.<ref>{{harvnb|Hatcher|2002|p=384}}</ref>
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