Editing Mayer–Vietoris sequence (section)

==Basic applications==

===''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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===Klein bottle===
[[Image:KleinBottle2D covered by Möbius strips.svg|thumb|200px|right|The Klein bottle ([[fundamental polygon]] with appropriate edge identifications) decomposed as two Möbius strips ''A'' (in blue) and ''B'' (in red).]]
A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the [[Klein bottle]] ''X''. One uses the decomposition of  ''X'' as the union of two [[Möbius strip]]s ''A'' and ''B'' [[Quotient space (topology)|glued]] along their boundary circle (see illustration on the right). Then ''A'', ''B'' and their intersection ''A''∩''B'' are [[Homotopy#Homotopy equivalence and null-homotopy|homotopy equivalent]] to circles, so the nontrivial part of the sequence yields<ref>{{harvnb|Hatcher|2002|p=151}}</ref>

:<math>
0 \rightarrow \tilde{H}_{2}(X) \rightarrow 
\mathbb{Z}\ \xrightarrow{\overset{}{\alpha}} \ \mathbb{Z} \oplus \mathbb{Z} 
\rightarrow \, \tilde{H}_1(X) \rightarrow 0
</math>

and the trivial part implies vanishing homology for dimensions greater than 2. The central map α sends 1 to (2, &minus;2) since the boundary circle of a Möbius band wraps twice around the core circle. In particular α is [[Injective function|injective]] so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, &minus;1) as a basis for '''Z'''<sup>2</sup>, it follows

:<math>\tilde{H}_n\left(X\right)\cong\delta_{1n}\,(\mathbb{Z}\oplus\mathbb{Z}_2)=
\begin{cases} 
\mathbb{Z}\oplus\mathbb{Z}_2 & \mbox{if } n=1,\\
0 & \mbox{if } n\ne1.    \end{cases}
</math>
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===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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===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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