Editing Mayer–Vietoris sequence (section)

===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>
{{-}}