= Integral Morse complex of the Klein bottle
{title2=$0\longrightarrow\mathbb Z\xrightarrow{(0,2)}\mathbb Z^2\xrightarrow{0}\mathbb Z\longrightarrow0$}
On the <Klein bottle> given by $(s,t)\sim(s+2\pi,-t)\sim(s,t+2\pi)$, the <Morse function> $f=\cos s+\varepsilon\cos t$, $0<\varepsilon<1$, has one minimum, two saddles, and one maximum. In its descending <Morse-Smale gradient flow>, the two branches from either saddle to the minimum cancel. From the maximum to the fiber saddle, reflection of the fiber under the base identification reverses the transverse orientation, making the two contributions add. The contributions to the base saddle cancel. With a choice of generator orientations the displayed <Morse chain complex> follows. Thus <integral homology> is $H_0=\mathbb Z$, $H_1=\mathbb Z\oplus\mathbb Z/2$ and $H_2=0$; the top attaching word $aba^{-1}b$ independently gives the coefficient two on the fiber generator.
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