= Euler characteristic of an annulus
{c}
{title2=$\chi(S^1\times[0,1])=0$}
Build a closed <annulus> from two nested $n$-sided <polygons>, $n\geq3$, with $n$ radial <edges> joining corresponding <vertices>. There are $2n$ <vertices>, $3n$ <edges> and $n$ quadrilateral <topological cells>, giving $\chi=2n-3n+n=0$. The central missing region and the exterior are not faces of the <annulus>. A <deformation retraction> onto a <circle> gives $(b_0,b_1,b_2)=(1,1,0)$, agreeing with the cell count. This is a useful reduction: the <annulus> is two-dimensional, yet its <homology> is that of a one-dimensional <cycle graph>.
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