= Six-vertex triangulation of the real projective plane
{title2=$V=6,\quad E=15,\quad F=10,\quad\chi=1$}
On vertices $1,\ldots,6$, take the <simplicial complex> generated by $123,124,135,146,156,236,245,256,345,346$. Every edge occurs in two facets. The vertex links, listed cyclically, are $23564$, $13654$, $12645$, $12536$, $13426$, and $14325$, respectively. Thus each vertex link is a <cycle graph>, and the realization is a compact connected <surface> without boundary. Its <Euler characteristic> is $6-15+10=1$, so the <classification theorem for surfaces> identifies it with the <real projective plane>. All fifteen possible edges occur, and the listed triangles are distinct three-element sets, so this is an honest abstract <simplicial complex>, not merely a polygon presentation.
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