Systems of curves on non-orientable surfaces

Abstract

We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is $\sim |\ddot{I}‡{|}^2$. In particular, we determine the exact cardinality of maximal complete $1$-systems of loops on punctured projective planes. To prove these results, we show that the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is $2|\ddot{I}‡|(|\ddot{I}‡|+1)$.

23 pages, 18 figures. Comments are welcome!