Systems of curves on non-orientable surfaces

TL;DR

The paper develops a comprehensive framework for maximal complete $1$-systems of loops and arcs on non-orientable surfaces, extending foundational results from orientable cases. It introduces and leverages geometric constructs such as lassos and honda paths for arcs, and a four-type decomposition of loops to derive tight lower and upper bounds, yielding quadratic growth in the Euler characteristic for non-orientable surfaces. It achieves an exact count for $n$-punctured projective planes and generalizes Przytycki’s arc bound to non-orientable settings, bridging a gap in the understanding of curve systems on non-orientable type surfaces. The results sharpen the contrast between orientable and non-orientable cases and provide explicit constructions and counting techniques that may inform further study of curve complexes and mapping class groups in non-orientable contexts.

Abstract

We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is $\sim |χ|^{2}$. 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 $2|χ|(|χ|+1)$.

Figures (17)

• Figure 1: Two depictions of $N_{7,2}$. The $\bigotimes$ in both pictures represents a cross-cap (i.e. an $\mathbb{S}^1$ with antipodal points identified).
• Figure 2: Here are some examples of simple essential curves (in green) and non-simple or non-essential curves (in red) on $N_{7,3}$. The curve $\alpha_1$ is a non-simple arc, $\alpha_2$ is an essential simple arc, $\gamma_1$ is an essential $1$-sided simple loop, $\gamma_2$ is a essential $2$-sided simple loop, $\alpha_3$ is a non-essential simple arc and $\gamma_3, \gamma_4$ are two non-essential simple loops, where $\gamma_3$ is non-primitive.
• Figure 3: $L_1$ and $L_2$ are two equivalent complete $1$-systems of loops on $S_{2,0}$.
• Figure 4: The regular neighbourhood of two $2$-sided loops intersecting once.
• Figure 5: A maximal complete $1$-system of loops on $S_{2,0}$.

Theorems & Definitions (59)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 2.1: Simple curves
• Definition 2.3: Regular neighbourhoods of curves
• Definition 2.4: Free isotopies
• Definition 2.5: Essential curves