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Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\Sigma$) such that $C'$ only intersects each $C_i$ at most twice. I believe I have a proof of this result that works for all surfaces, but I'm pretty sure this is classical stuff.

Indeed, in John Stillwell's book Classical Topology and Combinatorial Group Theory he mentions that the above result was proven by Lickorish (1962) for orientable surfaces. In the case of orientable surfaces, the homeomorphism can in fact be achieved via Dehn twists and isotopies. Unfortunately, the Lickorish proof doesn't work for non-orientable surfaces.

Question: Can someone please provide a reference of the above result for non-orientable surfaces?

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up vote 3 down vote accepted

There are two questions here.

1) The fact about Dehn twists and isotopies is really a consequence of the fact that the mapping class group of an orientable surface is generated by Dehn twists. For non-orientable surfaces, this is not true -- you also need the so-called "crosscap slides". For a discussion of this, see Lickorish's paper "Homeomorphisms of non-orientable two-manifolds" and Chillingworth's paper "A finite set of generators for the homeotopy group of a non-orientable surface".

2) For the fact about simple closed curves, there is a very general trick using the classification of surfaces that has become known as the "change of coordinates principle" for a surface. It can prove almost any statement of this type. For a nice discussion of it, see Section 1.3 of Farb and Margalit's book "A primer on mapping class groups", which is available here.

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Thanks a lot Andy. The book by Farb and Margalit is excellent. – Tony Huynh Aug 1 '10 at 21:52

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