Complements of Curves in Closed Surfaces: Homeomorphic?

[WWGD]

Hi, let $\alpha, \gamma$ be non-isotopic curves in a compact, oriented surface S. There is a result to the effect that $S-\alpha$ is homeo. to $S- \gamma$ . This is not true as stated; we can , e.g., remove a disk (trivial class) in a copy of S and then remove a meridian ( a generator of $H^1 (T^2)$). Does anyone know the corrected result? This is not true either if we only remove essential curves. It seems like if $\alpha, \gamma$ actually _were_ isotopic , then this would be true. Or maybe if they were homologous?
Thanks.

 

[homeomorphic]

I seem to remember coming across the result you are looking for in Farb and Margalit's Primer on Mapping Class Groups, in chapter 1, if I am not mistaken. I think maybe they call it the change of coordinates principle.

 

[homeomorphic]

http://www.math.ethz.ch/~bgabi/Farb%20Magalit%20January%202011%20version.pdf

 

[WWGD]

Thanks; in case anyone else is interested, here it is, towards the bottom of p.39:

There is an orientation-preserving homeo. of a surface S taking one s.c.c * $\gamma$ to $\beta$ iff the
corresponding cut surfaces $S -\gamma$ , $S- \beta$ are homeomorphic.

*s.c.c: Simple-Closed Curve.