Given two compact oriented surfaces that have the same number of genus and boundary components. How to construct a homeomorphism that sends one to another?



Construct a homeomorphism between two surfaces
A closed surface of genus g can be cut along 2g closed curves (all at one base point) to obtain a 4g-gon. Do this for both surfaces. Then choose a homeomorphism between the 4g-gons which matches the corresponding pairs of curves, so you get a well-defined homeomorphism of surfaces. (A 4g-gon is a cone over some interior cone point. You can first choose an appropriate homeomorphism of boundaries, then fix an interior „cone point“ for both 4g-goes, and then just take the cone map, that extends the chosen homeomorphism of boundaries into the interior, sending one cone point to the other.)


Similarly, surfaces with boundary can be cut into polygons, where now some edges of the polygon correspond to boundary curves.


