Simple Morse Functions on an Oriented
Surface with Boundary

Bohdana Hladysh

Taras Shevchenko National University of Kyiv, 4-e Akademika Glushkova Ave., Kyiv,
03127, Ukraine
E-mail: biv92@ukr.net

Alexandr Prishlyak

Taras Shevchenko National University of Kyiv, 4-e Akademika Glushkova Ave., Kyiv,
03127, Ukraine
E-mail: prishlyak@yahoo.com

Received July 16, 2017, revised June 19, 2019.

Abstract

In the paper, smooth functions with non-degenerate critical points on a smooth compact surface with boundary are considered. Firstly, it is shown that these functions are topologically equivalent to m-functions. The
equipped Reeb graph is used to describe their topological structure. Secondly,
the authors characterize the topological structure of all simple functions with at most 5 critical points. And finally, a formula for the genus of the surface based on the equipped Reeb graph is obtained.