NON-SINGULAR EXTENSIONS OF MORSE FUNCTIONS ON DISCONNECTED SURFACES

抄録

In this paper, we study non-singular extensions of Morse functions on closed orientable surfaces. By a non-singular extension of such a Morse function, we mean an extension to a function without critical points on some compact orientable 3-manifold having as boundary the given surface. In 1977, Curley characterized the existence of non-singular extensions of non-singular boundary germs in terms of combinatorics on associated labeled Reeb graphs. We apply Curley's result to show that every Morse function on a closed orientable (possibly disconnected) surface has a non-singular extension to a 3-manifold that is connected.