On reconstructing Morse functions with prescribed level sets on $3$-dimensional manifolds and a necessary and sufficient condition for the reconstruction

Abstract

We discuss a necessary and sufficient condition for reconstruction of Morse functions with prescribed (regular) level sets on $3$-dimensional manifolds. The present work strengthens a previous result of the author where only sufficient conditions are studied. Our new work is also regarded as a kind of addenda.

10 pages. A kind of addenda to a published article http://mfat.imath.kiev.ua/article/?id=1911: here we show that the sufficient condition for reconstructing Morse functions there is also a necessary condition. This is submitted to another refereed journal motivated by the acceptance of the article [5] for publication