标题： 在$3$维流形上具有指定原像的 Morse 函数的重构以及重构的充分必要条件 显示英文标题

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

摘要： 我们讨论在$3$维流形上具有指定原像的莫尔斯函数重构的必要充分条件。 目前的工作加强了作者之前的结论，其中仅研究了充分条件。 我们的新工作也被视为一种补充。 显示英文摘要

摘要： We discuss a necessary and sufficient condition for reconstruction of Morse functions with prescribed preimages 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 addendum.