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Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles

Naoki Kitazawa

Abstract

In this paper, we study Morse functions with regular level sets consisting of spheres, tori, or Klein Bottles on $3$-dimensional closed manifolds. We characterize $3$-dimensional manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$ of the circle $S^1$ and the sphere $S^2$, lens spaces, or non-orientable closed and connected manifolds of genus $1$ by a certain subclass of such Morse functions. This is a kind of extensions of the orientable case, by Saeki, in 2006. This is a variant of its extension by the author for $3$-dimensional orientable manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$, lens spaces, or torus bundles over $S^1$ by a certain class of Morse-Bott functions. We also classify Morse functions with given regular level sets consisting of $S^2$, $S^1 \times S^1$, or Klein Bottles in a certain sense, generalizing some previous work by the author.

Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles

Abstract

In this paper, we study Morse functions with regular level sets consisting of spheres, tori, or Klein Bottles on -dimensional closed manifolds. We characterize -dimensional manifolds represented by connected sums each of whose summands is the product of the circle and the sphere , lens spaces, or non-orientable closed and connected manifolds of genus by a certain subclass of such Morse functions. This is a kind of extensions of the orientable case, by Saeki, in 2006. This is a variant of its extension by the author for -dimensional orientable manifolds represented by connected sums each of whose summands is the product , lens spaces, or torus bundles over by a certain class of Morse-Bott functions. We also classify Morse functions with given regular level sets consisting of , , or Klein Bottles in a certain sense, generalizing some previous work by the author.

Paper Structure

This paper contains 4 sections, 6 theorems, 7 figures.

Table of Contents

  1. Introduction.
  2. Characterizations of $3$-dimensional manifolds of certain classes and one of our main result, Theorem \ref{['thm:4']}.
  3. Some classifications of Morse functions up to isomorphisms of ($m-1$)-labeled-digraphs with $m=2,3$.
  4. Conflict of interest and Data availability.

Key Result

Theorem 1

A $3$-dimensional closed, connected, and orientable manifold $M$ admits a Morse function $f:M \rightarrow \mathbb{R}$ such that regular level sets $f^{-1}(r)$ consist of surfaces diffeomorphic to $S^2$ or $S^1 \times S^1$ if and only if $M$ is diffeomorphic to a connected sum each of whose summand i

Figures (7)

  • Figure 1: Fundamental Morse functions in STEP 1. Their Reeb data are presented roughly: for the black colored edges $e$, $F_e=S^2$, and for the green colored edge $e$, $F_e=S^1 \times S^1,K^2$.
  • Figure 2: Construction of a new Morse function on a connected sum of given two manifolds, with given Morse functions, in STEP 1, where the Reeb digraphs are presented locally.
  • Figure 3: Deforming a function in Case 2-2 to a local simple Morse function in such a way that the 1st Betti number of the Reeb graph increases by $l_{v_2}$ and the Reeb digraphs are presented locally and the regular level sets of the resulting local Morse function are diffeomorphic to $S^2$ or $S^2 \sqcup S^2$.
  • Figure 4: A local representation of the Reeb data of a simple Morse function with regular level sets consisting of surfaces diffeomorphic to $S^2$, $S^1 \times S^1$, or $K^2$. Note that for the black colored edges $e$, $F_e=S^2$ always holds, and that either the following holds in addition. For the two green colored edges $e$ here, $F_e=S^1 \times S^1$ always hold or $F_e=K^2$ always hold.
  • Figure 5: An important local deformation of simple Morse functions for FIGURE \ref{['fig:4']}.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1: saeki4
  • Theorem 2: kitazawa8
  • Theorem 3
  • Theorem 4
  • proof : A proof of Theorem \ref{['thm:4']} (\ref{['thm:4.1']})
  • proof : A proof of Theorem \ref{['thm:4']} (\ref{['thm:4.2']})
  • Theorem 5
  • Theorem 6
  • proof