Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911570055921664 |
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| author | Kitazawa, Naoki |
| author_facet | Kitazawa, Naoki |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_04910 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles Kitazawa, Naoki Geometric Topology Combinatorics 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. |
| title | Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles |
| topic | Geometric Topology Combinatorics |
| url | https://arxiv.org/abs/2604.04910 |