On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds which may not be orientable
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866915078693978112 |
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| author | Kitazawa, Naoki |
| author_facet | Kitazawa, Naoki |
| contents | The Reeb space of a smooth function is a topological and combinatoric object and fundamental and important in understanding topological and geometric properties of the manifold of the domain. It is the graph and a topological space endowed with a natural topology. This is defined as the quotient space of the manifold of the domain where the equivalence relation is as follows: two points in the manifold are equivalent if and only if they are in a same connected component of a level set or a preimage. In considerable cases they are graphs (Reeb graphs): if the function is a so-called Morse(-Bott) functions for example, then this is the graph such that a point is a vertex if and only if the corresponding connected component of the level set contains some singular points.
The author previously constructed explicit smooth functions on suitable 3-dimensional connected, closed and orientable manifolds whose Reeb graphs are isomorphic to prescribed graphs and whose preimages are as prescribed types. This gives a new answer to so-called realization problems of graphs as Reeb graphs of smooth functions of suitable classes. The present paper concerns a variant in the case where the 3-dimensional manifolds may not be non-orientable extending the result before. \end{abstract} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_01300 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds which may not be orientable Kitazawa, Naoki General Topology Geometric Topology The Reeb space of a smooth function is a topological and combinatoric object and fundamental and important in understanding topological and geometric properties of the manifold of the domain. It is the graph and a topological space endowed with a natural topology. This is defined as the quotient space of the manifold of the domain where the equivalence relation is as follows: two points in the manifold are equivalent if and only if they are in a same connected component of a level set or a preimage. In considerable cases they are graphs (Reeb graphs): if the function is a so-called Morse(-Bott) functions for example, then this is the graph such that a point is a vertex if and only if the corresponding connected component of the level set contains some singular points. The author previously constructed explicit smooth functions on suitable 3-dimensional connected, closed and orientable manifolds whose Reeb graphs are isomorphic to prescribed graphs and whose preimages are as prescribed types. This gives a new answer to so-called realization problems of graphs as Reeb graphs of smooth functions of suitable classes. The present paper concerns a variant in the case where the 3-dimensional manifolds may not be non-orientable extending the result before. \end{abstract} |
| title | On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds which may not be orientable |
| topic | General Topology Geometric Topology |
| url | https://arxiv.org/abs/2108.01300 |