Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2020
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| _version_ | 1866916763185184768 |
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| author | Marzantowicz, Wacław Michalak, Łukasz Patryk |
| author_facet | Marzantowicz, Wacław Michalak, Łukasz Patryk |
| contents | We construct a correspondence between epimorphisms $φ\colon π_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto framed cobordism classes of such systems. In consequence, for closed manifolds any such $φ$ can be represented by the Reeb epimorphism of a Morse function $f\colon M \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map $M \to \mathcal{R}(f)$ onto the Reeb graph of $f$. Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups, providing a new purely geometrical-topological proof of the solution of this problem for surface groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_02388 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups Marzantowicz, Wacław Michalak, Łukasz Patryk Geometric Topology Group Theory 20F65 (Primary), 57M15, 57R90 (Secondary) We construct a correspondence between epimorphisms $φ\colon π_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto framed cobordism classes of such systems. In consequence, for closed manifolds any such $φ$ can be represented by the Reeb epimorphism of a Morse function $f\colon M \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map $M \to \mathcal{R}(f)$ onto the Reeb graph of $f$. Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups, providing a new purely geometrical-topological proof of the solution of this problem for surface groups. |
| title | Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups |
| topic | Geometric Topology Group Theory 20F65 (Primary), 57M15, 57R90 (Secondary) |
| url | https://arxiv.org/abs/2002.02388 |