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Combinatorial Modifications of Reeb Graphs and the Realization Problem

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Abstract

We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold M. Along the way, we show that the Reeb number \(\mathcal {R}(M)\), i.e., the maximum cycle rank among all Reeb graphs of functions on M, is equal to the corank of fundamental group \(\pi _1(M)\), thus extending a previous result of Gelbukh to the non-orientable case.

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  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, ul. Uniwersytetu Poznańskiego 4, 61-614, Poznań, Poland

    Łukasz Patryk Michalak

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  1. Łukasz Patryk Michalak
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Correspondence to Łukasz Patryk Michalak.

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The author was supported by the Polish Research Grants: NCN UMO-2015/19/B/ST1/01458 and NCN Sheng 1 UMO-2018/30/Q/ST1/00228.

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Michalak, Ł.P. Combinatorial Modifications of Reeb Graphs and the Realization Problem. Discrete Comput Geom 65, 1038–1060 (2021). https://doi.org/10.1007/s00454-020-00260-6

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  • DOI: https://doi.org/10.1007/s00454-020-00260-6

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