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Reeb graph invariants of Morse functions and $3$-manifold groups
- Publication Year :
- 2024
-
Abstract
- In this work we are focused on the existence of Morse functions on a closed manifold $M$ which are far from being ordered, i.e. whose Reeb graphs have positive first Betti number, especially the maximal possible, equals $\operatorname{corank}(\pi_1(M))$. In the case of $3$-manifolds we describe the minimal number of critical points needed to construct such functions, which is related with the number of vertices of degree $2$ in Reeb graphs. We define a new invariant of $3$-manifold groups and their presentations, and using Heegaard splittings we show its utility in determining occurrence of disordered Morse functions. In particular, the Freiheitssatz, a result for one-relator groups, allows us to calculate this invariant in the case of orientable circle-bundles over a surface, which provides an interesting example of the behaviour of Morse functions.<br />Comment: 13 pages
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2403.02291
- Document Type :
- Working Paper