Combinatorial modifications of Reeb graphs and the realization problem

Łukasz Patryk Michalak

Published 2018-11-20Version 1

We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the number of cycles 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 maximal number of cycles 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.