Real algebraic functions on closed manifolds whose Reeb spaces are given graphs

TL;DR

This paper constructs real algebraic functions on closed manifolds with prescribed Reeb graphs, extending previous work mainly on surfaces to more general manifolds, and addresses a longstanding question in the field.

Contribution

It provides a method to realize any given graph as the Reeb graph of a real algebraic function on a closed manifold, generalizing earlier results from surfaces to higher-dimensional manifolds.

Findings

Constructed real algebraic functions with prescribed Reeb graphs

Extended the realization problem to general manifolds

Connected the theory to algebraic domain constraints

Abstract

In our paper, we construct a real-algebraic function whose Reeb (Kronrod-Reeb) graph is a graph respecting some algebraic domain: a graph for this is called Poincar\'e-Reeb graph. The Reeb graph of a smooth function is defined as a natural graph which is the quotient space of the manifold of the domain under a natural equivalence relation for some wide and nice class of smooth functions. The vertex set is defined as the set of all connected components containing some singular points of the function: a singular point of a smooth function is a point where the differential vanishes. Morse-Bott functions give very specific cases. The relation is to contract each connected component of each preimage to a point. Sharko has asked a natural and important problem: can we construct a nice smooth function whose Reeb graph is a given graph? Explicit answers have been given first by Masumoto-Saeki in a generalized manner for closed surfaces. After that various answers have been presented by various researchers and most of them are essentially for functions on closed surfaces and Morse functions such that connected components of preimages containing no singular points are spheres. Recently the author has also considered questions and solved in cases the preimages are general manifolds.