Smooth functions which are Morse on preimages of values not being local extrema and constructing natural functions of the class on connected sums of manifolds admitting these functions

2025/10/01 by Kitazawa, Naoki
#Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.00464

Abstract

We discuss smooth functions which are Morse on preimages of values not being local extrema. We call such a function internally Morse or I-Morse. The Reeb graph of a smooth function is the space of all connected components of preimages of single points of it topologized with the natural quotient topology of the manifolds and a vertex of it is a point corresponding to a preimage with critical points. A smooth function is neat with respect to the Reeb graph or N-Reeb if the preimages of the vertices are the closed subsets in the manifolds of the domains with interiors being empty. We discuss I-Morse and N-Reeb functions, IN-Morse-Reeb functions. Our main result presents an IN-Morse-Reeb function respecting two such functions, on a connected sum of these given manifolds.

Citations

• Characterizing 3-dimensional manifolds represented as connected sums of Lens spaces, S² × S¹, and torus bundles over the circle by certain Morse-Bott functions
• Non-singular extensions of circle-valued Morse functions
• On smooth functions with two critical values
• Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages
• Reeb spaces of smooth functions on manifolds
• Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups
• Constructions of round fold maps on circle bundles

Related

• On Reeb graphs induced from smooth functions on closed or open manifolds
• Real algebraic functions on closed manifolds whose Reeb spaces are given graphs
• Maps on manifolds onto graphs locally regarded as the quotient maps onto Reeb spaces of some differentiable maps and a new construction problem
• Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages
• Smooth functions with simple structures on 3-dimensional closed manifolds with prescribed Reeb graphs and preimages