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

TL;DR

This work studies smooth functions that are Morse on preimages of non-extremal values (I-Morse) and introduces IN-Morse-Reeb (IN-M-R) functions alongside their Reeb digraphs. The main contribution is a constructive theorem: given two IN-M-R functions on closed connected $m$-manifolds ($m>1$), one can construct an IN-M-R function on the connected sum whose Reeb digraph is isomorphic to a prescribed graph $G_R$ obtained by gluing the original Reeb digraphs at two non-extremum points. The proof develops local models using $k$-handles, Ehresmann fibrations, and product bundles to realize the target Reeb digraph, including degree conditions at vertices and orientation considerations. These results advance the differential-topological study of singularities and Reeb graphs by enabling explicit synthesis of Morse-like functions with prescribed combinatorial structure on connected sums.

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.

Figures (5)

• Figure 1: A part of the manifold $M$ mapped to a vertex $v_{[p,q,],j}$ by the original quotient map $q_f$ and (a neighborhood of) $[p,q] \in \mathbb{R}$ by the original function $f$ in Proposition \ref{['prop:2']}. The manifold $B_{f,G} \subset M$ is shown in blue and the (interior of the) manifold $M-B_{f,G}$ is colored in green. Furthermore, red dots show critical points of the Morse function such that preimages of single points containing no critical point are disjoint unions of copies of $S^{m-1}$.
• Figure 2: Around each of the two product bundles, the product map of the original function $f_{a,b,s}$ and another suitable function is considered and regarded as a fold map whose singular points are regarded as the higher dimensional versions of critical points of index $0$. The (preimages of the) blue colored regions show the product bundles and we remove. After that, we glue the remaining functions (onto $[a,b]$) preserving the value at each point as we do in kitazawa6 and the preprint kitazawa7. We also consider the smoothing.
• Figure 3: We deform the resulting local fold map of FIGURE \ref{['fig:2']} to have our desired local Morse function.
• Figure 4: The Reeb digraph $W_f$ around a vertex $v$ of degree $1$ in $W_f$, mapped to a vertex of degree $1$ of $G$. The blue colored edge is originally a part of an edge of the graph $G$ containing the vertex $v \in W_f \subset G$. To each of blue colored two vertices, exactly one critical point of the local Morse function is mapped. Last, we can deform the function with the Reeb digraph by applying a well-known argument of canceling a pair of handles.
• Figure 5: Around each of the two product bundles around the vertex $v$ of degree $1$ of $W_f$, mapped into $G$. The product map of the original local function and another suitable function is considered in the upper figure and regarded as a fold map whose singular points are regarded as the higher dimensional versions of critical points of index $0$. Similar arguments are also in FIGURE \ref{['fig:2']}. The (preimages of the) blue colored regions in the upper figure show the product bundles and we remove their interiors as we do in FIGURE \ref{['fig:2']}. The blue dotted points show (the points whose preimages considered for the local fold map contain) critical points of the local Morse function presented in the upper part of FIGURE \ref{['fig:4']} by the Reeb graph. As presented in FIGURE \ref{['fig:3']}, the resulting local fold map is deformed to have our desired situation of FIGURE \ref{['fig:4']} (the lower figure here). The critical points shown in blue are mapped to the blue colored vertices of the graph of (the upper figure of) FIGURE \ref{['fig:4']}.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• Proposition 2
• proof
• proof : Our proof of Theorems \ref{['thm:1']} and \ref{['thm:2']}