A note on Reeb spaces of some explicit real analytic functions

This paper presents the Reeb spaces of explicit real analytic functions that are not finite graphs, extending previous studies on smooth functions by incorporating topological considerations of level sets and real algebraic constructions.

The Gist

Imagine you have a giant, complex piece of dough (a mathematical manifold). You want to understand its shape, but it's too big and twisted to look at all at once. So, you decide to slice it.

You take a knife and slice the dough horizontally, layer by layer. Each slice is a level set. If you look at just one slice, you might see a circle, a figure-eight, or a bunch of disconnected islands.

Now, imagine you take all those slices and stack them up, but instead of keeping them separate, you glue together any pieces of dough that are connected within the same slice. You squish each connected piece of dough down into a single dot.

The result of this squishing process is a new, simpler shape called the Reeb Space. It's like a shadow or a skeleton of your original dough, showing you how the pieces connect as you move from the bottom of the dough to the top.

The Big Question: What does this skeleton look like?

For a long time, mathematicians knew that if your dough is a nice, closed shape (like a sphere or a donut) and you slice it carefully, this skeleton usually looks like a finite graph. Think of a finite graph as a simple stick-figure drawing made of a limited number of dots (vertices) and lines (edges). It's tidy, countable, and easy to draw on a napkin.

Problem 1: Can we build a dough shape so that its skeleton is exactly the stick-figure drawing we want?
Answer: Yes, for simple drawings, we can.

Problem 2: Can we do this using "perfect" mathematical dough (called Real Analytic functions)?
Answer: This is harder. "Perfect" dough is rigid; you can't just glue pieces together with tape (like you can with smooth, flexible dough). It has to be one continuous, unbroken mathematical formula.

Problem 3 (The Focus of this Paper): What if the skeleton we want is infinite?
Imagine a stick-figure drawing that has an endless number of branches, like a tree that never stops growing, or a road that keeps splitting forever. Can we build a "perfect" dough shape that creates this infinite skeleton?

The Author's Solution: The Infinite Tree

Naoki Kitazawa, the author of this paper, says: "Yes, we can."

He constructs two specific examples to prove this:

Example 1: The Infinite Road (The Non-Compact Case)

Imagine a long, endless road that stretches to infinity. Along this road, there are infinitely many "rest stops" (critical points).

• The Dough: Kitazawa builds a shape that is infinite in size (non-compact).
• The Slices: As you move along the road, the slices of dough change shape. Sometimes they are circles, sometimes they disappear.
• The Skeleton: When you squish the connected parts together, you get an infinite graph. It looks like a line with infinitely many little loops or branches attached to it.
• The Magic: He proves that even though the shape is infinite, it is made of a single, perfect mathematical formula (a real analytic function). It's not a patchwork job; it's one continuous, elegant equation.

Example 2: The "Peano" Blob (The Compact Case)

This is the trickier part. Usually, if your dough is a closed, finite shape (like a ball), the skeleton must be a finite graph. But Kitazawa asks: What if we break the rules just a tiny bit?

He creates a closed, finite dough shape. However, the skeleton it produces is weird.

• The Shape: It looks like a finite graph (a tree), but at the very top, all the branches don't just end; they get infinitely close together, squeezing into a single point.
• The "Peano" Continuum: This shape is called a Peano continuum. Imagine a tree where the branches get smaller and smaller, infinitely, until they all merge into a fuzzy, dense point at the top.
• The Twist: If you remove that one fuzzy point at the top, the rest of the shape is a perfect, infinite tree (an E-graph). But with the point included, it's a new, strange object that has never been seen as a Reeb space before.
• The Construction: He uses a special mathematical function that behaves wildly near zero (oscillating like a sine wave that gets faster and faster) to create this "fuzzy" point.

Why Does This Matter?

Think of it like this:

• Old Math: "If you build a house, the blueprint must be a simple, finite drawing."
• Kitazawa's Math: "Actually, if you use the right kind of 'perfect' bricks (real analytic functions), you can build a house whose blueprint is an infinite, fractal-like structure that fits inside a finite box."

He shows that the world of "perfect" mathematical shapes is much more flexible and surprising than we thought. Even with the strict rules of real analytic functions, you can create structures that have infinite complexity, yet remain mathematically beautiful and precise.

In a Nutshell

Kitazawa took a complex mathematical tool (Reeb spaces), which usually produces simple, finite drawings, and showed that by using "perfect" mathematical formulas, you can generate infinite, complex skeletons. He built two models: one that stretches to infinity, and one that is finite but contains an infinite, fractal-like secret at its core. This expands our understanding of how shapes and functions relate to each other in the universe of mathematics.

Technical Summary

1. Problem Statement and Context

The paper addresses the Realization Problem for Reeb spaces. The Reeb space $R_c$ of a continuous map $c: X \to Y$ is defined as the quotient space of $X$ obtained by identifying points that lie in the same connected component of a level set $c^{-1}(q)$.

• Background: For smooth functions on compact manifolds (e.g., Morse functions), the Reeb space is often homeomorphic to a finite graph (the Reeb graph). Previous work by Sharko, Masumoto, Saeki, and Michalak established that any finite graph can be realized as the Reeb graph of a smooth function on a closed surface or manifold.
• The Gap: While the realization of finite graphs in the smooth category is well-studied, the realization of infinite graphs and non-graph topological spaces (specifically Peano continua) as Reeb spaces of explicit real analytic functions remains an open area.
• Specific Problems Addressed:
  1. Can we construct a smooth function whose Reeb space is an infinite graph?
  2. Can such constructions be performed within the real algebraic/analytic category (avoiding partitions of unity, which do not exist in this category)?
  3. Can we construct a compact manifold and a function where the Reeb space is a Peano continuum (a compact, connected, locally connected metric space) that is not homeomorphic to any graph or "E-graph" (a graph with ends)?

2. Methodology

The author employs a constructive approach using explicit real analytic functions and implicit function theorems to define manifolds as zero sets. The methodology avoids global gluing techniques (like partitions of unity) common in differential topology, relying instead on algebraic geometry and specific function behaviors.

• Manifold Construction: Manifolds are defined as zero sets of specific real analytic maps $e: \mathbb{R}^N \to \mathbb{R}^k$. The author ensures these zero sets are smooth manifolds by verifying the rank of the differential of the defining map is maximal (using the Implicit Function Theorem).
• Function Definition: The target function $c: X \to \mathbb{R}$ is defined as the restriction of the canonical projection $\pi_{N,1}(x_1, \dots, x_N) = x_1$ to the constructed manifold $X$.
• Level Set Analysis: The topology of the Reeb space is determined by analyzing the connected components of the level sets (contours). The author constructs the defining functions such that the level sets exhibit specific behaviors (e.g., discrete critical points, specific diffeomorphism types for regular contours).
• Key Technical Tools:
  • Explicit Functions: Use of functions like $\frac{\sin^2 x}{2(x^2+1)}$ and $e^{-1/x^2}\sin^2(1/x)$ to create infinite sequences of critical points accumulating at specific locations.
  • Geometric Constraints: Constructing regions $D_S$ bounded by disjoint real algebraic hypersurfaces to control the topology of the level sets.
  • Properness: Ensuring the quotient map $q_c: X \to R_c$ is proper (preimages of compact sets are compact) to maintain topological control.

3. Key Contributions and Results

The paper presents two main theorems providing affirmative answers to the open problems regarding infinite and non-graph Reeb spaces in the real analytic category.

Theorem 1: Realization of an Infinite Graph (Non-Compact Case)

• Construction: For any integer $m > 1$, the author constructs a non-compact, connected, smooth manifold $X_m \subset \mathbb{R}^{m+1}$ defined as the zero set of an explicit real analytic function.
• Function: A function $c_m: X_m \to \mathbb{R}$ is defined as the restriction of the projection $\pi_{m+1,1}$.
• Results:
  1. The Reeb space $R_{c_m}$ is isomorphic to a specific connected infinite graph $G_1$.
  2. The quotient map $q_{c_m}$ is proper.
  3. Every regular contour (level set component) is diffeomorphic to the sphere $S^{m-1}$.
  4. The critical points form a discrete set, and the structure of the Reeb space is an "E-graph" (a graph with ends) that does not depend on $m$ up to isomorphism.

Theorem 2: Realization of a Peano Continuum (Compact Case)

• Construction: For integers $m_1, m_2 > 1$, the author constructs a compact, connected, smooth manifold $X_{m_1, m_2} \subset \mathbb{R}^{m+2}$ (where $m=m_1+m_2$).
• Analyticity: The defining map is real analytic outside a set of Lebesgue measure zero.
• Results:
  1. The Reeb space $R_{c_{m_1, m_2}}$ is homeomorphic to a Peano continuum $G_2$.
  2. Non-Graph Property: $G_2$ is not homeomorphic to any E-graph.
  3. Structure: $G_2$ contains a unique point $p_{G_2}$ such that removing it yields a space homeomorphic to an E-graph. This point corresponds to the accumulation of infinitely many critical contours converging to a single point in the Reeb space.
  4. The manifold is compact, and the function is the restriction of a projection.

4. Significance and Implications

• Extension of Realization Theory: The paper significantly extends the scope of Reeb space realization from finite graphs on compact manifolds to infinite graphs and Peano continua. It demonstrates that the Reeb space of a "nice" (real analytic) function can be topologically complex, not just a finite graph.
• Real Analytic Category: A major contribution is achieving these results in the real analytic category. Unlike the smooth category, where partitions of unity allow for flexible global constructions, the real analytic category requires explicit, global definitions. The author proves that even with these rigid constraints, complex Reeb spaces can be realized.
• Topological Classification: The results clarify the boundary between Reeb spaces that are graphs and those that are not. Theorem 2 specifically constructs a compact manifold where the Reeb space fails to be a graph due to the accumulation of critical points, challenging the intuition that compactness implies a finite graph structure (which holds for Morse functions but not general smooth/analytic functions).
• Future Directions: The paper concludes by posing further problems (Problems 5 and 6) regarding the classification of Reeb spaces for compact manifolds with specific analytic properties and the existence of non-graph Reeb spaces for non-compact analytic manifolds with compact images.

Summary

Naoki Kitazawa's paper provides explicit constructions of real analytic functions on both non-compact and compact manifolds whose Reeb spaces are, respectively, an infinite graph and a Peano continuum that is not a graph. By utilizing explicit algebraic definitions and careful analysis of critical point accumulation, the author demonstrates that the topological complexity of Reeb spaces in the real analytic category is far richer than previously established for finite graphs, opening new avenues for the study of global topology via real analytic functions.