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Differentiable approximation of continuous locally definable maps that preserves the image

This paper extends previous results on the uniform approximation of continuous definable maps on compact sets to the Whitney topology setting for continuous locally definable maps on locally compact sets, preserving the image by combining o-minimal and PL geometry with Pawłucki's desingularization techniques.

Original authors: Antonio Carbone

Published 2026-08-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Antonio Carbone

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a constant tension between the smooth, flowing curves of the real world and the rigid, blocky structures of pure logic. For centuries, mathematicians have relied on a powerful idea called approximation: the ability to take a messy, continuous shape or movement and replace it with a simpler, smoother version that is close enough to be useful. Think of how a digital photograph, made of tiny square pixels, can look like a smooth, continuous image when you step back. This concept is so fundamental that it underpins much of modern geometry and physics. However, a stubborn problem has long plagued this field: what happens when you try to smooth out a shape but are forbidden from changing its destination? Imagine trying to smooth a crumpled piece of paper into a flat sheet, but you must ensure that every single point on the paper still lands on the exact same spot on the table as it did before. If the paper has holes, tears, or complex folds, standard smoothing techniques often fail, either tearing the paper or shifting the points to new locations. This is the specific challenge of preserving the image of a map while improving its smoothness.

The question becomes even more intricate when the shapes involved are not just simple geometric figures but belong to a special class of mathematical objects known as "definable" sets. These are shapes that can be described with precise logical rules, avoiding the chaotic, infinitely complex patterns that appear in some other areas of math. For a long time, mathematicians could successfully smooth these shapes if they were compact, meaning they were closed and bounded, like a solid sphere or a finite cube. But the real world is rarely so tidy; many important shapes are open-ended or stretch out infinitely, known as locally compact sets. Until now, it was unclear whether one could smooth these more complex, open-ended shapes while strictly keeping every point in its original destination. A recent paper by Antonio Carbone tackles this exact problem, proving that it is indeed possible to smooth these complex, locally defined maps without losing a single point of their original image, provided that the map itself is locally definable and its image is both locally compact and locally definable.

Carbone's work focuses on a specific type of mathematical map, which is simply a rule that assigns every point in one shape to a point in another. The goal was to take a continuous map that might be jagged or rough and replace it with a map that is differentiable, meaning it is smooth enough to have a well-defined slope at every point, without changing where the map sends the points. The difficulty lies in the fact that standard smoothing methods often act like a magnet, pulling points toward a center or stretching them out, which inevitably changes the final destination of the map. If the original map was surjective, meaning it covered every single point in the target area, a standard smoothing process might accidentally leave some points uncovered, effectively erasing parts of the image. Carbone's paper demonstrates that for a broad class of these definable shapes, this disaster can be avoided, as long as the map is locally definable and the target area satisfies the specific conditions of being locally compact and locally definable.

To achieve this, the author did not rely on a single, simple trick. Instead, the proof is a careful construction that weaves together two different branches of geometry. One branch deals with the smooth, continuous nature of the maps, while the other deals with the rigid, piecewise-linear structure of shapes made of flat triangles and their higher-dimensional equivalents. The strategy involves breaking the complex, open-ended shape into a collection of smaller, manageable, and compact pieces. For each of these small pieces, the mathematician applies a known technique that works perfectly for closed, bounded shapes. However, simply stitching these smoothed pieces back together would create new jagged edges where they meet. To solve this, the author introduces a sophisticated method of "desingularization," a technique originally developed by another mathematician to resolve sharp corners and self-intersections in geometric shapes. This method acts like a precise surgical tool, smoothing out the transitions between the pieces so that the entire map becomes a single, seamless, smooth surface.

The most critical part of the argument ensures that the final, smoothed map still hits every single target point that the original map hit. The author constructs a series of overlapping zones and uses a mathematical tool called a "partition of unity," which can be thought of as a way of blending different local solutions together without creating conflicts. By carefully controlling how these local solutions interact, the proof guarantees that the final result is not only smooth but also surjective, meaning it covers the entire target area just as the original did. The paper proves that if the original map is defined by logical rules, its image is locally compact and locally definable, one can always find a smooth version of that map that preserves the image exactly. This result extends previous findings that were limited to compact shapes, opening the door to applying these powerful smoothing techniques to a much wider range of mathematical and potentially physical problems where boundaries are not fixed or finite.

The significance of this work lies in its ability to bridge the gap between the rigid constraints of logical definability and the fluid requirements of smooth calculus. By showing that the image of a map can be preserved during the smoothing process, even in complex, non-compact settings, the paper removes a major obstacle in the field of real algebraic geometry. It confirms that the flexibility of smooth functions does not come at the cost of losing the structural integrity of the map's image. The proof is rigorous and complete, relying on established theorems and logical deductions rather than simulations or approximations. It stands as a definitive answer to a question that had remained open, providing a new toolkit for mathematicians who need to work with smooth maps on complex, definable sets. The result is a clearer understanding of how smoothness and structure can coexist, ensuring that even when we refine our mathematical models to be more elegant and differentiable, we do not lose sight of the reality they are meant to describe.

Technical Summary: Differentiable Approximation of Continuous Locally Definable Maps That Preserves the Image

Problem Statement
The paper addresses the problem of approximating continuous maps by differentiable maps within the context of o-minimal structures, specifically focusing on the preservation of the image. While classical results (such as those by Fernando and Ghiloni) establish that continuous definable maps on compact definable sets can be uniformly approximated by $C^p$ definable maps without altering their image, this property does not automatically extend to locally compact locally definable sets.

The core difficulty arises when the target space $Y$ possesses singularities or when the map $f: X \to Y$ is not proper. In general, approximating a continuous map $f$ by a $C^p$ map $g$ does not guarantee that $g(X) = f(X)$ . For instance, Sard's theorem implies that no surjective $C^p$ map exists from $\mathbb{R}$ to $\mathbb{R}^2$ for $p \ge 1$ , meaning a surjective continuous map cannot be approximated by a surjective $C^p$ map in the standard Euclidean setting. The paper seeks to resolve this for the specific category of locally definable maps where the image is also locally compact and locally definable.

Methodology
The proof of the main theorem relies on an interplay between o-minimal geometry and Piecewise Linear (PL) geometry, utilizing a multi-step reduction strategy:

Triangulation and Reduction to Polyhedra:
The author first utilizes a result by Pawłucki to triangulate the target space $Y$ (which is locally compact and locally definable) into a locally finite simplicial complex $L$ via a $C^p$ homeomorphism. This allows the problem to be reduced to approximating a map into the polyhedron $|L|$ . Similarly, the domain $X$ is triangulated.
Local Decomposition:
Since $X$ is locally compact but not necessarily compact, it is decomposed into a countable union of compact definable sets $\{Z_s\}_{s \ge 1}$ derived from the maximal simplices of the triangulation. The proof constructs a locally finite covering of $X$ and utilizes partitions of unity to glue local approximations together.
Desingularization and Weak Retractions:
A crucial component is the application of Pawłucki's desingularization techniques. The author constructs a family of "weak retractions" $\rho_s$ from a neighborhood of the target polyhedron $|L|$ back onto $|L|$ . These maps are $C^p$ , locally definable, and possess specific convexity properties (property (viii) in the proof) that ensure convex combinations of points in the image remain within the polyhedron.
Operator Construction:
A global approximation operator $\rho$ is defined using a partition of unity and the local weak retractions. This operator maps continuous functions to $C^p$ functions while ensuring the image remains within the target polyhedron $|L|$ .
Image Preservation via "Widening" and "Smoothing":
To ensure the approximating map is surjective onto the target (i.e., $g(X) = Y$ ), the author employs a technique involving the "widening" of the compact subsets $Z_s$ . They construct local $C^p$ maps that expand the domain slightly and then "smooth" these expansions. By carefully controlling the support of the partition of unity and the geometry of the triangulation, they ensure that the final global map covers the entire image $|L|$ .

Key Contributions and Results
The primary result is Theorem 1.2, which states:
Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be locally compact locally definable sets. Let $\varepsilon: X \to \mathbb{R}$ be a strictly positive continuous function and $p \ge 1$ an integer. If $f: X \to Y$ is a continuous locally definable map such that $f(X) = Y$ , then there exists a locally definable map $g: X \to Y$ of class $C^p$ such that:

$\|f(x) - g(x)\|_m < \varepsilon(x)$ for each $x \in X$ (uniform approximation in the strong Whitney topology).
$g(X) = Y$ (the image is preserved).

This result extends previous work on compact sets (specifically Theorem 1.4 from the author's prior work [2]) to the non-compact, locally definable setting. The paper explicitly notes that standard techniques for proper maps are insufficient for this general case, necessitating the new strategy involving desingularization and the specific handling of the image preservation constraint.

Significance
The paper claims significance by bridging a gap in the approximation theory of o-minimal structures. While differentiable approximation is well-understood for compact sets and for maps with specific properness conditions, this work demonstrates that the preservation of the image is achievable for the broader class of locally definable maps on locally compact sets. The methodology highlights the utility of combining o-minimal geometry with PL techniques and Pawłucki's desingularization, offering a robust framework for handling singular target spaces where tubular neighborhoods (standard in manifold theory) do not exist. The result ensures that within the tame category of o-minimal structures, the flexibility of differentiable approximation does not come at the cost of losing the topological property of surjectivity onto the target set.

Technical Summary: Differentiable Approximation of Continuous Locally Definable Maps That Preserves the Image

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