Core Concepts

The existence of a lifting of a non-degenerate piecewise linear map between graphs to an embedding can be characterized by the satisfiability of a 3-CNF boolean formula.

Abstract

The paper studies the problem of finding conditions for the existence of an embedding e f: P → Q × R such that f = prQ ◦ e f, where f: P → Q is a piecewise linear map between polyhedra. The focus is on non-degenerate maps between graphs, where non-degeneracy means that the preimages of points are finite sets.
The key results are:
Necessary conditions for the existence of a lifting:
All the covering maps pn: |K(n) f| → |e K(n) f|, n > 1 are trivial.
There are no n-obstructors for f for any n > 1.
Necessary and sufficient conditions for the existence of a lifting:
There exists an admissible collection of linear orders on the sets f^-1(v), v ∈ V(L).
If the covering map p2: |K(2) f| → |e K(2) f| is trivial, then the existence of a lifting is equivalent to the satisfiability of a 3-CNF formula Γ_f.
Connections to lifting of smooth immersions:
The author provides a counterexample to a result by V. Poénaru on lifting of smooth immersions to embeddings.
The author shows that for each 3-CNF formula Γ of a specific form, there exists a generic immersion g: S ↬ B of a surface S with boundary into a handlebody B such that the restriction of g to the preimage of the set of multiple points forms a map between graphs with Γ_f = Γ.
Approximation by embeddings:
For generic simplicial maps from a tree to a segment, the non-existence of 2-obstructors is a necessary and sufficient condition for the existence of a lifting.

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Key Insights Distilled From

by Alexey Gorel... at **arxiv.org** 04-19-2024

Deeper Inquiries

The results on lifting maps between graphs can be extended to higher-dimensional polyhedra or manifolds by considering the combinatorial techniques and conditions established in the paper. For higher-dimensional polyhedra, the concept of non-degenerate maps and the existence of liftings to embeddings can be generalized by adapting the conditions and necessary criteria outlined for graphs. This extension would involve defining appropriate equivalence relations, linear orders, and configurations specific to the higher-dimensional structures. By applying similar principles and combinatorial methods, the problem of lifting maps between higher-dimensional polyhedra or manifolds to embeddings can be addressed effectively.

Determining the satisfiability of the 3-CNF formula Γ_f and constructing the corresponding lifting can be approached using efficient algorithms and computational methods. One possible approach is to utilize SAT solvers, which are widely used in solving Boolean satisfiability problems. By encoding the conditions and constraints of the formula Γ_f into a SAT instance, the solver can efficiently determine the satisfiability of the formula. Additionally, techniques from computational topology and graph theory can be employed to analyze the structure of the formula and optimize the algorithm for constructing the lifting. By leveraging algorithmic tools and computational techniques, the process of determining the satisfiability of Γ_f and constructing the lifting can be streamlined and optimized.

The techniques developed in the paper have various potential applications beyond the specific problem of lifting maps to embeddings. Some of the applications include:
Computational Topology: The methods and results can be applied in computational topology to study the properties of simplicial complexes, graphs, and higher-dimensional structures. These techniques can aid in analyzing the connectivity, configurations, and embeddings of complex geometric spaces.
Graph Theory: The concepts and combinatorial techniques can be utilized in graph theory to study graph homomorphisms, colorings, and orders on vertices and edges. The results can contribute to the understanding of graph embeddings and structural properties of graphs.
Algorithm Design: The algorithms and computational methods developed for determining the satisfiability of 3-CNF formulas can be applied in various optimization problems, constraint satisfaction, and decision-making processes. These techniques have implications in algorithm design and complexity theory.
Data Analysis: The principles and results can be used in data analysis, network modeling, and pattern recognition tasks. By applying the combinatorial techniques to analyze data structures and relationships, insights can be gained into the underlying patterns and structures in complex datasets.
Overall, the techniques developed in the paper have broad applications in mathematics, computer science, and data analysis, offering valuable tools for studying complex systems and structures.

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