Frontiers of sphere recognition in practice

TL;DR

This paper explores the practical limits of sphere recognition in higher dimensions, presenting both positive and negative computational results and utilizing randomly constructed discrete Morse functions.

Contribution

It provides new insights into the computational complexity of sphere recognition in dimensions three to five, combining theoretical analysis with practical experiments.

Findings

Demonstrates the undecidability in dimensions five and above.

Identifies challenges in polynomial-time recognition in dimensions three and four.

Uses random discrete Morse functions to analyze recognition limits.

Abstract

Sphere recognition is known to be undecidable in dimensions five and beyond, and no polynomial time method is known in dimensions three and four. Here we report on positive and negative computational results with the goal to explore the limits of sphere recognition from a practical point of view. An important ingredient are randomly constructed discrete Morse functions.