Algorithms in 4-manifold topology
Abstract
We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric. In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.
Cite
@article{arxiv.2411.08775,
title = {Algorithms in 4-manifold topology},
author = {Stefan Bastl and Rhuaidi Burke and Rima Chatterjee and Subhankar Dey and Alison Durst and Stefan Friedl and Daniel Galvin and Alejandro García Rivas and Tobias Hirsch and Cara Hobohm and Chun-Sheng Hsueh and Marc Kegel and Frieda Kern and Shun Ming Samuel Lee and Clara Löh and Naageswaran Manikandan and Léo Mousseau and Lars Munser and Mark Pencovitch and Patrick Perras and Mark Powell and José Pedro Quintanilha and Lisa Schambeck and David Suchodoll and Martin Tancer and Annika Thiele and Paula Truöl and Matthias Uschold and Simona Veselá and Melvin Weiß and Magdalina von Wunsch-Rolshoven},
journal= {arXiv preprint arXiv:2411.08775},
year = {2025}
}
Comments
24 pages, 1 Figure; V2: Minor changes, version accepted for publication in Algebr. Geom. Topol