Algorithms in 4-manifold topology

TL;DR

This paper presents algorithms for classifying 4-manifolds, including deciding homeomorphism of simply connected topological 4-manifolds and stable diffeomorphism of certain smooth 4-manifolds, advancing computational topology in four dimensions.

Contribution

It introduces the first algorithms for determining homeomorphism of simply connected topological 4-manifolds and for stable classification of specific smooth 4-manifolds based on their fundamental groups.

Findings

Decidability of homeomorphism for simply connected topological 4-manifolds.

Algorithm for computing the Kirby-Siebenmann invariant from Kirby diagrams.

Decidability of stable diffeomorphism for certain classes of smooth 4-manifolds.

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.