Cataloguing PL 4-manifolds by gem-complexity

TL;DR

This paper presents an algorithm for classifying PL 4-manifolds using gem-complexity, successfully solving the PL-homeomorphism problem for manifolds up to gem-complexity 8 and exploring implications for exotic structures.

Contribution

The paper introduces a new algorithm for classifying PL 4-manifolds via coloured triangulations, achieving complete classification up to gem-complexity 8 and analyzing relationships among different topological categories.

Findings

Complete classification of PL 4-manifolds up to gem-complexity 8.

Proof of non-existence of exotic PL 4-manifolds within this complexity range.

Potential applications to recognizing triangulations of the K3-surface.

Abstract

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n=4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices). Possible interactions with the (not completely known) relationship among different classification in TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.