Small Triangulations of $4$-Manifolds: Introducing the $4$-Manifold Census

TL;DR

This paper develops a classification framework for triangulated 4-manifolds and applies it to enumerate all such manifolds with up to six pentachora, revealing a limited number of PL-types for most cases.

Contribution

The authors introduce a new framework for classifying PL-types of 4-manifolds and successfully classify all triangulations with up to six pentachora, except for some special cases.

Findings

Classified all triangulated 4-manifolds with up to six pentachora.

Found limited PL-types for most cases, with few exceptions.

Identified interesting combinatorial structures in resistant cases.

Abstract

We present a framework to classify PL-types of large censuses of triangulated $4$-manifolds, which we use to classify the PL-types of all triangulated $4$-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the $4$-sphere, $\mathbb{C}P^2$, and the rational homology sphere $QS^4(2)$, where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations -- which we deem interesting in their own rights.