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

Rhuaidi Antonio Burke; Benjamin A. Burton; Jonathan Spreer

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arXiv · 2412.04768

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

Rhuaidi Antonio Burke·Benjamin A. Burton·Jonathan Spreer

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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.

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Rhuaidi Antonio Burke, Benjamin A. Burton, Jonathan Spreer. 2024-12-06. Small Triangulations of $4$-Manifolds: Introducing the $4$-Manifold Census. https://arxiv.org/abs/2412.04768

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