Artin presentations, triangle groups, and 4-manifolds

TL;DR

This paper explores the connection between Artin presentations, triangle groups, and 4-manifolds, identifying specific presentations that yield certain simply-connected 4-manifolds with low second Betti number.

Contribution

It classifies all Artin presentations on two generators that present the trivial group and identifies corresponding 4-manifolds with small second Betti number.

Findings

All Artin presentations on two generators presenting the trivial group are found.

Classification of smooth, closed, simply-connected 4-manifolds with second Betti number ≤ 2 in this context.

Explicit correspondence between Artin presentations and certain 4-manifolds.

Abstract

Gonz{\'a}lez-Acu{\~n}a showed that Artin presentations characterize closed, orientable $3$-manifold groups. Winkelnkemper later discovered that each Artin presentation determines a smooth, compact, simply-connected $4$-manifold. We utilize triangle groups to find all Artin presentations on two generators that present the trivial group. We then determine all smooth, closed, simply-connected $4$-manifolds with second betti number at most two that appear in Artin presentation theory.