Artin presentations, triangle groups, and 4-manifolds
Jack S. Calcut, Jun Li

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 -manifold groups. Winkelnkemper later discovered that each Artin presentation determines a smooth, compact, simply-connected -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 -manifolds with second betti number at most two that appear in Artin presentation theory.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology
