Artin presentations, triangle groups, and 4-manifolds

Abstract

Gonz{á}lez-Acu{ñ}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.

13 pages, 10 figures. Included a proof of the folklore Theorem 2.11 (at the request of a referee), included more examples in Section 2, and made a few minor changes