Degree Year

2017

Document Type

Thesis

Degree Name

Bachelor of Arts

Department

Mathematics

Advisor(s)

Jack Calcut

Keywords

Open book decomposition, Artin presentation, Triangle group, 4-manifold

Abstract

In this paper we focus on the study of smooth closed simply-connected 4-manifolds. In particular, we study such manifolds arising from Artin presentations. Artin pre- sentations are intimately related to the pure braid group, and they characterize the fundamental groups of closed orientable 3-manifolds. Each Artin presentation r gives rise to a 4-manifold W4(r), whose boundary is a closed orientable 3-manifold M3(r) with fundamental group presented by r. In case the 4-manifold has bound- ary S3, we may close up W 4(r) by attaching a 4-handle to obtain W 4(r) ¿S3 D4. This closed 4-manifold has quadratic form represented by the exponent sum matrix of r.

Artin presentations arising from 2-strand pure braids are particularly tractable. Further, we show that they are naturally related to von Dyck (triangle) groups. These two facts play crucial roles in our determination of all closed 4-manifolds arising from Artin presentations on two generators.

Section 2 introduces the open book decomposition, which is an important con- struction of closed orientable 3-manifolds that leads to their correspondence with Artin presentations. Section 3 explains how Artin presentations give rise to 4- manifolds, specifically how the 3-manifold boundary of a 4-manifold can be studied with an Artin presentation group as their fundamental group. Then in section 4 we find all possible closed, smooth, simply-connected 4-manifolds from Artin pre- sentations on two generators.

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Mathematics Commons

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