# Homotopy groups of contact 3-manifolds

 dc.contributor.advisor Chairperson, Graduate Committee: David Ayala; Ryan Grady (co-chair) en dc.contributor.author Perry, Daniel George en dc.date.accessioned 2020-02-06T16:46:40Z dc.date.available 2020-02-06T16:46:40Z dc.date.issued 2019 en dc.description.abstract A contact 3-manifold (M, xi) is an three-dimensional manifold endowed with a completely nonintegrable distribution. In studying such a space, standard homotopy groups, which are defined using continuous/smooth maps, are not useful as they are not sensitive to the distribution. To remedy this, we consider horizontal homotopy groups which are defined using horizontal maps, i.e., smooth maps that lie tangent to the distribution at every point. Due to the distribution being completely nonintegrable, horizontal maps into (M, xi) have rank at most 1. This is used to show that the first horizontal homotopy group is uncountably generated and indicates that the higher horizontal homotopy groups are trivial. We also consider Lipschitz homotopy groups which are defined using Lipschitz maps. We first endow (M, xi) with a metric that is sensitive to the distribution, the Carnot-Caratheodory metric. With respect to this metric structure, the contact 3-manifold is purely 2-unrectifiable. This is used to show that the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Furthermore, over the contact 3-manifold is a metric space, called the universal path space, that acts as a universal cover of the contact 3-manifold in that the universal path space is Lipschitz simply-connected and has a unique lifting property. Homotopy groups, horizontal homotopy groups, and Lipschitz homotopy groups are all instances of homotopy groups of sheaves, which are defined. en dc.identifier.uri https://scholarworks.montana.edu/handle/1/15621 en dc.language.iso en en dc.publisher Montana State University - Bozeman, College of Letters & Science en dc.rights.holder Copyright 2019 by Daniel George Perry en dc.subject.lcsh Topology en dc.subject.lcsh Geometry, Differential en dc.subject.lcsh Group theory en dc.title Homotopy groups of contact 3-manifolds en dc.type Dissertation en mus.data.thumbpage 207 en thesis.degree.committeemembers Members, Graduate Committee: Lukas Geyer; Tomas Gedeon. en thesis.degree.department Mathematical Sciences. en thesis.degree.genre Dissertation en thesis.degree.name PhD en thesis.format.extentfirstpage 1 en thesis.format.extentlastpage 325 en

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