Show simple item record

dc.contributor.advisorChairperson, Graduate Committee: David Ayalaen
dc.contributor.authorPerry, Daniel Georgeen
dc.date.accessioned2020-02-06T16:46:40Z
dc.date.available2020-02-06T16:46:40Z
dc.date.issued2019en
dc.identifier.urihttps://scholarworks.montana.edu/xmlui/handle/1/15621
dc.description.abstractA 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.language.isoenen
dc.publisherMontana State University - Bozeman, College of Letters & Scienceen
dc.subject.lcshTopology.en
dc.subject.lcshGeometry, Differential.en
dc.subject.lcshGroup theory.en
dc.titleHomotopy groups of contact 3-manifoldsen
dc.typeDissertationen
dc.rights.holderCopyright 2019 by Daniel George Perryen
thesis.degree.committeemembersMembers, Graduate Committee: David Ayala (chairperson); Ryan Grady (co-chair); Lukas Geyer; Tomas Gedeon.en
thesis.degree.departmentMathematical Sciences.en
thesis.degree.genreDissertationen
thesis.degree.namePhDen
thesis.format.extentfirstpage1en
thesis.format.extentlastpage325en
mus.data.thumbpage207en


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record