# Homotopy Groups of Contact 3-Manifolds

## Talk by Dan Perry (Graduate Student in Mathematical Sciences, MSU) - Public PhD Dissertation Defense

### 7/5/2019 Wilson Hall 1-144 10:10-11:00am

#### 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-Carath\'{e}odory 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.