Date of Award

May 2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Boris Okun

Committee Members

Craig Guilbault, Chris Hruska, Jonah Gaster, Jeb Willenbring

Keywords

Coarse Adjunction, Coarse Homotopy, Coarse Quotient, Coarse Whitehead

Abstract

A pair (X, A) has the homotopy extension property if any homotopy of A the extends overX × {0} can be extended to a homotopy of X. The main goal of this dissertation is to define a coarse analog of the homotopy extension property for coarse homotopies and prove coarse versions of results from algebraic topology involving this property. First, we define a notion of a coarse adjunction metric for constructing coarse adjunction spaces. We use this to redefine coarse CW complexes and to construct a coarse version of the mapping cylinder. We then prove various pairs of spaces have the coarse homotopy extension property. In particular, pairs of coarse CW complexes. We then prove results involving the coarse homotopy extension property, leading to the result that a coarse map f : X → Y is a coarse homotopy equivalence if and only if the coarse mapping cylinder coarse deformation retracts onto its copy of X. We use this to prove our main result, a coarse version of Whitehead’s Theorem: If a cellular coarse map f between coarse CW complexes induces isomorphisms between coarse homotopy groups, then f is a coarse homotopy equivalence.

Download

Included in

Mathematics Commons

Share

COinS