Coarse Homotopy Extension Property and its Applications

Mr. William Braubach
University of Wisconsin-Milwaukee

A pair (X, A) has the homotopy extension property if any homotopy of A 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 from X into 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.

Advisor: Prof. Boris Okun

Committee Members:
Profs. Boris Okun, Craig Guilbault, Jeb Willenbring, Jonah Gaster, and Chris Hruska