Coarse Homotopy Theory and Shape Theory
This thesis establishes a connection between the coarse homotopy theory of Euclidean cones and the shape theory of compact metric spaces. For a compact metric space $X$, the coarse homotopy groups of the Euclidean cone $cX$ are determined by pointed shape invariants of $X$, fitting into a $\varprojlim^1$ sequence involving the \v{C}ech homotopy groups. Using inverse mapping telescopes and strong shape theory, we prove that two compact metric spaces are strong shape equivalent if and only if their Euclidean cones are coarsely homotopy equivalent. We also prove a version of the coarse Whitehead theorem for Euclidean cones. In contrast, we derive a similar $\varprojlim^1$ sequence for arbitrary proper metric spaces, and construct a counterexample showing that the coarse Whitehead theorem fails in this general setting.