Transfinitely iterated wild sets

Abstract

In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each $n\ge 0$, the "$\ddot{I}{€}_n$-wild set" ${\mathbf{w}}_n(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. Since the operator ${\mathbf{w}}_n$ permits iteration, every given space $X$ yields a descending transfinite sequence of nested subspaces $\{{\mathbf{w}}_n^{\hat{I}}º(X){\}}_{\hat{I}}º$ that stabilizes at some smallest ordinal ${\mathbf{wrk}}_n(X)$ called the "$\ddot{I}{€}_n$-wild rank" of $X$. We show that the entire transfinite sequence $\{ho({\mathbf{w}}_n^{\hat{I}}º(X)){\}}_{\hat{I}}º$ of homotopy types is a homotopy invariant of $X$ and that ${\mathbf{wrk}}_n(X)$ can be an arbitrary countable ordinal when $X$ is an $n$-dimensional Peano continuum. It remains open if there exists a continuum $X$ with uncountable $\ddot{I}{€}_n$-wild rank. This difficulty motivates the parallel study a basepoint-free version ${\mathbf{fwrk}}_n(X)$, called the "free $\ddot{I}{€}_n$-wild rank" of $X$. We show that for every continuum $X$, ${\mathbf{fwrk}}_n(X)$ is always countable and can be any countable ordinal.

16 pages, 1 figure