Finite-dimensional spaces in resolving classes

TL;DR

This paper uses resolving classes to connect properties of CW complexes with their mapping spaces, providing a new proof of the Sullivan conjecture by linking conditions on $X$ to the triviality of maps into finite-dimensional complexes.

Contribution

It establishes a novel connection between resolving classes and the triviality of mapping spaces, offering an elementary proof of the Sullivan conjecture.

Findings

If $X$ is a CW complex of finite type with trivial maps into spheres, then maps into all finite-dimensional complexes are trivial.

Under mild conditions on $ ext{pi}_1(X)$, the triviality extends to all finite-dimensional complexes.

Provides a new, elementary proof of the Sullivan conjecture using resolving classes.

Abstract

Using the theory of resolving classes, we show that if $X$ is a CW complex of finite type such that $\map_*(X, S^{2n+1})\sim *$ for all sufficiently large $n$, then $\map_*(X, K) \sim *$ for every simply-connected finite-dimensional CW complex $K$; and under mild hypotheses on $\pi_1(X)$, the same conclusion holds for \textit{all} finite-dimensional complexes $K$. Since it is comparatively easy to prove the former condition for $X = B\ZZ/p$ (we give a proof in an appendix), this result can be applied to give a new, more elementary proof of the Sullivan conjecture.