Spaces which invert weak homotopy equivalences

TL;DR

This paper characterizes spaces that invert weak homotopy equivalences, showing that only contractible spaces have this property, thus clarifying a fundamental aspect of homotopy theory.

Contribution

It proves that the only spaces which invert weak equivalences are the contractible ones, resolving a question posed by Strom and Goodwillie.

Findings

Only contractible spaces invert weak equivalences.

Non-empty spaces do not invert weak equivalences unless contractible.

Provides a complete characterization of such spaces.

Abstract

It is well known that if $X$ is a CW-complex, then for every weak homotopy equivalence $f:A\to B$, the map $f_*:[X,A]\to [X,B]$ induced in homotopy classes is a bijection. For which spaces $X$ is $f^*:[B,X]\to [A,X]$ a bijection for every weak equivalence $f$? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.