Smallest weakly contractible non-contractible topological spaces

Abstract

We characterize the topological spaces of minimum cardinality which are weakly contractible but not contractible. This is equivalent to finding the non-dismantlable posets of minimum cardinality such that the geometric realization of their order complexes are contractible. Specifically, we prove that all weakly contractible topological spaces with fewer than nine points are contractible. We also prove that there exist (up to homeomorphism) exactly two topological spaces of nine points which are weakly contractible but not contractible.

Realization of Permutation Modules via Alexandroff Spaces

“…1. The space L 1 is weakly contractible but not contractible, [4], with aut(L 1 ) ∼ = Z/2 and ht(L 1 ) = 2. This space was key in [6] to realize groups as the group of homeomorphisms as well as group of self-homotopy equivalences of Alexandroff spaces.…”