Whitehead's theorem for minimal finite models

2026/08/06 by David Mosquera-Lois
Mathematics · #math.AT #msc:55P10 #msc:55P15 #msc:06A07 #msc:57Q10

paper · pdf

14 pages, 1 figure. Comments welcome

arxiv created 2026/08/06 · arxiv updated 2026/08/07

Abstract

We investigate the extent to which Whitehead's theorem remains valid for minimal finite models. We show that it fails in this setting, answering negatively a question posed by Barmak. More precisely, for every n≥ 2, we construct a weak homotopy equivalence between two (2n+4)-point minimal finite models of Sⁿ\vee S^n-1\vee S^n-1 which are not homotopy equivalent. The minimality of these examples follows from a near-extremal classification theorem: if a connected finite space has at most 2n+3 points and nonzero nth homology over a field, then its order complex is homotopy equivalent either to Sⁿ or to Sⁿ\vee S^k for some 1≤ k≤ n. Finally, we prove a positive Whitehead-type result: under a natural cohomological rigidity hypothesis, every weak homotopy equivalence between minimal finite models is a homotopy equivalence.

Citations

• Minimal Finite Model of Wedge Sum of Spheres
• Simple Homotopy Types and Finite Spaces
• Singular homology groups and homotopy groups of finite topological spaces