Whitehead's theorem for minimal finite models

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 $n\geq 2$, we construct a weak homotopy equivalence between two $(2n+4)$-point minimal finite models of $S^n\vee S^{n-1}\vee S^{n-1}$ 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 $n$th homology over a field, then its order complex is homotopy equivalent either to $S^n$ or to $S^n\vee S^k$ for some $1\leq k\leq n$. 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.