Minimal finite models of wedges of spheres with at most eight points

Abstract

A minimal finite model of a topological space is a finite $T_0$-space of least cardinality that is weak homotopy equivalent to it. We determine, up to homeomorphism, the minimal finite models of eight wedges of one- and two-dimensional spheres. The spaces $S^2\vee S^1$ and $S^2\vee S^2$ have exactly two and three minimal finite models, respectively, all with seven points, whereas $S^1\vee S^1\vee S^2$ has exactly seven minimal finite models, $S^1\vee S^1\vee S^1\vee S^2$ has one, $S^1\vee S^2\vee S^2$ has six, $S^2\vee S^2\vee S^2$ has five, $S^2\vee S^2\vee S^2\vee S^2$ has three, and $S^1\vee S^1\vee S^2\vee S^2$ has two, all with eight points. To prove this we classify the connected finite $T_0$-spaces of height $2$ without beat points that have seven or eight points, and we identify their weak homotopy types by means of McCord's theorem and explicit collapse and suspension arguments for their two-dimensional order complexes. Together with the known results for spheres and wedges of circles, this determines, up to weak homotopy equivalence, all path-connected spaces whose minimal finite models have at most eight points.