Minimal Finite Model of Wedge Sum of Spheres

TL;DR

This paper classifies minimal finite models of certain topological spaces, showing specific models for the Möbius band and various wedge sums of spheres with precise point counts.

Contribution

It provides a classification of minimal finite models for wedge sums of spheres and the Möbius band, identifying exact point counts for these models.

Findings

Möbius band has the same minimal finite model as the circle.

Certain wedge sums of spheres have minimal models on exactly seven or eight points.

Explicit minimal models are identified for multiple wedge sums of spheres.

Abstract

We classify minimal finite models of the M\"{o}bius band and several wedge sums of spheres. In particular, we show that the minimal finite model of the M\"{o}bius band coincides with that of the circle $S^{1}$. Furthermore, we prove that both $S^{2}\vee S^{1}$ and $S^{2}\vee S^{2}$ admit minimal finite models on exactly seven points, and that each of $S^{1}\vee S^{1}\vee S^{2}$, $S^{1}\vee S^{1}\vee S^{1}\vee S^{2}$, $S^{1}\vee S^{2}\vee S^{2}$, $S^{2}\vee S^{2}\vee S^{2}$, and $S^{2}\vee S^{2}\vee S^{2}\vee S^{2}$ admits a minimal finite model on exactly eight points.