Minimal Finite Model of Wedge Sum of Spheres
Ponaki Das, Sainkupar Marwein Mawiong
TL;DR
This work classifies minimal finite $T_0$-spaces that are weakly homotopy equivalent to wedge sums of spheres, providing a complete catalog up to eight points. The Möbius band is shown to share its four-point minimal model with $S^{1}$, while $S^{2} obreakigvee obreak S^{1}$ and $S^{2} obreakigvee obreak S^{2}$ require seven-point cores (two and three non-homeomorphic models, respectively). For eight-point height-two spaces, the authors enumerate all minimal cores for $S^{1} obreakigvee obreak S^{1} obreakigvee obreak S^{2}$, $S^{1} obreakigvee obreak S^{1} obreakigvee obreak S^{1} obreakigvee obreak S^{2}$, $S^{1} obreakigvee obreak S^{2} obreakigvee obreak S^{2}$, $S^{2} obreakigvee obreak S^{2} obreakigvee obreak S^{2}$, and $S^{2} obreakigvee obreak S^{2} obreakigvee obreak S^{2} obreakigvee obreak S^{2}$, including the role of beat points and duality in consolidating distinct models. The results illuminate how discrete poset structures encode classical homotopy types and establish a framework for extending minimal finite models to larger wedge sums and higher dimensions.
Abstract
We classify minimal finite models of the Möbius band and several wedge sums of spheres. In particular, we show that the minimal finite model of the Möbius 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.