Combinatorial Structure of Manifolds with Poincar\'e Conjecture

TL;DR

This paper introduces a combinatorial graph-based framework to analyze manifold structures, leading to a new characterization of fundamental groups and a proof that all homotopy spheres are homeomorphic to standard spheres, including Perelman's 3-sphere result.

Contribution

It develops a novel combinatorial approach linking labeled graphs to manifold topology, providing new insights into fundamental groups and the Poincaré conjecture.

Findings

Characterization of fundamental groups via labeled graphs

Classification of locally compact manifolds with finite loops

Proof that all homotopy spheres are homeomorphic to spheres

Abstract

A manifold $M^n$ inherits a labeled $n$-dimensional graph $\widetilde{M}[G^L]$ structure consisting of its charts. This structure enables one to characterize fundamental groups of manifolds, classify those of locally compact manifolds with finite non-homotopic loops by that of labeled graphs $G^L$. As a by-product, this approach also concludes that {\it every homotopy $n$-sphere is homeomorphic to the sphere $S^n$ for an integer $n\geq 1$}, particularly, the Perelman's result for $n=3$.