Combinatorial Structure of Manifolds with Poincaré Conjecture

Abstract

A manifold $M^n$ inherits a labeled $n$-dimensional graph $\tilde{M}[G^L]$ 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 $n\ge 1$}, particularly, the Perelman's result for $n=3$.

15 pages, 5 figures

References in corpus (1)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds