Combinatorial Structure of Manifolds with Poincaré Conjecture

Details Combinatorial Structure of Manifolds with Poincaré Conjecture Combinatorial Structure of Manifolds with Poincaré Conjecture

2010-04-08

arxiv.org/abs/1004.1231v2

Mathematics

General Mathematics

15 pages, 5 figures

Scientific paper

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 $n=3$.

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