New Approach to the Generalized Poincare Conjecture

Alexander A.Ki Ermolits

doi:10.4236/am.2013.49183

Abstract

Using our proof of the Poincare conjecture in dimension three and the method of mathematical induction a short and transparent proof of the generalized Poincare conjecture (the main theorem below) has been obtained. Main Theorem. Let Mn be a n-dimensional, connected, simply connected, compact, closed, smooth manifold and there exists a smooth finite triangulation on Mn which is coordinated with the smoothness structure of Mn. If Sn is the n-dimensional sphere then the manifolds Mn and Sn are homemorphic.

Highlights

We can fix some Riemannian metric g on a manifold Mn of dimension n which defines the length of arc of a piecewise smooth curve and the continuous function x; y of the distance between two points x, y M n
The topology defined by the function of distance is the same as the topology of the manifold Mn [1]
With the help of this algorithm we get that every compact, connected, closed manifold Mn of dimension n having the triangulation above can be represented as a union of a n-dimensional cell Cn and a connected union

Summary

Introduction

We can fix some Riemannian metric g on a manifold Mn of dimension n which defines the length of arc of a piecewise smooth curve and the continuous function x; y of the distance between two points x, y M n. With the help of this algorithm we get that every compact, connected, closed manifold Mn of dimension n having the triangulation above can be represented as a union of a n-dimensional cell Cn and a connected union. Using the method of mathematical induction and the algorithms we retract all the simplexes from K n 1 to a point x0, obtained and Mn is a decomposition homeomorphic to

On Algorithm of Extension of Coordinate Neighborhood

Proof of the Main Theorem