New Approach to the Generalized  Poincare Conjecture

Alexander A.ki Ermolits

doi:10.4236/am.2013.49183

Home > Journals > Physics & Mathematics > AM

Applied Mathematics > Vol.4 No.9, September 2013

New Approach to the Generalized Poincare Conjecture ()

Alexander A.ki Ermolits

IIT-BSUIR, Minsk, Belarus.

DOI: 10.4236/am.2013.49183 PDF HTML 2,956 Downloads 4,308 Views Citations

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.

Keywords

Compact Smooth Manifolds; Riemannian Metric; Smooth Triangulation; Homotopy-Equivalence; Algorithms

Share and Cite:

A. Ermolits, "New Approach to the Generalized Poincare Conjecture," Applied Mathematics, Vol. 4 No. 9, 2013, pp. 1361-1365. doi: 10.4236/am.2013.49183.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. Gromoll, W. Klingenberg and W. Meyer, “Riemannsche Geometrie im Grossen,” Springer, Berlin, 1968.
[2]	A. A. Ermolitski, “On a Geometric Black Hole of a Compact Manifold,” Intellectual Archive Journal, Vol. 1, No. 1, 2012, pp. 101-108.
[3]	J. R. Munkres, “Elementary Differential Topology,” Princeton University Press, Princeton, 1966.
[4]	A. A. Ermolitski, “Three-Dimensional Compact Manifold and the Poincare Conjecture,” Intellectual Archive Journal, Vol. 1, No. 4, 2012, pp. 51-62.
[5]	D. B. Fuks and V. A. Rohlin, “Beginner’s Course in Topology/Geometric Chapters,” Nauka, Moscow, 1977.
[6]	M. W. Hirsch, “Differential Topology,” Springer, New York, Heigelberg, Berlin, 1976.