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On 3-manifolds
TL;DR: In this paper, the boundary of a polyhedron P with boundary faces identified in pairs was studied and it was shown that (P)/~ is a finite number of internally flat complexes attached to each other along the edges of a finite graph that contains at least one closed circuit.
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Abstract: It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P with the polygonal faces identified in pairs leads us to the following conclusion: either a three dimensional manifold is homeomorphic to a sphere or to a polyhedron P with its boundary faces identified in pairs so that (\partial P)/~ is a finite number of internally flat complexes attached to each other along the edges of a finite graph that contains at least one closed circuit. Each of those internally flat complexes is obtained from a polygon where each side may be identified with more than one different sides. Moreover, Euler characteristic of (\partial P)/~ is equal to one and the fundamental group of (\partial P)/~ is not trivial.
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References
•Book
Algebraic Topology: An Introduction
William S. Massey
- 02 Nov 1977
TL;DR: Professor Massey's book developed from lecture notes of courses taught to Yale undergraduate and graduate students over a period of several years, and is the author of numerous research articles on algebraic topology and related topics.