Structures of Three-Manifilds

We give a simple proof on the Poincar\'e's conjecture which states that every compact smooth $3-$manifold which is homotopically equivalent to $S^3$ is diffeomorphic to $S^3$.

We consider the operation to crush a subset of a manifold to one-point when the result of the crushing also be a manifold. Then the Poincare conjecture is split to two problems; for any closed orientable 3-manifold which is not homeomorphic to the sphere, one is that this operation preserve simply-connectedness, and another one is that we can get a non-simply connected space by applying the operation. We show these propositions in this paper.

v1: In this paper, we will give an elementary proof by the Heegaard splittings of the 3-dimentional Poincare conjecture in point of view of PL topology. This paper is of the same theory in [4](1983) excluding the last three lines of the proof of the main theorem. v2: This paper gives the basic result of [1](1997), i.e., a handle sliding and a band move of Heegaard diagrams correspond to a replacement and a substitution in relations of the fundamental groups derived from Heega...

In this paper give a survey about L^2-invariants focusing on 3-manifolds.

In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the classification of surfaces as it relates to the Poincare and Thurston Geometrization conjectures. Finally, we survey Perelman's results on Ricci flows with surgery...

We describe theoretical backgrounds for a computer program that recognizes all closed orientable 3-manifolds up to complexity 8. The program can treat also not necessarily closed 3-manifolds of bigger complexities, but here some unrecognizable (by the program) 3-manifolds may occur.

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...

In this paper, we prove a geometrization conjecture, every orientable smooth closed 3-manifold with finite fundamental group is homeomorphic to $S^3/G$ for some finite cyclic subgroup $G\subset {Isom}^+(S^3)$.

The title says it all.

We prove Poincare's Conjecture that every simply connected, closed three-manifold is topologically equivalent to the three-sphere. The proof is founded on the algebraic formulation discovered by J. Stallings.