Structures of Three-Manifilds

The aim of the work is to prove the following main theorem. Theorem. Let M3 be a three-dimensional, connected, simple-connected, closed, compact, smooth manifold. Tnen the manifold M3 is diffeomorphic to the three-dimensional sphere.

The paper gives a review of progress towards extending the Thurston programme to the Poincare duality case. For a full abstract, see the published version at the above link.

We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.

This paper proves that any compact, closed, simply connected and connected three dimensional stellar manifold is stellar equivalent to the three dimensional sphere.

In this paper, by use of techniques associated to cobordism theory and Morse theory,we give a simple proof of Poincare conjecture, i.e. Every compact smooth simply connected 3-manifold is homeomorphic to 3-sphere.

This is the announcement of an alternative approach to the 3-dimensional Poincar\'e Conjecture, different from Perelman's big and spectacular breakthrough. No claim concerning the other parts of the Thurston Geometrization Conjecture, come with our purely 4-dimensional line of argument.

This paper poses some basic questions about instances (hard to find) of a special problem in 3-manifold topology. "Important though the general concepts and propositions may be with the modern industrious passion for axiomatizing and generalizing has presented us...nevertheless I am convinced that the special problems in all their complexity constitute the stock and the core of mathematics; and to master their difficulty requires on the whole the harder labor." Hermann Weyl 1...

In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincar\'e conjecture due to Hamilton and Perelman.

In May 2015, a conference entitled "Groups, Geometry, and 3-manifolds" was held at the University of California, Berkeley. The organizers asked participants to suggest problems and open questions, related in some way to the subject of the conference. These have been collected here, roughly divided by topic. The name (or names) attached to each question is that of the proposer, though many of the questions have been asked before.

This is a survey article on finite type invariants of 3-manifolds written for the Encyclopedia of Mathematical Physics to be published by Elsevier.