Topology of 3-dimensional manifolds - Eaman Eftekhary (ریاضی و جامعه)
This is the second paper from a trio which reviews some of the progress in low dimensional topology in the past century. Starting with the work of Poincare in the last years of 19th century and first few years of 20th century, we review the major steps in putting 3-manifolds and their algebraic topology in a solid mathematical framework, and the important theorems which strengthened the understanding of 3-manifolds, including the prime decomposition theorem and JSJ decomposition of 3-manifolds. Highlighting the significance of hyperbolic 3-manifolds, proving the monster theorem and formulating the geometrization conjecture by Thurston has been a turning point in 3-manifold topology. The proof of geometrization conjecture by Perelman, using Ricci flow of Hamilton, affirmed that the fundamental group is an almost perfect invariant of closed 3-manifolds. Yet, it is not clear how geometric properties are reflected in the fundamental group, and its is difficult to verify whether two group presentations give isomorphic fundamental groups or not. Alternative approaches to the study of 3-manifolds and 4-dimensional cobordisms between them using abelian groups include, in particular, the theories which are formulated as topological quantum field theories (TQFTs). These approaches are also reviewed in the paper. In particular, a theorem of the author which addresses the strength of the later invariants in distinguishing 3-manifolds from the standard 3-sphere is discussed.