Five lectures on 3-manifold topology

Abstract

The article consists of five parts intended for graduate students of mathematics or researchers seeking to extend their mathematical knowledge in mathematics, in particular, in 3-manifold topology. The first section is devoted to describing the most popular classical ways for presenting 3-manifolds: triangulations, Heegaard splittings, surgery along framed links, and special spines. The latter is used in the next section for an elementary description of “quantum” or “state sum” invariants of 3-manifolds discovered by V. Turaev and O. Viro. This description includes an explicit geometric construction of the first nontrivial invariant of TV-type, which can be calculated by looking at a special spine of a given 3-manifold. The third section is short but informative. It contains a very clear explanation of the famous JSJ-decomposition theorem, including its relation to the Thurston geometrization conjecture solved by G. Perelman. The next section is an exposition of the Haken theory of normal surfaces, which is one of the main tools for investigating 3-manifolds. It is very important since it explains how to construct the JSJ-decompositions algorithmically. The last section is devoted to description of the main steps in the proof of the algorithmic classification theorem for sufficiently large 3-manifolds, which includes algorithmic classification of knots.