The essence of the complexity theory of 3-manifolds is the following. To each compact 3-manifold M we associate a non-negative integer с(M), called the complexity of M such that it adequately reflects complexity of M in the informal sense of this expression. It turns out that this conception is very useful for tabulation of 3-manifolds and rigorous formulations of different conjectures. In the talk we describe basic notions and results of this theory as well as new methods for exact calculation of complexity. The main problem in composing tables of 3-manifolds consists in casting out duplicates. We shall describe several quantum invariants of 3-manifolds capable to solve this problem and show how they work.
The author is partially supported by Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020).