This is an introductory talk on 3-manifold topology, with a focus on constructive methods. It is part of the workshop of Low-dimensional topology and quantum algebra, which continues for the rest of the week.
Why be happy when you could be normal?
Computation, experimentation and conjectures have played a driving role in low-dimensional topology right from the cradle of modern topology in the work of Poincare. There is a successful history of replacing non-constructive existence proofs by practical solutions, and heuristic methods by rigorous algorithms.
In this introductory talk, I will describe some of the key techniques used to study a 3-dimensional space, including finding essential surfaces in the space, constructing a geometric structure on it, and computing key invariants. If time permits, I will also outline some open problems on 3-manifolds and related fields.