By something called surgery on a link in the 3-sphere. Unfortunately, surgery on different links can produce the same 3-manifold, so one needs to know exactly when this happens. It turns out that the answer to that question is Kirby calculus. The talk will not assume serious knowledge of topology and will hopefully motivate the study of topology. We give an introduction to the theory of virtual knots, initiated recently by L. Kauffman.
Virtual knots provide an unexpected generalization of classical knots.
We discuss some surprising new features of virtual knots and the corresponding new theory of finite type invariants. Octav Cornea, Universite de Lille 1, France Complexity of connecting manifolds and smoothing problems. In the general setting the study of this problem has been initiated by John Franks. We will see that to these moduli spaces one can associate naturally certain bordims classes. These classes can be identified with some Hopf invariants which makes them computable.