Abstract:

Kneser's conjecture states that if the fundamental group of a closed 3-manifold is a free product of two non-trivial groups then the closed 3-manifold is a non-trivial connected sum of two closed 3-manifolds. In this talk, we will discuss a generalized Kneser conjecture for 3-manifolds with boundary. I will start with some basic notions in 3-manifold topology which motivates Kneser's conjecture and its generalizations, and then outline our proof for a generalized Kneser's conjecture. If time permits, a classification problem for knotted surfaces in the 3-sphere, the main application of the conjecture we have in mind, will be presented in the end. It is a joint project with Bellettini and Paolini.