Topological Surgery and its Dynamics

Topological surgery occurs in natural phenomena where two points are selected and attracting or repelling forces are applied. The two points are connected via an invisible `thread'. In order to model topologically such phenomena we introduce dynamics in 1-, 2- and 3-dimensional topological surgery, by means of attracting or repelling forces between two selected points in the manifold, and we address examples. We also introduce the notions of solid 1- and 2-dimensional topological surgery, and of truncated 1-, 2- and 3-dimensional topological surgery, which are more appropriate for modelling natural processes. On the theoretical level, these new notions allow to visualize 3-dimensional surgery and to connect surgeries in different dimensions. We hope that through this study, topology and dynamics of many natural phenomena as well as topological surgery may now be better understood.