Connection Matrices in Combinatorial Topological Dynamics [Paperback]

This book provides an introduction to the theory of connection matrices in the context of combinatorial multivector fields. The theory of connection matrices was proposed by Conley and Franzosa for classical continuous-time dynamical systems as a tool for studying connecting orbits between isolated invariant sets. It generalizes the Morse complex in Morse theory, and has found numerous applications in dynamics. Connection matrices have been and still are a challenging topic to study, as there are no complete introductory texts, and both their intricate definition and properties are scattered over numerous research papers.

In recent years, dynamical concepts have found their way into a combinatorial context. Starting with combinatorial vector fields, introduced by Forman to generalize classical Morse theory, it has been realized that this transfer of ideas can lead to important applications. Similarly, Conley's theory of isolated invariant sets has been transferred to the combinatorial setting. This, when combined with the concept of multivector fields, opens the door to a complete combinatorial dynamical theory.