New Topological Invariants for Real- and Angle-Valued Maps: An Alternative to Morse-Novikov Theory

This book is about new topological invariants of real- and angle-valued maps inspired by Morse Novikov theory, a chapter of topology, which has recently raised interest outside of mathematics; for example, in data analysis, shape recognition, computer science and physics. They are the backbone of what the author proposes as a computational alternative to Morse Novikov theory, referred to in this book as AMN-theory.

These invariants are on one side analogues of rest points, instantons and closed trajectories of vector fields and on the other side, refine basic topological invariants like homology and monodromy. They are associated to tame maps, considerably more general than Morse maps, that are defined on spaces which are considerably more general than manifolds. They are computable by computer implementable algorithms and have strong robustness properties. They relate the dynamics of flows that admit the map as "Lyapunov map" to the topology of the underlying space, in a similar manner as Morse Novikov theory does.

Readership: Graduate students and researchers in geometry and topology, topologists, geometers, experts in dynamics, computer scientists and data analysts.

This book presents an alternative to what the topologists refer to as "Morse Novikov theory," a mathematical theory which belongs to the fields of geometry and topology. The theory presented has interest in topology and dynamics, has provided inspiration and has applications outside of mathematics, especially in data analysis and shape recognitions in physics and computer science.

It describes a new class of invariants associated with a generic continuous real or angle valued map defined on a compact metrizable space based on homology with coefficients in a given field. The invariants are finite, computable by implementable algorithms in case the underlying space of the map has a triangulation and the map is simplicial, and are, in some sense, the analogues of the set of trajectories between rest points and of closed trajectories of a generic vector field (which admits a Lyapunov closed one form) on a smooth manifold. These can be used to conclude existence of such trajectories even for flows on compact metric spaces which are not smooth and cannot be described via differential calculus. Two alternative definitions of these invariants based on different mathematics and algorithms for their calculation will be described. The book presents remarkable properties (stability, Poincar -duality) of these invariants and will relate them to the global algebraic topology of the space the map is defined on, in the spirit of Morse Novikov theory.