Algebraic topological invariants for Morse-Smale vector fields

The goal of this thesis is to introduce new algebraic methods for the topological analysis of fields. We consider different special cases like gradient fields, gradient-like Morse-Smale vector fields, or Morse-Smale vector fields with closed orbits. We describe different ways for assigning a chain complex to a vector field and discuss connections to known invariants from topological data analysis. We start by studying the category of tame epimorphic parametrized chain complexes, factored chain complexes for short, generalizing results from persistent homology to this setting, namely the structure theorem and the isometry theorem. We present a pipeline that produces a factored chain complex from a weighted based chain complex and apply this to the Morse complex in both the smooth and discrete settings. In combination with the structure theorem, this yields a tagged barcode for gradient-like Morse-Smale vector f ields on compact Riemannian manifolds. We prove local stability and combinatorial approximation results, and describe connections to the classical persistence barcode in the special case of the gradient of a scalar field. Going beyond the gradient-like case, we recall a method by Franks [25] to replace a closed orbit by a pair of fixed points via a local perturbation. We show that there are mul tiple non-equivalent ways of following this procedure and and describe the consequences of this non-uniqueness to the endeavour of assigning CW complexes or chain complexes to Morse-Smale vector fields in a canonical way. In order to assign a chain complex to any Morse-Smale vector field, we use the filtration of the underlying manifold by unstable manifolds used by Smale in [46] and consider the spectral sequence in Čech homology associated with this filtration. We show that the terms on the first page of this spectral sequence admit canonical bases corresponding to the fixed points and closed orbits of the vector field. In the 2D case, we present a method to rearrange the algebraic information of the spectral sequence so as to obtain a canonical chain complex, whose homology agrees with the singular homology of the manifold. We derive the Morse inequalities from [46] from this chain complex and present a method for endowing it with bases in each degree.