Tagged barcodes for Morse-Smale vector fields

In this talk I will present some new invariants for Morse-Smale vector fields. In the first part of the talk we consider the gradient-like case, where we construct an invariant called ‘tagged barcode’. We start by considering the Morse complex, which is a chain complex defined in terms of singular points and flow lines of the vector field and whose homology is isomorphic to the homology of the underlying manifold. We then identify a sequence of pairs of singular points along which we can simplify the Morse complex. Recording the distances between the pairs we simplified yields the tagged barcode. 
In the second part of the talk we include closed orbits into our analysis, presenting a different method where we use a filtration of the manifold by unstable manifolds first described by Smale. We consider the spectral sequence associated with this filtration and then rearrange the algebraic information in order to obtain a chain complex. 
This is joint work with Claudia Landi. The content of the first part can be found on arXiv (identifier 2401.08466), the content of the second part is work in progress.