From Morse Theory to Homological Discrete Vector Fields

Yann-Situ Gazull (LIS)

This talk follows Alexandra Bac's introduction to computational homology and cohomology, and will focus on Morse Theory and Homological Discrete Vector Fields (HDVF). There exists many classical ways of actually computing homology and cohomology groups, from Smith normal forms to the categorical approach of effective homology. One well known combinatorial approach is dicrete Morse Theory. We present here a more general, and Made in Marseile, approach : Homological Discrete Vector Fields or HDVF. Time and audience requests depending, we will also present

• HDVSs (how to move from one HDVF to another through HDVFs operations, connectedness of HDVFs space, equivalence between HDVFs and state-of-art methods (SNF, persistent homology, tri-partitions).
• Homology bases "that can be computed". In a recent result, we have proved that only specific bases can be computed by state-of-art methods and we characterized them (explicit bases).
• Constructive Alexander duality. Another result, based on HDVFs, give a new (constructive) proof of Alexander isomorphism between the homology of an object and that of its complementary in S^n. HDVFs provide the isomorphism.
• Homological configurations (works in progress...). Starting from previous works, Yann-Situ Gazull introduced a new notion, that of homological configurations, classifying HDVFs (ie. homology computations) over a given object. These configurations are not homotopy invariants but they provide an interesting tool to analyse the topology/geometry link.