Homology Sequences and Theorems in Persistence Setup
Project no. 21882 · Status: Kapanmış · 2020
Summary
Topology together with algebra provides a large field, called Algebraic Topology, where we have numerous tools to examine spaces and classify them. Such tools we use are homotopy, homology, cohomology and so on. By applying these methods we derive information about spaces and do our classification with respect to them. In some cases we may only have limited information about a space, for example a set of points sampled from it, but we may still want to know how the space actually looks like or what kind of properties it has such as its connectedness or number of holes it has etc. Thanks to the persistence approach, we use the tools we mentioned above to not only compare spaces but also have significant information about a space. We first build a structure on the limited data we have and apply those usual methods of algebraic topology. Throughout this thesis, the tool we shall examine is homology. Homology has some structures such as the exact sequences and theorems to simplify relatively complicated calculations. We see how not only the method but also those exact sequences and theorems based on usual homology are applied to the persistence approach.