WEIGHTED CUBICAL HOMOLOGY

Abstract

In this thesis, we first introduce the definition of weighted cubical complex by adding a weight function to cubical complex. We also check the structure of weighted cubical complex such as weighted boundary map and weighted cubical complex. After this concept, the weighted cubical homology is generated by weighted cubical complex. Some examples are also given to show the difference between general cubical homology and weighted cubical homology. Then we prove an important theorem named Mayer-Vietories sequence for weighted cubical homology by zig-zag lemma. Near the end of this thesis, we are going to talk about weighted cubical persistent homology. Besides the definition, we also check some properties. In the end, an idea of algorithm by AT model for weighted cubical complex will be given as future work.