The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various directions. However, no attempt has been made so far to extend the concept of topological persistence to a generalization of 'maps' such as cocycles which are discrete counterparts of closed differential degree one-forms, a well known concept in differential geometry. We define a notion of topological persistence for 1-cocycles in this paper and show how to compute its relevant numbers. It turns out that, instead of the standard persistence, one of its variants which we call level persistence can be leveraged for this purpose. It is worth mentioning that 1-cocycles appear in practice such as in data ranking or in discrete vector fields.
Recent work in REF defines a modified version of topological persistence (level persistence) for 1-cocycles, and shows that such 1-cocycles can be interpreted as a circle valued map.
Defining and Computing Topological Persistence for 1-cocycles
ArXiv
ArXiv
Mathematics Computer Science