Parametrized homology via zigzag persistence

Gunnar Carlsson, Vin de Silva, Sara Kališnik, Dmitriy Morozov
2019 Algebraic and Geometric Topology  
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or, equivalently, as a persistence diagram. By defining persistence in terms of finite rectangle measures, we classify barcode intervals into four classes. Each of these conveys how the homological features perish at both ends of the interval over which they are defined.
doi:10.2140/agt.2019.19.657 fatcat:7msdmumyprh25h5ty47r5thlay