Parametrized Homology via Zigzag Persistence

Algebraic and Geometric Topology, vol. 19, pages 657–700, 2019.
PDF Preprint
DOI: 10.2140/agt.2019.19.657
arXiv: 1604.03596
This paper introduces parametrized homology, a continuous-parameter generalization of levelset zigzag persistent homology that captures the behavior of the homology of the fibers of a real-valued function on a topological space. This information is encoded as a "barcode" of real intervals, each corresponding to a homological feature supported over that interval; or, equivalently, as a persistence diagram. Points in the persistence diagram are classified algebraically into four classes; geometrically, the classes identify the distinct ways in which homological features perish at the boundaries of their interval of persistence. We study the conditions under which spaces fibered over the real line have a well-defined parametrized homology; we establish the stability of these invariants and we show how the four classes of persistence diagram correspond to the four diagrams that appear in the theory of extended persistence.
Gunnar Carlsson, Vin de Silva, and Dmitriy Morozov. Zigzag persistent homology and real-valued functions. Proceedings of the Annual Symposium on Computational Geometry, pages 247–256, 2009.