Geodesic
Parametrized homology via zigzag persistence
Carlsson, Gunnar1 ; de Silva, Vin2 ; Kališnik, Sara3 ; Morozov, Dmitriy4
1 Department of Mathematics, Stanford University, Stanford, CA, United States
2 Department of Mathematics, Pomona College, Claremont, CA, United States
3 Department of Mathematics Wesleyan University, Middletown, CT, United States
4 Lawrence Berkeley National Laboratory, Berkeley Institute for Data Science, Berkeley, CA, United States
Algebraic and Geometric Topology, Tome 19 (2019) no. 2, pp. 657-700

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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.

DOI : 10.2140/agt.2019.19.657
Classification : 55N35, 55N99
Keywords: persistent homology, zigzag persistence, levelset zigzag persistence, extended persistence

Carlsson, Gunnar 1 ; de Silva, Vin 2 ; Kališnik, Sara 3 ; Morozov, Dmitriy 4

1 Department of Mathematics, Stanford University, Stanford, CA, United States
2 Department of Mathematics, Pomona College, Claremont, CA, United States
3 Department of Mathematics Wesleyan University, Middletown, CT, United States
4 Lawrence Berkeley National Laboratory, Berkeley Institute for Data Science, Berkeley, CA, United States 
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     title = {Parametrized homology via zigzag persistence},
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Carlsson, Gunnar; de Silva, Vin; Kališnik, Sara; Morozov, Dmitriy. Parametrized homology via zigzag persistence. Algebraic and Geometric Topology, Tome 19 (2019) no. 2, pp. 657-700. doi: 10.2140/agt.2019.19.657

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