Persistent homology with non-contractible preimages

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dotsc,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent to $S^1$. We also consider the space of functions on $Y$-shaped (resp., starshaped) trees with a $2$-point persistence diagram, and show that this space is homotopy equivalent to $S^1$ (resp., to a bouquet of circles).

Keywords

persistent homology, persistence diagram, homotopy, poset

2010 Mathematics Subject Classification

55P15

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K. Mischaikow was partially supported by NSF grants 1521771, 1622401, 1839294, 1841324, 1934924, by NIH-1R01GM126555-01 as part of the Joint DMS/NIGMS Initiative to Support Research at the Interface of the Biological and Mathematical Science, and a grant from the Simons Foundation.

C. Weibel was supported by NSF grants, and the Simonyi Endowment at IAS.

Received 17 May 2021

Received revised 28 November 2021

Accepted 29 November 2021

Published 14 September 2022