Floer-type bipersistence modules and rectangle barcodes

Kanta Koeda, Ryuma Orita, Kanon Yashiro
arXiv:2312.07847

Overview

#

Floer homology can be studied within an action window, with both a lower and an upper cutoff. Varying these two bounds gives a two-parameter persistence module. For Floer-type complexes over a field with trivial value group, we show that these modules decompose into rectangles, producing a geometric picture of how the homology changes with the window.

We establish an explicit correspondence between the resulting graded rectangle barcodes and the classical graded one-parameter barcodes that preserves the bottleneck distance. The rectangles display spectral invariants and boundary depth, while their corners record critical points and one-periodic Hamiltonian orbits. Stability results control how the picture changes when the Morse function or Hamiltonian diffeomorphism varies. This makes the rectangle barcode a useful geometric presentation of the information in the classical barcode.

@misc{koeda2023floertype,
  author        = {Koeda, Kanta and Orita, Ryuma and Yashiro, Kanon},
  title         = {{Floer-type bipersistence modules and rectangle barcodes}},
  year          = {2023},
  eprint        = {2312.07847},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/2312.07847}
}