nLab persistent homotopy

Idea

Persistent homotopy studies homotopy types (of topological spaces) as a parameter varies, hence filtered homotopy types (of filtered topological spaces), with focus on which elements of homotopy groups at given stage persist how far through the filtering to later stages. A key application is to Vietoris-Rips complexes of discrete subsets in a metric space.

Notably in topological data analysis (TDA) these VR complexes arise as “point clouds” of datapoints, and the corresponding persistent homotopy is thought to detect relevant structure hidden in such data. As such, persistent homotopy refines the traditional use of persistent homology in TDA.

In general, persistent homotopy theory is to persistent homology as homotopy theory is to homology theory: homotopy is a finer invariant than homology, the former sees the full homotopy type of a topological space, the latter at most the underlying stable homotopy type.

In other words, homology involves a kind of linearization or abelianization which loses information that is retained in the homotopy type (see the Hurewicz theorem). Therefore persistent homotopy is in general a finer invariant of filtered topological spaces than persistent homology. In fact, traditional persistent homology considers only ordinary homology which is the coarsest of all generalized homology invariants. Hence in between the coarse invariant of persistent homology and the fine invariants of persistent homotopy will be intermediate invariants that would deserve to be called persistent generalized homology – but these have not yet found much attention, certainly not in the context of topological data analysis.

However, besides homology there is, dually, also cohomology, whose analogous homotopy theoretic refinement is (non-abelian cohomology theories, but in particular:) co-homotopy. The generalization of cohomotopy to the context of persistence lends itself to the analysis of persistence of level sets of continuous functions: see at persistent cohomotopy (and see the references below).

	coarse	intermediate	fine
homology	ordinary homology	generalized homology	homotopy
cohomology	ordinary cohomology	generalized cohomology	cohomotopy
persistent homology	persistent ordinary homology	persistent generalized homology	persistent homotopy
persistent cohomology	persistent ordinary cohomology	persistent generalized cohomology	persistent cohomotopy

Related concepts

• Vietoris-Rips complex

• homotopy theory

• persistence object

• parameterized homotopy theory

• persistent homology

• filtered topological space

References

General

Original articles with focus on establishing the homotopy-version of the stability theorem and the persistent version of Whitehead's theorem:

• Andrew J. Blumberg, Michael Lesnick, Universality of the Homotopy Interleaving Distance $[$arXiv:1705.01690$]$

• J. F. Jardine, Data and homotopy types $[$arXiv:1908.06323$]$

• Edoardo Lanari, Luis Scoccola, Rectification of interleavings and a persistent Whitehead theorem, Algebraic & Geometric Topology (to appear), $[$arXiv:2010.05378$]$

Review:

• J. F. Jardine, Persistent homotopy theory (2020) $[$pdf, pdf$]$

Further discussion:

• Facundo Mémoli, Ling Zhou, Persistent Homotopy Groups of Metric Spaces $[$arXiv:1912.12399, talk$]$

Discussion with focus on the van Kampen theorem, excision and the Hurewicz theorem in persistent homotopy:

• Mehmet Ali Batan, Mehmetcik Pamuk, Hanife Varli, Persistent Homotopy $[$arXiv:1909.08865$]$

Cohomotopy in topological data analysis

Introducing persistent cohomotopy as a tool in topological data analysis, improving on the use of well groups from persistent homology:

• Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom 56 (2016) 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)

• Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, 19 2 (2017) (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)

• Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology 1 (2018) 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)

Review:

• Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 $[$pdf$]$

• Urs Schreiber on joint work with Hisham Sati: New Foundations for TDA – Cohomotopy, (May 2022)