Multiparameter persistent homology via generalized Morse theory

We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space. In the case of smooth functions on a compact manifold, we apply cobordism theory and Cerf theory to study the resulting persistence modules. We give examples in which we obtain a complete description of the persistence module as a direct sum of indecomposable summands and provide a corresponding visualization.

[1]  R. Ghrist.  Barcodes: The persistent topology of data , 2007 .

[2]  Afra Zomorodian,et al.  The Theory of Multidimensional Persistence , 2007, SCG '07.

[3]  N. Mishachev,et al.  Wrinkling of smooth mappings and its applications. I , 1997 .

[4]  J. Wagoner,et al.  Pseudo-isotopies of compact manifolds , 1973 .

[5]  Michael Lesnick,et al.  Interactive Visualization of 2-D Persistence Modules , 2015, ArXiv.

[6]  Andrew Ranicki,et al.  Morse theory for manifolds with boundary , 2012, 1207.3066.

[7]  D. Braess,et al.  Morse-Theorie für berandete Mannigfaltigkeiten , 1974 .

[8]  Ezra Miller.  Homological algebra of modules over posets. , 2020, 2008.00063.

[9]  Jun Yu Li,et al.  Advances in geometric analysis , 2012 .

[10]  Robion Kirby,et al.  A calculus for framed links inS3 , 1978 .

[11]  Bei Wang,et al.  A Kernel for Multi-Parameter Persistent Homology , 2018, Comput. Graph. X.

[12]  Kiyoshi Igusa.  The space of framed functions , 1987 .

[13]  M. Morse.  Relations between the critical points of a real function of $n$ independent variables , 1925 .

[14]  Peter Bubenik,et al.  Homological Algebra for Persistence Modules , 2019, Found. Comput. Math..

[15]  Oliver Vipond,et al.  Multiparameter Persistence Landscapes , 2018, J. Mach. Learn. Res..

[16]  Heather A. Harrington,et al.  Stratifying Multiparameter Persistent Homology , 2017, SIAM J. Appl. Algebra Geom..

[17]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[18]  M. Irwin.  On the Stable Manifold Theorem , 1970 .

[19]  A. Patel,et al.  Persistent local systems , 2018, 1805.02539.

[20]  Steve Oudot,et al.  Exact computation of the matching distance on 2-parameter persistence modules , 2018, SoCG.

[21]  J. Cerf.  La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie , 1970 .

[22]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[23]  Anna Schenfisch,et al.  Natural Stratifications of Reeb Spaces and Higher Morse Functions , 2020, ArXiv.

[24]  B. Hajduk.  Minimal m-functions , 1981 .

[25]  D. joyce.  On manifolds with corners , 2009, 0910.3518.