## Geometry & Topology

### Persistent homology and Floer–Novikov theory

#### Abstract

We construct “barcodes” for the chain complexes over Novikov rings that arise in Novikov’s Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. Moreover, we prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and we use this to prove a $C0$–robustness result for the fixed points of Hamiltonian diffeomorphisms. Our approach, which is rather different from the standard methods of persistent homology, is based on a nonarchimedean singular value decomposition for the boundary operator of the chain complex.

#### Article information

Source
Geom. Topol., Volume 20, Number 6 (2016), 3333-3430.

Dates
Revised: 9 December 2015
Accepted: 3 January 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859087

Digital Object Identifier
doi:10.2140/gt.2016.20.3333

Mathematical Reviews number (MathSciNet)
MR3590354

Zentralblatt MATH identifier
1359.53070

Subjects
Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 55U15: Chain complexes

#### Citation

Usher, Michael; Zhang, Jun. Persistent homology and Floer–Novikov theory. Geom. Topol. 20 (2016), no. 6, 3333--3430. doi:10.2140/gt.2016.20.3333. https://projecteuclid.org/euclid.gt/1510859087

#### References

• M Audin, M Damian, Morse theory and Floer homology, Springer, London (2014)
• S A Barannikov, The framed Morse complex and its invariants, from “Singularities and bifurcations” (V I Arnol'd, editor), Adv. Soviet Math. 21, Amer. Math. Soc., Providence, RI (1994) 93–115
• U Bauer, M Lesnick, Induced matchings and the algebraic stability of persistence barcodes, J. Comput. Geom. 6 (2015) 162–191
• P Biran, O Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009) 2881–2989
• D Burghelea, T K Dey, Topological persistence for circle-valued maps, Discrete Comput. Geom. 50 (2013) 69–98
• D Burghelea, S Haller, Topology of angle valued maps, bar codes and Jordan blocks, preprint (2013)
• G Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009) 255–308
• F Chazal, D Cohen-Steiner, M Glisse, L Guibas, S Oudot, Proximity of persistence modules and their diagrams, from “Computational geometry”, ACM (2009) 237–246
• F Chazal, V de Silva, M Glisse, S Oudot, The structure and stability of persistence modules, preprint (2012)
• D Cohen-Steiner, H Edelsbrunner, J Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007) 103–120
• O Cornea, A Ranicki, Rigidity and gluing for Morse and Novikov complexes, J. Eur. Math. Soc. 5 (2003) 343–394
• W Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl. 14 (2015) art. id. 1550066, 8 pp.
• M Entov, L Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 2003 (2003) 1635–1676
• M Farber, Topology of closed one-forms, Mathematical Surveys and Monographs 108, Amer. Math. Soc., Providence, RI (2004)
• A Floer, An instanton-invariant for $3$–manifolds, Comm. Math. Phys. 118 (1988) 215–240
• A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513–547
• A Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989) 575–611
• A Floer, H Hofer, Symplectic homology, I: Open sets in $\mathbb{C}\sp n$, Math. Z. 215 (1994) 37–88
• U Frauenfelder, The Arnold–Givental conjecture and moment Floer homology, Int. Math. Res. Not. 2004 (2004) 2179–2269
• K Fukaya, Y-G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory: anomaly and obstruction, Volume I, AMS/IP Studies in Advanced Mathematics 46, Amer. Math. Soc., Providence, RI (2009)
• K Fukaya, Y-G Oh, H Ohta, K Ono, Displacement of polydisks and Lagrangian Floer theory, J. Symplectic Geom. 11 (2013) 231–268
• K Fukaya, K Ono, Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999) 933–1048
• R Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. 45 (2008) 61–75
• H Hofer, D A Salamon, Floer homology and Novikov rings, from “The Floer memorial volume” (H Hofer, C H Taubes, A Weinstein, E Zehnder, editors), Progr. Math. 133, Birkhäuser, Basel (1995) 483–524
• H Hofer, E Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser, Basel (1994)
• V Humilière, R Leclercq, S Seyfaddini, Coisotropic rigidity and $C\sp 0$–symplectic geometry, Duke Math. J. 164 (2015) 767–799
• K S Kedlaya, $p$–adic differential equations, Cambridge Studies in Advanced Mathematics 125, Cambridge Univ. Press (2010)
• D Le Peutrec, F Nier, C Viterbo, Precise Arrhenius law for $p$–forms: the Witten Laplacian and Morse–Barannikov complex, Ann. Henri Poincaré 14 (2013) 567–610
• Y-J Lee, Reidemeister torsion in Floer–Novikov theory and counting pseudo-holomorphic tori, I, J. Symplectic Geom. 3 (2005) 221–311
• G Liu, G Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998) 1–74
• A F Monna, T A Springer, Sur la structure des espaces de Banach non-archimédiens, Nederl. Akad. Wetensch. Proc. Ser. A 27 (1965) 602–614
• M Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ. 18, Amer. Math. Soc., New York (1934)
• S P Novikov, Multivalued functions and functionals: an analogue of the Morse theory, Dokl. Akad. Nauk SSSR 260 (1981) 31–35 In Russian
• Y-G Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, from “The breadth of symplectic and Poisson geometry” (J E Marsden, T S Ratiu, editors), Progr. Math. 232, Birkhäuser, Boston (2005) 525–570
• J Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol. 20 (2016) 779–1034
• L Polterovich, E Shelukhin, Autonomous Hamiltonian flows, Hofer's geometry and persistence modules, Selecta Math. 22 (2016) 227–296
• J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827–844
• D Salamon, Lectures on Floer homology, lecture notes (1997) Available at \setbox0\makeatletter\@url https://people.math.ethz.ch/~salamon/PREPRINTS/floer.pdf {\unhbox0
• M Schwarz, Morse homology, Progress in Mathematics 111, Birkhäuser, Basel (1993)
• M Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000) 419–461
• V de Silva, D Morozov, M Vejdemo-Johansson, Dualities in persistent (co)homology, Inverse Problems 27 (2011) art. id. 124003, 1–17
• M Usher, Spectral numbers in Floer theories, Compos. Math. 144 (2008) 1581–1592
• M Usher, Duality in filtered Floer–Novikov complexes, J. Topol. Anal. 2 (2010) 233–258
• M Usher, Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds, Israel J. Math. 184 (2011) 1–57
• M Usher, Hofer's metrics and boundary depth, Ann. Sci. Éc. Norm. Supér. 46 (2013) 57–128
• A Zomorodian, G Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005) 249–274