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On the Hofer-Zehnder conjecture. (English) Zbl 07583001

Author’s abstract: We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of J. Franks [Invent. Math. 108, No. 2, 403–418 (1992; Zbl 0766.53037)], as conjectured by H. Hofer and E. Zehnder [Symplectic invariants and Hamiltonian dynamics. Basel: Birkhäuser (1994; Zbl 0805.58003)].

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
37J11 Symplectic and canonical mappings
37J46 Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems
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[1] Abouzaid, Mohammed, Symplectic cohomology and {V}iterbo’s theorem. Free Loop Spaces in Geometry and Topology, IRMA Lect. Math. Theor. Phys., 24, 271-485 (2015) · Zbl 1385.53078
[2] Abouzaid, Mohammed; Kragh, Thomas, Simple homotopy equivalence of nearby {L}agrangians, Acta Math.. Acta Mathematica, 220, 207-237 (2018) · Zbl 1396.53104
[3] Arnol\cprime{d}, V. I., A stability problem and ergodic properties of classical dynamical systems. Proc. {I}nternat. {C}ongr. {M}ath., 387-392 (1968)
[4] Arnol\cprime{d}, Vladimir, Sur une propri\'{e}t\'{e} topologique des applications globalement canoniques de la m\'{e}canique classique, C. R. Acad. Sci. Paris. Comptes Rendus Hebdomadaires des S\'{e}ances de l’Acad\'{e}mie des Sciences, 261, 3719-3722 (1965) · Zbl 0134.42305
[5] Bauer, Ulrich; Lesnick, Michael, Induced matchings and the algebraic stability of persistence barcodes, J. Comput. Geom.. Journal of Computational Geometry, 6, 162-191 (2015) · Zbl 1405.68398
[6] Biran, Paul; Cornea, Octav, Rigidity and uniruling for {L}agrangian submanifolds, Geom. Topol.. Geometry & Topology, 13, 2881-2989 (2009) · Zbl 1180.53078
[7] Biran, Paul; Cornea, Octav; Shelukhin, Egor, Lagrangian Shadows and Triangulated Categories, Ast\'{e}risque. Ast\'{e}risque, 426, 128 pp. (2021) · Zbl 1491.53091
[8] Birkhoff, George D., Proof of {P}oincar\'{e}’s geometric theorem, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 14, 14-22 (1913) · JFM 44.0761.01
[9] Birkhoff, George D., An extension of {P}oincar\'{e}’s last geometric theorem, Acta Math.. Acta Mathematica, 47, 297-311 (1926) · JFM 52.0573.02
[10] Borel, Armand, Seminar on Transformation Groups, Ann. of Math. Stud., 46, vii+245 pp. (1960) · Zbl 0091.37202
[11] Bramham, B.; Hofer, H., First steps towards a symplectic dynamics. Surveys in Differential Geometry. {V}ol. {XVII}, Surv. Differ. Geom., 17, 127-177 (2012) · Zbl 1382.53023
[12] Bredon, Glen E., Introduction to Compact Transformation Groups, Pure Appl. Math., 46, xiii+459 pp. (1972) · Zbl 0246.57017
[13] Brown, M.; Neumann, W. D., Proof of the {P}oincar\'{e}-{B}irkhoff fixed point theorem, Michigan Math. J.. Michigan Mathematical Journal, 24, 21-31 (1977) · Zbl 0402.55001
[14] Carlsson, Gunnar, Topology and data, Bull. Amer. Math. Soc. (N.S.). Amer. Math. Soc.. Bulletin. New Series, 46, 255-308 (2009) · Zbl 1172.62002
[15] Carlsson, Gunnar; Zomorodian, A. Collins, A.; Guibas, L. J., R:Z tence barcodes for shapes, Internat. J. Shape Modeling, 11, 149-187 (2005) · Zbl 1092.68688
[16] Chazal, Fr\'{e}d\'{e}ric; de Silva, Vin; Glisse, Marc; Oudot, Steve, The Structure and Stability of Persistence Modules, SpringerBriefs Math., x+120 pp. (2016) · Zbl 1362.55002
[17] Chazal, Fr\'{e}d\'{e}ric; Steiner, D. C.; Glisse, M.; Guibas, L. J.; Oudot, S. Y., Proximity of persistence modules and their diagrams. Proceedings of the 25th Annual Symposium on Computational Geometry, SCG, 2009, 237-246 (2009) · Zbl 1380.68387
[18] \c{C}ineli, Erman; Ginzburg, Viktor L., On the iterated {H}amiltonian {F}loer homology, Commun. Contemp. Math.. Communications in Contemporary Mathematics, 23, 2050026-23 (2021) · Zbl 1458.53090
[19] Cohen-Steiner, David; Edelsbrunner, Herbert; Harer, John, Stability of persistence diagrams, Discrete Comput. Geom.. Discrete & Computational Geometry. An International Journal of Mathematics and Computer Science, 37, 103-120 (2007) · Zbl 1117.54027
[20] Collier, Brian; Kerman, Ely; Reiniger, Benjamin M.; Turmunkh, Bolor; Zimmer, Andrew, A symplectic proof of a theorem of {F}ranks, Compos. Math.. Compositio Mathematica, 148, 1969-1984 (2012) · Zbl 1267.53093
[21] Conley, Charles; Zehnder, Eduard, Morse-type index theory for flows and periodic solutions for {H}amiltonian equations, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 37, 207-253 (1984) · Zbl 0559.58019
[22] Conley, C.; Zehnder, E., Subharmonic solutions and {M}orse theory. Mathematical physics, VII (Boulder, Colo., 1983), Phys. A. Physica A, 124, 649-657 (1984) · Zbl 0605.58015
[23] Conley, C.; Zehnder, E., A global fixed point theorem for symplectic maps and subharmonic solutions of {H}amiltonian equations on tori. Nonlinear Functional Analysis and its Applications, {P}art 1, Proc. Sympos. Pure Math., 45, 283-299 (1986) · Zbl 0559.58019
[24] Conley, C. C.; Zehnder, E., The {B}irkhoff-{L}ewis fixed point theorem and a conjecture of {V}. {I}. {A}rnol\cprime d, Invent. Math.. Inventiones Mathematicae, 73, 33-49 (1983) · Zbl 0516.58017
[25] Crawley-Boevey, William, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl.. Journal of Algebra and its Applications, 14, 1550066-8 (2015) · Zbl 1345.16015
[26] Edelsbrunner, Herbert; Letscher, David; Zomorodian, Afra, Topological persistence and simplification. Discrete and Computational Geometry and Graph Drawing, Discrete Comput. Geom., 28, 511-533 (2002) · Zbl 1011.68152
[27] {Eliashberg}, Y., Estimates on the number of fixed points of area preserving transformations (1979)
[28] Entov, Michael; Polterovich, Leonid, Calabi quasimorphism and quantum homology, Int. Math. Res. Not.. International Mathematics Research Notices, 1635-1676 (2003) · Zbl 1047.53055
[29] Entov, Michael; Polterovich, Leonid, Symplectic quasi-states and semi-simplicity of quantum homology. Toric Topology, Contemp. Math., 460, 47-70 (2008) · Zbl 1146.53066
[30] Entov, Michael; Polterovich, Leonid, Rigid subsets of symplectic manifolds, Compos. Math.. Compositio Mathematica, 145, 773-826 (2009) · Zbl 1230.53080
[31] Floer, Andreas, Proof of the {A}rnol\cprime d conjecture for surfaces and generalizations to certain {K}\"{a}hler manifolds, Duke Math. J.. Duke Mathematical Journal, 53, 1-32 (1986) · Zbl 0607.58016
[32] Floer, Andreas, Morse theory for fixed points of symplectic diffeomorphisms, Bull. Amer. Math. Soc. (N.S.). Amer. Math. Soc.. Bulletin. New Series, 16, 279-281 (1987) · Zbl 0617.53042
[33] Floer, Andreas, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys.. Communications in Mathematical Physics, 120, 575-611 (1989) · Zbl 0755.58022
[34] Floer, Andreas, Witten’s complex and infinite-dimensional {M}orse theory, J. Differential Geom.. Journal of Differential Geometry, 30, 207-221 (1989) · Zbl 0678.58012
[35] Floyd, E. E., On periodic maps and the {E}uler characteristics of associated spaces, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 72, 138-147 (1952) · Zbl 0046.16603
[36] Fortune, Barry, A symplectic fixed point theorem for {\({\bf C}{\rm P}^n\)}, Invent. Math.. Inventiones Mathematicae, 81, 29-46 (1985) · Zbl 0547.58015
[37] Fortune, Barry; Weinstein, Alan, A symplectic fixed point theorem for complex projective spaces, Bull. Amer. Math. Soc. (N.S.). Amer. Math. Soc.. Bulletin. New Series, 12, 128-130 (1985) · Zbl 0566.58013
[38] Franks, John, Geodesics on {\(S^2\)} and periodic points of annulus homeomorphisms, Invent. Math.. Inventiones Mathematicae, 108, 403-418 (1992) · Zbl 0766.53037
[39] Franks, John, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math.. New York Journal of Mathematics, 2, 1-19 (1996) · Zbl 0891.58033
[40] Franks, John; Handel, Michael, Periodic points of {H}amiltonian surface diffeomorphisms, Geom. Topol.. Geometry and Topology, 7, 713-756 (2003) · Zbl 1034.37028
[41] Frauenfelder, Urs, The {A}rnold-{G}ivental conjecture and moment {F}loer homology, Int. Math. Res. Not.. International Mathematics Research Notices, 2179-2269 (2004) · Zbl 1088.53058
[42] Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru, Displacement of polydisks and {L}agrangian {F}loer theory, J. Symplectic Geom.. The Journal of Symplectic Geometry, 11, 231-268 (2013) · Zbl 1292.53056
[43] Fukaya, Kenji; Ono, Kaoru, Arnold conjecture and {G}romov-{W}itten invariant, Topology. Topology. An International Journal of Mathematics, 38, 933-1048 (1999) · Zbl 0946.53047
[44] Fukaya, Kenji; Ono, Kaoru, Arnold conjecture and {G}romov-{W}itten invariant for general symplectic manifolds. The {A}rnoldfest, Fields Inst. Commun., 24, 173-190 (1999) · Zbl 0946.53047
[45] Ghrist, Robert, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. (N.S.). Amer. Math. Soc.. Bulletin. New Series, 45, 61-75 (2008) · Zbl 1391.55005
[46] Ginzburg, Viktor L., The {C}onley conjecture, Ann. of Math. (2). Annals of Mathematics. Second Series, 172, 1127-1180 (2010) · Zbl 1228.53098
[47] Ginzburg, Viktor L.; G\"{u}rel, Ba\c{s}ak Z., Action and index spectra and periodic orbits in {H}amiltonian dynamics, Geom. Topol.. Geometry & Topology, 13, 2745-2805 (2009) · Zbl 1172.53052
[48] Ginzburg, Viktor L.; G\"{u}rel, Ba\c{s}ak Z., Local {F}loer homology and the action gap, J. Symplectic Geom.. The Journal of Symplectic Geometry, 8, 323-357 (2010) · Zbl 1206.53087
[49] Ginzburg, Viktor L.; G\"{u}rel, Ba\c{s}ak Z., Conley conjecture for negative monotone symplectic manifolds, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 1748-1767 (2012) · Zbl 1242.53100
[50] Ginzburg, Viktor L.; G\"{u}rel, Ba\c{s}ak Z., Hyperbolic fixed points and periodic orbits of {H}amiltonian diffeomorphisms, Duke Math. J.. Duke Mathematical Journal, 163, 565-590 (2014) · Zbl 1408.53112
[51] Ginzburg, Viktor L.; G\"{u}rel, Ba\c{s}ak Z., Non-contractible periodic orbits in {H}amiltonian dynamics on closed symplectic manifolds, Compos. Math.. Compositio Mathematica, 152, 1777-1799 (2016) · Zbl 1375.53106
[52] Ginzburg, Viktor L.; G\"{u}rel, Ba\c{s}ak Z., Hamiltonian pseudo-rotations of projective spaces, Invent. Math.. Inventiones Mathematicae, 214, 1081-1130 (2018) · Zbl 1447.53073
[53] Ginzburg, Viktor L.; G\"{u}rel, Ba\c{s}ak Z., Conley conjecture revisited, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 761-798 (2019) · Zbl 1428.53085
[54] Gromov, M., Pseudo holomorphic curves in symplectic manifolds, Invent. Math.. Inventiones Mathematicae, 82, 307-347 (1985) · Zbl 0592.53025
[55] G\"{u}rel, Ba\c{s}ak Z., On non-contractible periodic orbits of {H}amiltonian diffeomorphisms, Bull. Lond. Math. Soc.. Bulletin of the London Mathematical Society, 45, 1227-1234 (2013) · Zbl 1283.37061
[56] Hein, Doris, The {C}onley conjecture for irrational symplectic manifolds, J. Symplectic Geom.. The Journal of Symplectic Geometry, 10, 183-202 (2012) · Zbl 1275.37026
[57] Hingston, Nancy, Subharmonic solutions of {H}amiltonian equations on tori, Ann. of Math. (2). Annals of Mathematics. Second Series, 170, 529-560 (2009) · Zbl 1180.58009
[58] Hofer, H., On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 115, 25-38 (1990) · Zbl 0713.58004
[59] Hofer, Helmut; Zehnder, Eduard, Symplectic Invariants and {H}amiltonian Dynamics, Birkh\"auser Advanced Texts: {B}asler Lehrb\"{u}scher (1994) · Zbl 0805.58003
[60] Hsiang, Wu Yi, Cohomology Theory of Topological Transformation Groups, Ergeb. Math. Grenzgeb., 85, x+164 pp. (1975) · Zbl 0429.57011
[61] Jacobson, Nathan, Basic Algebra. {II}, xviii+686 pp. (1989) · Zbl 0694.16001
[62] Kaledin, Dmitry, Cartier isomorphism and {H}odge theory in the non-commutative case. Arithmetic Geometry, Clay Math. Proc., 8, 537-562 (2009) · Zbl 1205.19004
[63] Kislev, Asaf; Shelukhin, Egor, Bounds on spectral norms and barcodes, Geom. Topol.. Geometry & Topology, 25, 3257-3350 (2021) · Zbl 1485.57023
[64] Lalonde, Fran\c{c}ois; McDuff, Dusa, The geometry of symplectic energy, Ann. of Math. (2). Annals of Mathematics. Second Series, 141, 349-371 (1995) · Zbl 0829.53025
[65] Le Calvez, Patrice, Periodic orbits of {H}amiltonian homeomorphisms of surfaces, Duke Math. J.. Duke Mathematical Journal, 133, 125-184 (2006) · Zbl 1101.37031
[66] Le Calvez, Patrice, Pourquoi les points p\'{e}riodiques des hom\'{e}omorphismes du plan tournent-ils autour de certains points fixes?, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 41, 141-176 (2008) · Zbl 1168.37010
[67] Le Roux, Fr\'{e}d\'{e}ric; Seyfaddini, Sobhan; Viterbo, Claude, Barcodes and area-preserving homeomorphisms, Geom. Topol.. Geometry & Topology, 25, 2713-2825 (2021) · Zbl 1494.37040
[68] Leclercq, R\'{e}mi; Zapolsky, Frol, Spectral invariants for monotone {L}agrangians, J. Topol. Anal.. Journal of Topology and Analysis, 10, 627-700 (2018) · Zbl 1398.57037
[69] Liu, Gang, Associativity of quantum multiplication, Comm. Math. Phys.. Communications in Mathematical Physics, 191, 265-282 (1998) · Zbl 0891.53022
[70] Liu, Gang; Tian, Gang, Floer homology and {A}rnold conjecture, J. Differential Geom.. Journal of Differential Geometry, 49, 1-74 (1998) · Zbl 0917.58009
[71] Markl, Martin, Ideal perturbation lemma, Comm. Algebra. Communications in Algebra, 29, 5209-5232 (2001) · Zbl 0994.55012
[72] McDuff, Dusa; Salamon, Dietmar, {\(J\)}-Holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, xiv+726 pp. (2012) · Zbl 1272.53002
[73] McLean, Mark, Local {F}loer homology and infinitely many simple {R}eeb orbits, Algebr. Geom. Topol.. Algebraic & Geometric Topology, 12, 1901-1923 (2012) · Zbl 1253.53078
[74] Oh, Yong-Geun, Spectral invariants, analysis of the {F}loer moduli space, and geometry of the {H}amiltonian diffeomorphism group, Duke Math. J.. Duke Mathematical Journal, 130, 199-295 (2005) · Zbl 1113.53054
[75] Oh, Yong-Geun, Symplectic {T}opology and {F}loer {H}omology. {V}ol. 1. Symplectic Geometry and Pseudoholomorphic Curves, New Math. Monogr., 28, xxiii+395 pp. (2015) · Zbl 1338.53004
[76] Orita, Ryuma, Non-contractible periodic orbits in {H}amiltonian dynamics on tori, Bull. Lond. Math. Soc.. Bulletin of the London Mathematical Society, 49, 571-580 (2017) · Zbl 1372.53089
[77] Orita, Ryuma, On the existence of infinitely many non-contractible periodic orbits of {H}amiltonian diffeomorphisms of closed symplectic manifolds, J. Symplectic Geom.. The Journal of Symplectic Geometry, 17, 1893-1927 (2019) · Zbl 1434.53084
[78] Ostrover, Yaron, Calabi quasi-morphisms for some non-monotone symplectic manifolds, Algebr. Geom. Topol.. Algebraic & Geometric Topology, 6, 405-434 (2006) · Zbl 1114.53070
[79] Pardon, John, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol.. Geometry & Topology, 20, 779-1034 (2016) · Zbl 1342.53109
[80] Piunikhin, S.; Salamon, D.; Schwarz, M., Symplectic {F}loer-{D}onaldson theory and quantum cohomology. Contact and Symplectic Geometry, Publ. Newton Inst., 8, 171-200 (1996) · Zbl 0874.53031
[81] Poincar{\'e}, H., Sur un th{\'e}or{\`e}me de g{\'e}om{\'e}trie, Rendiconti del Circolo Matematico di Palermo (1884-1940), 33, 375-407 (1912) · JFM 43.0757.03
[82] Polterovich, Leonid; Rosen, Daniel, Function Theory on Symplectic Manifolds, CRM Monogr. Ser., 34, xii+203 pp. (2014) · Zbl 1310.53002
[83] Polterovich, Leonid; Shelukhin, Egor, Autonomous {H}amiltonian flows, {H}ofer’s geometry and persistence modules, Selecta Math. (N.S.). Selecta Mathematica. New Series, 22, 227-296 (2016) · Zbl 1337.53094
[84] Polterovich, Leonid; Shelukhin, Egor; Stojisavljevi\'{c}, Vuka\v{s}in, Persistence modules with operators in {M}orse and {F}loer theory, Mosc. Math. J.. Moscow Mathematical Journal, 17, 757-786 (2017) · Zbl 1422.53074
[85] Po\'{z}niak, Marcin, Floer homology, {N}ovikov rings and clean intersections. Northern {C}alifornia {S}ymplectic {G}eometry {S}eminar, Amer. Math. Soc. Transl. Ser. 2, 196, 119-181 (1999) · Zbl 0948.57025
[86] Robbin, Joel; Salamon, Dietmar, The {M}aslov index for paths, Topology. Topology. An International Journal of Mathematics, 32, 827-844 (1993) · Zbl 0798.58018
[87] Ruan, Yongbin, Virtual neighborhoods and pseudo-holomorphic curves. Proceedings of 6th {G}\"{o}kova {G}eometry-{T}opology {C}onference, Turkish J. Math.. Turkish Journal of Mathematics, 23, 161-231 (1999) · Zbl 0967.53055
[88] Ruan, Yongbin; Tian, Gang, A mathematical theory of quantum cohomology, Math. Res. Lett.. Mathematical Research Letters, 1, 269-278 (1994) · Zbl 0860.58006
[89] Ruan, Yongbin; Tian, Gang, A mathematical theory of quantum cohomology, J. Differential Geom.. Journal of Differential Geometry, 42, 259-367 (1995) · Zbl 0860.58005
[90] Salamon, Dietmar, Lectures on {F}loer homology. Symplectic Geometry and Topology, IAS/Park City Math. Ser., 7, 143-229 (1999) · Zbl 1031.53118
[91] Salamon, Dietmar; Zehnder, Eduard, Morse theory for periodic solutions of {H}amiltonian systems and the {M}aslov index, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 45, 1303-1360 (1992) · Zbl 0766.58023
[92] Schwarz, Matthias, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math.. Pacific Journal of Mathematics, 193, 419-461 (2000) · Zbl 1023.57020
[93] Seidel, Paul, Symplectic {F}loer homology and the mapping class group, Pacific J. Math.. Pacific Journal of Mathematics, 206, 219-229 (2002) · Zbl 1061.53065
[94] Seidel, Paul, Fukaya Categories and {P}icard-{L}efschetz Theory, Zurich Lect. Adv. Math., viii+326 pp. (2008) · Zbl 1159.53001
[95] Seidel, Paul, The equivariant pair-of-pants product in fixed point {F}loer cohomology, Geom. Funct. Anal.. Geometric and Functional Analysis, 25, 942-1007 (2015) · Zbl 1331.53119
[96] Shelukhin, E., Viterbo conjecture for {Z}oll symmetric spaces (2018)
[97] Shelukhin, E.; Zhao, J., The {\({\Z}/p{\Z}\)}-equivariant product-isomorphism in fixed point {F}loer homology
[98] Smith, P. A., Transformations of finite period, Ann. of Math. (2). Annals of Mathematics. Second Series, 39, 127-164 (1938) · JFM 64.1275.01
[99] Tonkonog, Dmitry, Commuting symplectomorphisms and {D}ehn twists in divisors, Geom. Topol.. Geometry & Topology, 19, 3345-3403 (2015) · Zbl 1332.53106
[100] Usher, Michael, The sharp energy-capacity inequality, Commun. Contemp. Math.. Communications in Contemporary Mathematics, 12, 457-473 (2010) · Zbl 1200.53077
[101] Usher, Michael, Boundary depth in {F}loer theory and its applications to {H}amiltonian dynamics and coisotropic submanifolds, Israel J. Math.. Israel Journal of Mathematics, 184, 1-57 (2011) · Zbl 1253.53085
[102] Usher, Michael, Hofer’s metrics and boundary depth, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 46, 57-128 (2013) · Zbl 1271.53076
[103] Usher, Michael; Zhang, Jun, Persistent homology and {F}loer-{N}ovikov theory, Geom. Topol.. Geometry & Topology, 20, 3333-3430 (2016) · Zbl 1359.53070
[104] Viterbo, Claude, Symplectic topology as the geometry of generating functions, Math. Ann.. Mathematische Annalen, 292, 685-710 (1992) · Zbl 0735.58019
[105] Witten, Edward, Two-dimensional gravity and intersection theory on moduli space. Surveys in Differential Geometry, 243-310 (1991) · Zbl 0757.53049
[106] Zapolsky, F., The {L}agrangian {F}loer-quantum-{PSS} package and canonical orientations in {F}loer theory
[107] Zomorodian, Afra; Carlsson, Gunnar, Computing persistent homology, Discrete Comput. Geom.. Discrete & Computational Geometry. An International Journal of Mathematics and Computer Science, 33, 249-274 (2005) · Zbl 1069.55003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.