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Computing the Rabinowitz Floer homology of tentacular hyperboloids
1. | Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany |
2. | Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland |
We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids $ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $. Using an embedding of a compact sphere $ \Sigma_0\simeq S^{2k-1} $ into the hypersurface $ \Sigma $, we construct a chain map from the Floer complex of $ \Sigma $ to the Floer complex of $ \Sigma_0 $. In contrast to the compact case, the Rabinowitz Floer homology groups of $ \Sigma $ are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.
References:
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A. Abbondandolo and W. J. Merry,
Floer homology on the time-energy extended phase space, J. Symplectic Geom., 16 (2018), 279-355.
doi: 10.4310/JSG.2018.v16.n2.a1. |
[2] |
A. Abbondandolo and M. Schwarz,
Estimates and computations in Rabinowitz-Floer homology, J. Topol. Anal., 1 (2009), 307-405.
doi: 10.1142/S1793525309000205. |
[3] |
A. Abbondandolo and M. Schwarz,
Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722.
doi: 10.2140/gt.2010.14.1569. |
[4] |
P. Albers and U. Frauenfelder, Rabinowitz Floer homology: a survey, Global Differential Geometry, Springer Proc. Math., 17, Springer, Heidelberg, 2012,437–461.
doi: 10.1007/978-3-642-22842-1_14. |
[5] |
P. Albers, U. Fuchs and W. J. Merry,
Orderability and the Weinstein conjecture, Compos. Math., 151 (2015), 2251-2272.
doi: 10.1112/S0010437X15007642. |
[6] |
P. Albers, U. Fuchs and W. J. Merry,
Positive loops and $l^{\infty }$ contact systolic inequalities, Selecta Math. (N.S.), 23 (2017), 2491-2521.
doi: 10.1007/s00029-017-0338-2. |
[7] |
P. Albers and J. Kang,
Vanishing of Rabinowitz Floer homology on negative line bundles, Math. Z., 285 (2017), 493-517.
doi: 10.1007/s00209-016-1718-6. |
[8] |
P. Albers and W. J. Merry,
Orderability, contact non-squeezing, and Rabinowitz Floer homology, J. Symplectic Geom., 16 (2018), 1481-1547.
doi: 10.4310/JSG.2018.v16.n6.a1. |
[9] |
M. Audin and M. Damian, Morse Theory and Floer Homology, Universitext, Springer, London; EDP Sciences, Les Ulis, 2014.
doi: 10.1007/978-1-4471-5496-9. |
[10] |
B. Chantraine, V. Colin and G. D. Rizell,
Positive Legendrian isotopies and Floer theory, Ann. Inst. Fourier (Grenoble), 69 (2019), 1679-1737.
doi: 10.5802/aif.3279. |
[11] |
K. Cieliebak and U. A. Frauenfelder,
A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251-316.
doi: 10.2140/pjm.2009.239.251. |
[12] |
K. Cieliebak and U. Frauenfelder,
Morse homology on noncompact manifolds, J. Korean Math. Soc., 48 (2011), 749-774.
doi: 10.4134/JKMS.2011.48.4.749. |
[13] |
K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz Floer homology and symplectic homology, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 957–1015.
doi: 10.24033/asens.2137. |
[14] |
K. Cieliebak, U. Frauenfelder and G. P. Paternain,
Symplectic topology of Mañé's critical values, Geom. Topol., 14 (2010), 1765-1870.
doi: 10.2140/gt.2010.14.1765. |
[15] |
K. Cieliebak, Y. Eliashberg and L. Polterovich,
Contact orderability up to conjugation, Regul. Chaotic Dyn., 22 (2017), 585-602.
doi: 10.1134/S1560354717060028. |
[16] |
K. Cieliebak and A. Oancea,
Symplectic homology and the Eilenberg–Steenrod axioms, Algebr. Geom. Topol., 18 (2018), 1953-2130.
doi: 10.2140/agt.2018.18.1953. |
[17] |
A. C. da Silva, Lectures on Symplectic Geometry, Lectures Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-540-45330-7. |
[18] |
A. Fauck,
Rabinowitz-Floer homology on Brieskorn spheres, Int. Math. Res. Not. IMRN, 2015 (2015), 5874-5906.
doi: 10.1093/imrn/rnu109. |
[19] |
A. Fauck, Rabinowitz-Floer Homology on Brieskorn Manifolds, Ph.D thesis, Humboldt-Universität zu Berlin, 2016. |
[20] |
A. Floer,
Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.
doi: 10.1007/BF01260388. |
[21] |
M. Fraser, L. Polterovich and D. Rosen, On Sandon-type metrics for contactomorphism groups, Ann. Math. Qué., 42 (2018), 191–214.
doi: 10.1007/s40316-017-0092-z. |
[22] |
U. Frauenfelder,
The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not. IMRN, 2004 (2004), 2179-2269.
doi: 10.1155/S1073792804133941. |
[23] |
S. Ganatra, J. Pardon and V. Shende, Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math. Inst. Hautes Études Sci., 131 (2020), 73–200.
doi: 10.1007/s10240-019-00112-x. |
[24] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989. |
[25] |
Y. Groman, Floer theory and reduced cohomology on open manifolds, preprint, arXiv: 1510.04265. |
[26] |
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
![]() ![]() |
[27] |
L. Hörmander,
Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.
doi: 10.1007/BF02572374. |
[28] |
F. Laudenbach, Symplectic Geometry and Floer Homology, Soc. Brasil. Mat., Rio de Janeiro, 2004. |
[29] |
D. McDuff and D. Salamon, Introduction to Symplectic Topology, 3rd edition, Oxford Graduate Texts in Mathematics, 27, Oxford University Press, 2017. |
[30] |
W. J. Merry,
On the Rabinowitz Floer homology of twisted cotangent bundles, Calc. Var. Partial Differential Equations, 42 (2011), 355-404.
doi: 10.1007/s00526-011-0391-1. |
[31] |
E. Miranda and C. Oms, The singular Weinstein conjecture, preprint, arXiv: 2005.09568. |
[32] |
F. Pasquotto, R. C. Vandervorst and J. Wiśniewska, Rabinowitz Floer homology for tentacular Hamiltonians, Int. Math. Res. Not. IMRN, 6 (2020).
doi: 10.1093/imrn/rnaa132. |
[33] |
F. Pasquotto and J. Wiśniewska,
Bounds for tentacular Hamiltonians, J. Topol. Anal., 12 (2020), 209-265.
doi: 10.1142/S179352531950047X. |
[34] |
A. F. Ritter,
Topological quantum field theory structure on symplectic cohomology, J. Topol., 6 (2013), 391-489.
doi: 10.1112/jtopol/jts038. |
[35] |
J. Robbin and D. Salamon,
The Maslov index for paths, Topology, 32 (1993), 827-844.
doi: 10.1016/0040-9383(93)90052-W. |
[36] |
S. Suhr and K. Zehmisch,
Linking and closed orbits, Abh. Math. Semin. Univ. Hambg., 86 (2016), 133-150.
doi: 10.1007/s12188-016-0118-5. |
[37] |
J. B. van den Berg, F. Pasquotto, T. Rot and R. C. A. M. Vandervorst,
On periodic orbits in cotangent bundles of non-compact manifolds, J. Symplectic Geom., 14 (2016), 1145-1173.
doi: 10.4310/JSG.2016.v14.n4.a6. |
[38] |
J. B. van den Berg, F. Pasquotto and R. C. Vandervorst,
Closed characteristics on non-compact hypersurfaces in $\mathbb{R}^{2n}$, Math. Ann., 343 (2009), 247-284.
doi: 10.1007/s00208-008-0271-y. |
[39] |
S. Venkatesh,
Rabinowitz Floer homology and mirror symmetry, J. Topol., 11 (2018), 144-179.
doi: 10.1112/topo.12050. |
[40] |
C. Viterbo,
A proof of Weinstein's conjecture in $\mathbb{R}^{2n}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 337-356.
doi: 10.1016/S0294-1449(16)30363-8. |
[41] |
J. J. Wiśniewska, Rabinowitz Floer Homology for Tentacular Hamiltonians, Ph.D thesis, Vrije Universiteit Amsterdam, 2017. |
show all references
References:
[1] |
A. Abbondandolo and W. J. Merry,
Floer homology on the time-energy extended phase space, J. Symplectic Geom., 16 (2018), 279-355.
doi: 10.4310/JSG.2018.v16.n2.a1. |
[2] |
A. Abbondandolo and M. Schwarz,
Estimates and computations in Rabinowitz-Floer homology, J. Topol. Anal., 1 (2009), 307-405.
doi: 10.1142/S1793525309000205. |
[3] |
A. Abbondandolo and M. Schwarz,
Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722.
doi: 10.2140/gt.2010.14.1569. |
[4] |
P. Albers and U. Frauenfelder, Rabinowitz Floer homology: a survey, Global Differential Geometry, Springer Proc. Math., 17, Springer, Heidelberg, 2012,437–461.
doi: 10.1007/978-3-642-22842-1_14. |
[5] |
P. Albers, U. Fuchs and W. J. Merry,
Orderability and the Weinstein conjecture, Compos. Math., 151 (2015), 2251-2272.
doi: 10.1112/S0010437X15007642. |
[6] |
P. Albers, U. Fuchs and W. J. Merry,
Positive loops and $l^{\infty }$ contact systolic inequalities, Selecta Math. (N.S.), 23 (2017), 2491-2521.
doi: 10.1007/s00029-017-0338-2. |
[7] |
P. Albers and J. Kang,
Vanishing of Rabinowitz Floer homology on negative line bundles, Math. Z., 285 (2017), 493-517.
doi: 10.1007/s00209-016-1718-6. |
[8] |
P. Albers and W. J. Merry,
Orderability, contact non-squeezing, and Rabinowitz Floer homology, J. Symplectic Geom., 16 (2018), 1481-1547.
doi: 10.4310/JSG.2018.v16.n6.a1. |
[9] |
M. Audin and M. Damian, Morse Theory and Floer Homology, Universitext, Springer, London; EDP Sciences, Les Ulis, 2014.
doi: 10.1007/978-1-4471-5496-9. |
[10] |
B. Chantraine, V. Colin and G. D. Rizell,
Positive Legendrian isotopies and Floer theory, Ann. Inst. Fourier (Grenoble), 69 (2019), 1679-1737.
doi: 10.5802/aif.3279. |
[11] |
K. Cieliebak and U. A. Frauenfelder,
A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251-316.
doi: 10.2140/pjm.2009.239.251. |
[12] |
K. Cieliebak and U. Frauenfelder,
Morse homology on noncompact manifolds, J. Korean Math. Soc., 48 (2011), 749-774.
doi: 10.4134/JKMS.2011.48.4.749. |
[13] |
K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz Floer homology and symplectic homology, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 957–1015.
doi: 10.24033/asens.2137. |
[14] |
K. Cieliebak, U. Frauenfelder and G. P. Paternain,
Symplectic topology of Mañé's critical values, Geom. Topol., 14 (2010), 1765-1870.
doi: 10.2140/gt.2010.14.1765. |
[15] |
K. Cieliebak, Y. Eliashberg and L. Polterovich,
Contact orderability up to conjugation, Regul. Chaotic Dyn., 22 (2017), 585-602.
doi: 10.1134/S1560354717060028. |
[16] |
K. Cieliebak and A. Oancea,
Symplectic homology and the Eilenberg–Steenrod axioms, Algebr. Geom. Topol., 18 (2018), 1953-2130.
doi: 10.2140/agt.2018.18.1953. |
[17] |
A. C. da Silva, Lectures on Symplectic Geometry, Lectures Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-540-45330-7. |
[18] |
A. Fauck,
Rabinowitz-Floer homology on Brieskorn spheres, Int. Math. Res. Not. IMRN, 2015 (2015), 5874-5906.
doi: 10.1093/imrn/rnu109. |
[19] |
A. Fauck, Rabinowitz-Floer Homology on Brieskorn Manifolds, Ph.D thesis, Humboldt-Universität zu Berlin, 2016. |
[20] |
A. Floer,
Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.
doi: 10.1007/BF01260388. |
[21] |
M. Fraser, L. Polterovich and D. Rosen, On Sandon-type metrics for contactomorphism groups, Ann. Math. Qué., 42 (2018), 191–214.
doi: 10.1007/s40316-017-0092-z. |
[22] |
U. Frauenfelder,
The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not. IMRN, 2004 (2004), 2179-2269.
doi: 10.1155/S1073792804133941. |
[23] |
S. Ganatra, J. Pardon and V. Shende, Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math. Inst. Hautes Études Sci., 131 (2020), 73–200.
doi: 10.1007/s10240-019-00112-x. |
[24] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989. |
[25] |
Y. Groman, Floer theory and reduced cohomology on open manifolds, preprint, arXiv: 1510.04265. |
[26] |
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
![]() ![]() |
[27] |
L. Hörmander,
Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.
doi: 10.1007/BF02572374. |
[28] |
F. Laudenbach, Symplectic Geometry and Floer Homology, Soc. Brasil. Mat., Rio de Janeiro, 2004. |
[29] |
D. McDuff and D. Salamon, Introduction to Symplectic Topology, 3rd edition, Oxford Graduate Texts in Mathematics, 27, Oxford University Press, 2017. |
[30] |
W. J. Merry,
On the Rabinowitz Floer homology of twisted cotangent bundles, Calc. Var. Partial Differential Equations, 42 (2011), 355-404.
doi: 10.1007/s00526-011-0391-1. |
[31] |
E. Miranda and C. Oms, The singular Weinstein conjecture, preprint, arXiv: 2005.09568. |
[32] |
F. Pasquotto, R. C. Vandervorst and J. Wiśniewska, Rabinowitz Floer homology for tentacular Hamiltonians, Int. Math. Res. Not. IMRN, 6 (2020).
doi: 10.1093/imrn/rnaa132. |
[33] |
F. Pasquotto and J. Wiśniewska,
Bounds for tentacular Hamiltonians, J. Topol. Anal., 12 (2020), 209-265.
doi: 10.1142/S179352531950047X. |
[34] |
A. F. Ritter,
Topological quantum field theory structure on symplectic cohomology, J. Topol., 6 (2013), 391-489.
doi: 10.1112/jtopol/jts038. |
[35] |
J. Robbin and D. Salamon,
The Maslov index for paths, Topology, 32 (1993), 827-844.
doi: 10.1016/0040-9383(93)90052-W. |
[36] |
S. Suhr and K. Zehmisch,
Linking and closed orbits, Abh. Math. Semin. Univ. Hambg., 86 (2016), 133-150.
doi: 10.1007/s12188-016-0118-5. |
[37] |
J. B. van den Berg, F. Pasquotto, T. Rot and R. C. A. M. Vandervorst,
On periodic orbits in cotangent bundles of non-compact manifolds, J. Symplectic Geom., 14 (2016), 1145-1173.
doi: 10.4310/JSG.2016.v14.n4.a6. |
[38] |
J. B. van den Berg, F. Pasquotto and R. C. Vandervorst,
Closed characteristics on non-compact hypersurfaces in $\mathbb{R}^{2n}$, Math. Ann., 343 (2009), 247-284.
doi: 10.1007/s00208-008-0271-y. |
[39] |
S. Venkatesh,
Rabinowitz Floer homology and mirror symmetry, J. Topol., 11 (2018), 144-179.
doi: 10.1112/topo.12050. |
[40] |
C. Viterbo,
A proof of Weinstein's conjecture in $\mathbb{R}^{2n}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 337-356.
doi: 10.1016/S0294-1449(16)30363-8. |
[41] |
J. J. Wiśniewska, Rabinowitz Floer Homology for Tentacular Hamiltonians, Ph.D thesis, Vrije Universiteit Amsterdam, 2017. |
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