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  2021, 17: 353-399. doi: 10.3934/jmd.2021013

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Alexander Fauck 1, , Will J. Merry 2, and  Jagna Wiśniewska 2,

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

Received  October 09, 2020 Revised  April 19, 2021 Published  September 2021

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.

Keywords: Rabinowitz Floer homology, non-compact hypersurfaces, Weinstein conjecture.
Mathematics Subject Classification: Primary: 53D40, 57R17; Secondary: 37J50, 70H05.
Citation: Alexander Fauck, Will J. Merry, Jagna Wiśniewska. Computing the Rabinowitz Floer homology of tentacular hyperboloids. Journal of Modern Dynamics, 2021, 17: 353-399. doi: 10.3934/jmd.2021013
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.

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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