Computing the Rabinowitz Floer homology of tentacular hyperboloids

Alexander Fauck; Will J. Merry; Jagna Wiśniewska

doi:10.3934/jmd.2021013

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

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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

Received: October 09, 2020

Revised: April 19, 2021

Published: September 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 53D40, 57R17; Secondary: 37J50, 70H05.

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[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.