Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 20, 2015

Brieskorn manifolds, positive Sasakian geometry, and contact topology

Charles P. Boyer, Leonardo Macarini and Otto van Koert
From the journal Forum Mathematicum

Abstract

Using S1-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn–Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of S2×S3 and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki–Einstein metrics on certain homotopy spheres. Finally, a new family of Sasaki–Einstein metrics of real dimension 20 on S5 is exhibited.

MSC 2010: 53D40; 53D42; 53C25

Communicated by Jörg Brüdern


Funding source: Simons Foundation

Award Identifier / Grant number: 245002

Funding statement: The first author was partially supported by a grant (#245002) from the Simons Foundation. The second author was partially supported by CNPq, Brazil. The third author was supported by a stipend from the Humboldt Foundation.

This work was born at the AIM Workshop on Transversality in Contact Homology in Palo Alto, CA, December 8–12, 2014 and we would like to thank the American Institute of Mathematics for its hospitality. We also thank Frédéric Bourgeois and Jean Gutt for interesting and helpful discussions. In addition the first author thanks Chi Li and Song Sun for discussions concerning Remark 7.

References

[1] Abe K., On a generalization of the Hopf fibration. I. Contact structures on the generalized Brieskorn manifolds, Tôhoku Math. J. (2) 29 (1977), no. 3, 335–374. 10.2748/tmj/1178240604Search in Google Scholar

[2] Abe K. and Erbacher J., Nonregular contact structures on Brieskorn manifolds, Bull. Amer. Math. Soc. 81 (1975), 407–409. 10.1090/S0002-9904-1975-13759-1Search in Google Scholar

[3] Abouzaid M., Symplectic cohomology and Viterbo’s theorem, preprint 2013, http://arxiv.org/abs/1312.3354. Search in Google Scholar

[4] Bogomolov F. A. and de Oliveira B., Stein small deformations of strictly pseudoconvex surfaces, Birational Algebraic Geometry (Baltimore 1996), Contemp. Math. 207, American Mathematical Society, Providence (1997), 25–41. 10.1090/conm/207/02717Search in Google Scholar

[5] Bourgeois F. and Oancea A., Fredholm theory and transversality for the parametrized and for the S1-invariant symplectic action, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1181–1229. 10.4171/JEMS/227Search in Google Scholar

[6] Bourgeois F. and Oancea A., The Gysin exact sequence for S1-equivariant symplectic homology, J. Topol. Anal. 5 (2013), no. 4, 361–407. 10.1142/S1793525313500210Search in Google Scholar

[7] Boyer C. P. and Galicki K., Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235, electronic. 10.2140/gt.2006.10.2219Search in Google Scholar

[8] Boyer C. P. and Galicki K., Sasakian Geometry, Oxford Math. Monogr., Oxford University Press, Oxford, 2008. 10.1093/acprof:oso/9780198564959.001.0001Search in Google Scholar

[9] Boyer C. P., Galicki K. and Kollár J., Einstein metrics on spheres, Ann. of Math. (2) 162 (2005), no. 1, 557–580. 10.4007/annals.2005.162.557Search in Google Scholar

[10] Boyer C. P., Galicki K. and Nakamaye M., Sasakian geometry, homotopy spheres and positive Ricci curvature, Topology 42 (2003), no. 5, 981–1002. 10.1016/S0040-9383(02)00027-7Search in Google Scholar

[11] Boyer C. P. and Nakamaye M., On Sasaki–Einstein manifolds in dimension five, Geom. Dedicata 144 (2010), 141–156. 10.1007/s10711-009-9393-ySearch in Google Scholar

[12] Boyer C. P. and Pati J., On the equivalence problem for toric contact structures on S3-bundles over S2, Pacific J. Math. 267 (2014), no. 2, 277–324. 10.2140/pjm.2014.267.277Search in Google Scholar

[13] Boyer C. P. and Tønnesen-Friedman C. W., On positivity in Sasakian geometry, in preparation. Search in Google Scholar

[14] Cieliebak K. and Eliashberg Y., From Stein to Weinstein and Back, Amer. Math. Soc. Colloq. Publ. 59, American Mathematical Society, Providence, 2012. 10.1090/coll/059Search in Google Scholar

[15] Conti D., Cohomogeneity one Einstein–Sasaki 5-manifolds, Comm. Math. Phys. 274 (2007), no. 3, 751–774. 10.1007/s00220-007-0286-3Search in Google Scholar

[16] El Kacimi-Alaoui A., Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compos. Math. 73 (1990), no. 1, 57–106. Search in Google Scholar

[17] Fauck A., Rabinowitz–Floer homology on Brieskorn spheres, Int. Math. Res. Not. IMRN 2015 (2015), no. 14, 5874–5906. 10.1093/imrn/rnu109Search in Google Scholar

[18] Frauenfelder U., Schlenk F. and van Koert O., Displaceability and the mean Euler characteristic, Kyoto J. Math. 52 (2012), no. 4, 797–815. 10.1215/21562261-1728866Search in Google Scholar

[19] Futaki A., Ono H. and Wang G., Transverse Kähler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds, J. Differential Geom. 83 (2009), no. 3, 585–635. 10.4310/jdg/1264601036Search in Google Scholar

[20] Gauntlett J. P., Martelli D., Sparks J. and Yau S.-T., Obstructions to the existence of Sasaki–Einstein metrics, Comm. Math. Phys. 273 (2007), no. 3, 803–827. 10.1007/s00220-007-0213-7Search in Google Scholar

[21] Ghigi A. and Kollár J., Kähler–Einstein metrics on orbifolds and Einstein metrics on spheres, Comment. Math. Helv. 82 (2007), no. 4, 877–902. 10.4171/CMH/113Search in Google Scholar

[22] Grothendieck A., Séminaire de géométrie algébrique du Bois Marie 1962. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), North-Holland Publishing, Amsterdam, 1968. Search in Google Scholar

[23] Gutt J., On the minimal number of periodic Reeb orbits on a contact manifold Dissertation, Universite de Strasbourg, 2014. Search in Google Scholar

[24] Gutt J., The positive equivariant symplectic homology as an invariant for some contact manifolds, preprint 2015, http://arxiv.org/abs/1503.01443. 10.4310/JSG.2017.v15.n4.a3Search in Google Scholar

[25] He W., Isometry group of Sasaki–Einstein metric, C. R. Math. Acad. Sci. Paris 352 (2014), no. 1, 71–73. 10.1016/j.crma.2013.10.037Search in Google Scholar

[26] Kollár J., Einstein metrics on five-dimensional Seifert bundles, J. Geom. Anal. 15 (2005), no. 3, 445–476. 10.1007/BF02930981Search in Google Scholar

[27] Kwon M. and van Koert O., Brieskorn manifolds in contact topology, preprint 2015, http://arxiv.org/abs/1310.0343v3; to appear in Bull. Lond. Math. Soc.. 10.1112/blms/bdv088Search in Google Scholar

[28] Li C., Numerical solutions of Kähler–Einstein metrics on 2 with conical singularities along a smooth quadric curve, J. Geom. Anal. 25 (2015), no. 3, 1773–1797. 10.1007/s12220-014-9493-2Search in Google Scholar

[29] Li C. and Sun S., Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 (2014), no. 3, 927–973. 10.1007/s00220-014-2123-9Search in Google Scholar

[30] Lutz R. and Meckert C., Structures de contact sur certaines sphères exotiques, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 11, 591–593. Search in Google Scholar

[31] Marinescu G. and Yeganefar N., Embeddability of some strongly pseudoconvex CR manifolds, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4757–4771, electronic. 10.1090/S0002-9947-07-04047-0Search in Google Scholar

[32] Martelli D., Sparks J. and Yau S.-T., Sasaki–Einstein manifolds and volume minimisation, Comm. Math. Phys. 280 (2008), no. 3, 611–673. 10.1007/s00220-008-0479-4Search in Google Scholar

[33] Milnor J. and Orlik P., Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385–393. 10.1016/0040-9383(70)90061-3Search in Google Scholar

[34] Morita S., A topological classification of complex structures on S1×S2n-1, Topology 14 (1975), 13–22. 10.1016/0040-9383(75)90030-0Search in Google Scholar

[35] Nitta Y. and Sekiya K., Uniqueness of Sasaki–Einstein metrics, Tôhoku Math. J. (2) 64 (2012), no. 3, 453–468. 10.2748/tmj/1347369373Search in Google Scholar

[36] Nozawa H., Deformation of Sasakian metrics, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2737–2771. 10.1090/S0002-9947-2013-06020-5Search in Google Scholar

[37] Randell R. C., The homology of generalized Brieskorn manifolds, Topology 14 (1975), no. 4, 347–355. 10.1016/0040-9383(75)90019-1Search in Google Scholar

[38] Sasaki S. and Hsu C. J., On a property of Brieskorn manifolds, Tôhoku Math. J. (2) 28 (1976), no. 1, 67–78. 10.2748/tmj/1178240879Search in Google Scholar

[39] Sato H., Remarks concerning contact manifolds, Tôhoku Math. J. 29 (1977), no. 4, 577–584. 10.2748/tmj/1178240494Search in Google Scholar

[40] Smale S., On the structure of 5-manifolds, Ann. of Math. (2) 75 (1962), 38–46. 10.2307/1970417Search in Google Scholar

[41] Takahashi T., Deformations of Sasakian structures and its application to the Brieskorn manifolds, Tôhoku Math. J. (2) 30 (1978), no. 1, 37–43. 10.2748/tmj/1178230095Search in Google Scholar

[42] Uebele P., Symplectic homology of some Brieskorn manifolds, preprint 2015, http://arxiv.org/abs/1502.04547. 10.1007/s00209-015-1596-3Search in Google Scholar

[43] Ustilovsky I., Infinitely many contact structures on S4m+1, Int. Math. Res. Not. IMRN 1999 (1999), no. 14, 781–791. 10.1155/S1073792899000392Search in Google Scholar

[44] van Koert O., Open books for contact five-manifolds and applications of contact homology, Dissertation, Universität zu Köln, 2005. Search in Google Scholar

[45] van Koert O., Contact homology of Brieskorn manifolds, Forum Math. 20 (2008), no. 2, 317–339. 10.1515/FORUM.2008.016Search in Google Scholar

Received: 2015-7-19
Revised: 2015-10-10
Published Online: 2015-11-20
Published in Print: 2016-9-1

© 2016 by De Gruyter