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Brieskorn manifolds, positive Sasakian geometry, and contact topology

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


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.


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

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