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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 28, Issue 5

# Brieskorn manifolds, positive Sasakian geometry, and contact topology

Charles P. Boyer
• Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, United States of America
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/ Leonardo Macarini
• Corresponding author
• Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária, Rio de Janeiro, Brazil, Postal Code 21941-909
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/ Otto van Koert
• Department of Mathematics and Research Institute of Mathematics, Seoul National University, Building 27, room 402, San 56-1, Sillim-dong, Gwanak-gu, Seoul, South Korea, Postal Code 151-747
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Published Online: 2015-11-20 | DOI: https://doi.org/10.1515/forum-2015-0142

## Abstract

Using ${S}^{1}$-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 ${S}^{2}×{S}^{3}$ 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 ${S}^{5}$ is exhibited.

MSC 2010: 53D40; 53D42; 53C25

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Revised: 2015-10-10

Published Online: 2015-11-20

Published in Print: 2016-09-01

Funding Source: Simons Foundation

Award identifier / Grant number: 245002

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.

Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 943–965, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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