Brieskorn Manifolds, Positive Sasakian Geometry, and Contact Topology

TL;DR

This paper demonstrates the existence of infinitely many inequivalent contact structures and positive Sasakian structures on certain manifolds using symplectic homology, and introduces new Sasaki-Einstein metrics, enriching contact and Sasakian geometry.

Contribution

It establishes the existence of infinitely many contact and positive Sasakian structures on specific manifolds via symplectic homology techniques, and constructs new Sasaki-Einstein metrics.

Findings

Infinitely many inequivalent contact structures on manifolds including connected sums of S^2×S^3.

Moduli space of positive Sasakian structures has infinitely many components.

New family of 20-dimensional Sasaki-Einstein metrics on S^5.

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 $S^2\times S^3$ 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.