Brieskorn manifolds, positive Sasakian geometry, and contact topology

Abstract

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.

Contact Structures of Sasaki Type and Their Associated Moduli

“…if it exists. The following corollary, which follows from Proposition 4.21 of [BMvK16], was suggested by Otto van Koert For details regarding the invariance and computation of the mean Euler characteristic we refer to Lemma 5.15 and Appendix C in [KvK16].…”

“…In this survey we concentrate mainly on (2). Here we give a brief review referring to [BMvK16,BO13] for details. As mentioned previously a contact manifold of Sasaki type is holomorphically fillable; however, to compute our invariants we need a stronger condition on the filling.…”

“…where I(t, u, z) is a complex structure on the Stein manifold W ∞ , and S ± are S 1 -orbits of critical points of A N . Suffice it to say (see Section 4.2 of [BMvK16] for details) that from this data one obtains an S 1 equivariant Floer chain complex SC S 1 ,N and from this its S 1 equivariant homology SH S 1 ,N (W ). However, this homology depends on the choice of Hamiltonian as well as on N. The dependence on N can be removed by taking the direct limit as N− − →∞.…”

“…In Example 5.1 and Lemma 5.2 of [BMvK16] an example of contact manifolds that cannot be distinguished by χ m (W ) but can by their homology groups SH S 1 ,+ is given.…”

“…A first question may be: what is π 0 (M cl γ )? For example we have the following from [BMvK16] giving examples of 5-manifolds whose moduli space of positive Sasaki classes have an infinite number of components: Theorem 6.1. For each k = 1, 2, .…”