Brieskorn manifolds in contact topology

Abstract

Abstract. In this survey, we give an overview of Brieskorn manifolds and varieties, and their role in contact topology. We discuss open books, fillings and invariants such as contact and symplectic homology. We also present some new results involving exotic contact structures, invariants and orderability. The main tool for the required computations is a version of the Morse-Bott spectral sequence. We provide a proof for the particular version that is useful for us. A brief historical overview and introductionBrieskorn varieties are affine varieties of the formand are a natural generalization of Fermat varieties. They became popular after it was observed by Hirzebruch that links of singular Brieskorn varieties at 0, meaning sets of the formcan sometimes be homeomorphic, but not diffeomorphic to spheres. Some essential ingredients for the necessary computations were worked out by Pham, [P], and a rather complete picture was worked out by Brieskorn, [Br]. The above links Σ(a 0 , . . . , a n ) are now known as Brieskorn manifolds.In contact geometry, Brieskorn manifolds were first recognized as contact manifolds by AbeErbacher, Lutz-Meckert and Sasaki-Hsu around 1975-1976, [AE, LM, Sa]. In particular, their work showed that at least some exotic spheres admit contact structures. Around the same time, Thomas [T] also used Brieskorn manifolds to establish existence results for contact structures on simplyconnected 5-manifolds. With hindsight, the role of Brieskorn manifolds in early constructions of contact 5-manifolds can be clarified; a classification result for simply-connected 5-manifolds by Smale, [Sm], combined with a homology computation, see Section 3.4, shows that every simplyconnected spin 5-manifold is actually the connected sum of Brieskorn manifolds.Since Brieskorn manifolds also include many spheres with the standard smooth structure, these manifolds can be used to show that there are non-standard contact structures on S 2n−1 following Eliashberg, [E]. The main ingredient is a result of Eliashberg-Gromov-McDuff stating that aspherical symplectic fillings for the standard contact sphere (S 2n−1 , ξ 0 ) are diffeomorphic to D 2n . Smooth Brieskorn varieties, natural symplectic fillings for Brieskorn manifolds, are symplectically aspherical, but typically have a lot of homology. Hence the contact structure on the boundary, a Brieskorn manifold, is non-standard provided none of the exponents equals 1.Later on, Ustilovsky, [U1], showed that there are infinitely many non-isomorphic contact structures on S 4n+1 with the same formal homotopy class of almost contact structures as the standard contact sphere.Because Brieskorn manifolds have a rich and rather understandable structure, they are still of interest, and in this note we will highlight some of their more recent applications to contact and symplectic topology, including some new results on orderability and exotic contact structures.

Two closed orbits for non-degenerate Reeb flows

“…Lemma 8.9 in[36] and Lemma 7.1 in[43]). Suppose that α is Morse-Bott and, if α is degenerate, assume furthermore that the linearized Reeb flow is complex linear with respect to a unitary trivialization of the contact structure along every periodic orbit of α. ThenHF loc pB p,c , Hq -H˚`µ pBp,cq pB p,c ; Λq where µpB p,c q is the Robbin-Salamon index of any closed orbit in B p,c .…”

Complex singularities and contact topology

“…Ustilovsky [192] proved the following property of a special class of Pham-Brieskorn singularities (see Definition 2.11), using the so-called contact homology (see also Kwon and van Koert [98] But Brieskorn [20] had proved that all the previous singularity boundaries are diffeomorphic to the standard sphere S 4m+1 (see also Hirzebruch [82] or Hirzebruch and Mayer [84]). Therefore, the smooth structure on the boundary of an isolated singularity of dimension ≥ 3 does not determine the associated contact structure.…”

“…A survey of the properties of the contact boundaries of Pham-Brieskorn singularities was written by Kwon and van Koert in [98].…”

Koszul duality patterns in Floer theory

“…For example, if charK = 3, the spectral sequence has to be degenerate but otherwise there has to be a non-trivial differential. See also the appendix of [47] for a similar spectral sequence obtained via another natural choice of a contact form on the lens space L(n + 1, n).…”

“…n odd) as parametrized by S 1 S 1 . This leads to a standard Morse-Bott type spectral sequence converging to SH * (X Γ ) (see [57] and/or [47] for a more recent exposition). For example, for n = 2, the E 1 page is given by Comparing this with our results from Sec.…”