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.

For a Liouville domain W whose boundary admits a periodic Reeb flow, we can consider the connected component [τ ] ∈ π0(Symp c ( W )) of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, of the component [τ ] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [τ ] has infinite order in π0(Symp c ( W )) if there is an admissible Lagrangian L in W whose wrapped Floer homology is infinite dimensional.We apply our results to fibered twists coming from the Milnor fibers of A k -type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse-Bott spectral sequences.

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex star-shaped hypersurfaces in ${\mathbb {R}}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.

We prove uniqueness, up to diffeomorphism, of symplectically aspherical fillings of certain unit cotangent bundles, including those of higherdimensional tori.Theorem 1.1. The diffeomorphism type of symplectically aspherical fillings of ST * T n is unique.Remark 1.2. While the present paper was in preparation, Theorem 1.1 has independently been obtained by Bowden-Gironella-Moreno [2] as an application of

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