Open Access

Peter Uebele

• We discuss various versions of symplectic homology in the context of Brieskorn manifolds and their Stein fillings. Under a certain index condition, some flavors of symplectic homology, namely SH⁺ and SˇH, are independent of the filling. Thus, they can be used to distinguish the contact structures on Brieskorn manifolds. We will examine their relative advantages and disadvantages compared to other invariant such as the formal homotopy class, contact homology and the mean Euler characteristic. On a concrete example, Σ(2l, 2, 2, 2), we will show that SH⁺ and SˇH actually contain more information than contact homology. On the other hand, symplectic homology is always very hard to compute, and we can do so for Σ(2l, 2, 2, 2) only with the help of a symmetry of the manifold. For more complicated examples, where the full computation of symplectic homology remains elusive, we will turn our attention to the mean Euler characteristic, which is a quantity derived from positiveWe discuss various versions of symplectic homology in the context of Brieskorn manifolds and their Stein fillings. Under a certain index condition, some flavors of symplectic homology, namely SH⁺ and SˇH, are independent of the filling. Thus, they can be used to distinguish the contact structures on Brieskorn manifolds. We will examine their relative advantages and disadvantages compared to other invariant such as the formal homotopy class, contact homology and the mean Euler characteristic. On a concrete example, Σ(2l, 2, 2, 2), we will show that SH⁺ and SˇH actually contain more information than contact homology. On the other hand, symplectic homology is always very hard to compute, and we can do so for Σ(2l, 2, 2, 2) only with the help of a symmetry of the manifold. For more complicated examples, where the full computation of symplectic homology remains elusive, we will turn our attention to the mean Euler characteristic, which is a quantity derived from positive S¹-equivariant symplectic homology. While it contains significantly less information, its computation is essentially a matter of combinatorics. With this tool, we can prove that there exist infinitely many exotic but homotopically trivial contact structures on S⁷, S¹¹ and S¹⁵, which was previously known only for spheres of dimension 4m+1. Moreover, SˇH has the algebraic structure of a unital, graded commutative ring, where multiplication is given by the pair-of-pants product. Again, the full ring structure is extremely hard to compute. However, for a large class of examples, we achieve a partial result in this direction: There is a generator s such that the combination of all products with s gives SˇH the structure of a free and finitely generated module over the ring of Laurent polynomials Z/2Z[s, s⁻¹]. In particular, SˇH is finitely generated as a Z/2Z-algebra, although its vector space dimension is infinite. Similarly, the usual symplectic homology SH of the filling can be given the structure of a finitely generated module over the polynomial ring Z/2Z[s].…