Higher-dimensional contact manifolds with infinitely many Stein fillings

Abstract

For any integer n ≥ 2, we construct an infinite family of (4n − 1)dimensional contact manifolds each of which admits infinitely many pairwise homotopy inequivalent Stein fillings.

Contact Manifolds with Flexible Fillings

“…As we discussed after the statement of Theorem 1.9 in the Introduction, claim (1) does not always hold for n = 2; hence the condition n ≥ 3 is crucial. We also note that (2) holds for certain (Y 2n−1 , ξ); Ozbagci and Stipsicz [57] constructed examples for n = 2 and Oba [56] constructed examples for n ≥ 4 and even.…”

“…The proof of Proposition 1.6 works more generally than just for flexible Weinstein domains; any contact manifold with a Liouville filling that symplectically embeds into a subcritical domain remembers the homology of its fillings. But not any Weinstein domain that has an almost symplectic embedding into a subcritical domain has a genuine symplectic embedding there (and not every contact manifold has a unique filling [56]). So the flexible condition cannot be removed.…”

“…If n ≥ 4 is odd, then either (1)' Y has infinitely many contact structures that are almost contactomorphic to (Y, ξ) and admit almost symplectomorphic Liouville fillings, or (2)' there exists a contact structure on Y that is almost contactomorphic to (Y, ξ) and has infinitely many almost symplectomorphic fillings with different symplectomorphism types. [56,57], fillings of the same contact manifold are distinguished by algebraic topology. It is also not known whether all almost contact manifolds have a fillable contact structure.…”

“…However not much was known beyond the subcritical case. There are contact manifolds with many different fillings [56,57] so the results in the subcritical setting do not hold in general. The main purpose of this paper is to extend those results to flexible fillings, which generalize subcritical fillings and also satisfy an h-principle; see Section 2.…”

Maximal contact and symplectic structures

“…Controlling the topology of the fillings is quite subtle in this dimension but there are now examples of contact 3‐manifolds with infinitely many homeomorphic but not diffeomorphic fillings [5], fillings with arbitrary fundamental group [6], ‘large’ fillings with unbounded Euler characteristic and signature [8], and ‘small’ fillings with [4]. The first example of a high‐dimensional contact manifold with infinitely many non‐homotopy equivalent Weinstein fillings is due to Oba [48]. Many of these constructions use open book decompositions of contact manifolds and construct fillings by finding different factorizations of the open book monodromies into positive Dehn twists.…”