Existence of a dilation for W_k when k ≥ 2

Determine whether the Stein manifold W_k, obtained as the double bubble plumbing of two cotangent bundles T* S^3 along an unknotted circle with identification parameter k ≥ 2, admits a dilation in symplectic cohomology over some field K; equivalently, establish whether there exists a degree-1 class b ∈ SH^1(W_k; K) such that the Batalin–Vilkovisky operator Δ(b) = 1 ∈ SH^0(W_k; K).

Background

The paper recalls that previous works established a quasi-dilation for W_0 over characteristic zero and a genuine dilation for W_1 over the field with three elements. For k ≥ 2, the authors note that an affine realization enabling known constructions is unavailable, and they therefore introduce and use the weaker notion of cyclic quasi-dilation to obtain their main classification results.

A dilation is a particularly strong feature of symplectic cohomology, represented by a degree-1 class whose BV image equals the unit, and it imposes significant geometric restrictions on Lagrangian topology. Whether such a class exists for W_k when k ≥ 2 remains unclear and is explicitly stated as an uncertainty by the authors.