A Negative Answer to the Strong Geography Question of Alfieri and Binns

Alfieri and Binns say that an F[U]-module M satisfies the strong geography restriction if it has a direct summand F[U]/U^ℓ ⊕ F[U]/U^(ℓ−1) ⊕ ⋯ ⊕ F[U]/U, where ℓ is the least integer with U^ℓ M_red = 0. They showed that HF⁻(Y) satisfies it when Y is surgery on a knot in S³ or large surgery on a link, and asked whether it holds for every rational homology sphere Y. We show that the answer is no. For the Brieskorn sphere Y = Σ(30,47,83), with either orientation, the reduced Heegaard Floer homology is T_16 ⊕ T_14^10 ⊕ T_13^16 ⊕ ⋯ ⊕ T_1^92, where T_k = F[U]/U^k. So ℓ = 16, but F[U]/U^15 is not a direct summand of HF⁻(Y). The proof combines the Ozsváth–Szabó description of HF⁺ for plumbed manifolds, Némethi's reduction to the graded root of an explicit function τ, and an exact computer calculation. We give a certificate, the 1707 turning points of τ, from which the summand lengths can be rechecked by a short program. The manifold Y is an irreducible integral homology sphere, is not an L-space, and satisfies Lin's weaker restriction. It also shows that the word "large" cannot be removed from the link-surgery theorem of Alfieri and Binns. Computer searches with two independent programs show that among Seifert fibred integral homology spheres Σ(a_1,…,a_n), the counterexamples with the smallest product a_1⋯a_n are Σ(30,47,83) and three spheres with four singular fibres, all of product 117030. This is an unrefereed note.