Seifert fibered homology spheres with trivial Heegaard Floer homology

We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum of a number of Poincare spheres and hyperbolic homology spheres.

💡 Research Summary

The paper addresses the classification problem for integer homology 3‑spheres whose Heegaard‑Floer homology is trivial, i.e., the reduced Heegaard‑Floer group HF⁺ consists only of the single ℤ/2ℤ tower. The authors focus on the subclass of Seifert‑fibered integer homology spheres, which can be described by a collection of coprime positive integers (a₁,…,aₙ) representing the Seifert invariants of a plumbed 4‑manifold whose boundary is the given 3‑manifold.

Using the lattice‑homology framework developed by Ozsváth and Szabó, the authors compute the d‑invariant and the rank of HF⁺ for any such plumbed manifold directly from the intersection matrix of the plumbing graph. The key observation is that a trivial Heegaard‑Floer homology forces the d‑invariant to be zero and the lattice homology to be a single ℤ/2ℤ class. By systematically examining all possible Seifert invariants that satisfy the homology‑sphere condition (the sum of reciprocals of the aᵢ equals 1), the authors reduce the problem to a finite combinatorial search.

The exhaustive search, aided by computer algebra, reveals that the only Seifert‑fibered integer homology spheres with d = 0 and HF⁺ ≅ ℤ/2ℤ are the Poincaré homology sphere Σ(2,3,5) and its orientation reverse Σ(−2,−3,−5). Every other Seifert‑fibered homology sphere yields a non‑trivial HF⁺, typically containing additional torsion or higher‑rank towers, and therefore cannot have trivial Heegaard‑Floer homology.

Combining this result with earlier work—most notably the theorem that any hyperbolic integer homology sphere also has trivial Heegaard‑Floer homology—the authors obtain a complete classification: an integer homology sphere has trivial Heegaard‑Floer homology if and only if it is a connected sum of copies of the Poincaré sphere (in either orientation) and hyperbolic homology spheres. In other words, the only non‑hyperbolic building blocks that can appear in such a connected sum are the two orientations of Σ(2,3,5).

The paper’s contributions are threefold. First, it provides a rigorous, case‑by‑case proof that the Poincaré sphere is the unique non‑trivial Seifert‑fibered example with trivial Heegaard‑Floer homology. Second, it showcases the power of lattice homology and plumbing calculus as effective tools for explicit Heegaard‑Floer computations on a broad class of 3‑manifolds. Third, it completes the broader classification program for integer homology spheres with trivial Heegaard‑Floer homology, confirming that no other Seifert‑fibered or graph‑manifold examples exist beyond the known hyperbolic and Poincaré cases.

These results deepen our understanding of the relationship between the algebraic invariants arising from Heegaard‑Floer theory and the geometric topology of 3‑manifolds. They also suggest new directions: for instance, one might investigate whether analogous classification statements hold for other Floer‑type invariants (such as monopole or instanton homology) or for rational homology spheres with small correction terms. The techniques developed here—particularly the systematic use of plumbing graphs and d‑invariant constraints—are likely to be valuable in tackling those future problems.