[Paper Review] On the non-existence of L-space surgery structure

Motoo Tange|ArXiv.org|Jul 2, 2007

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper proves that the reversed Poincaré homology sphere, $\overline{\Sigma(2,3,5)}$, does not admit any positive integral Dehn surgery yielding a lens space. Using Ozsváth and Szabó's contact invariant and the non-existence of tight contact structures on $\overline{\Sigma(2,3,5)}$ established by Etnyre and Honda, the author shows that the absence of tight contact structures obstructs the existence of positive L-space surgery structures, leading to a non-existence result for lens space surgeries on this manifold.

ABSTRACT

We exhibit homology spheres which never yield lens spaces by any integral Dehn surgery by using Ozsvath Szabo's contact invariant.

Motivation & Objective

• To investigate whether certain homology spheres, particularly $\overline{\Sigma(2,3,5)}$, can yield lens spaces via integral Dehn surgery.
• To explore the relationship between the existence of tight contact structures and the possibility of L-space surgeries on homology spheres.
• To extend results on lens space surgery to L-space homology spheres by analyzing the role of contact invariants and knot fibrations.
• To provide a topological obstruction to positive L-space surgery using the non-vanishing of the Ozsváth-Szabó contact invariant.

Proposed method

• Utilizes Ozsváth and Szabó's contact invariant $c(\xi)$ to detect tight contact structures on 3-manifolds.
• Applies the surgery exact triangle in Heegaard Floer homology to relate $HF^+$ groups of $Y$, $Y_0(K)$, and $Y_p(K)$.
• Uses the fact that $\widehat{HFK}(Y,K,g) \cong \mathbb{Z}_2$ for fibered knots in L-space homology spheres to deduce fiberedness of $K$ via Ni's result.
• Relies on the correspondence between open book decompositions and contact structures via the Thurston-Winkelnkemper construction.
• Applies Theorem 3.1: if $c(\xi) \neq 0$, then $\xi$ is tight, so non-vanishing contact invariant implies existence of tight contact structure.
• Combines the non-existence of tight contact structures on $\overline{\Sigma(2,3,5)}$ (from Etnyre and Honda) with the contact invariant argument to rule out positive L-space surgeries.

Experimental results

Research questions

• RQ1 Can the reversed Poincaré homology sphere $\overline{\Sigma(2,3,5)}$ yield a lens space via positive integral Dehn surgery?
• RQ2 Is there a topological obstruction to positive L-space surgery on homology spheres that lack tight contact structures?
• RQ3 Does the non-vanishing of the Ozsváth-Szabó contact invariant imply the existence of a tight contact structure on a 3-manifold?
• RQ4 Can the contact invariant be used to rule out L-space surgery structures on homology spheres with trivial or non-tight contact structures?
• RQ5 Is there a general principle linking the existence of tight contact structures to the possibility of positive L-space surgeries?

Key findings

• The reversed Poincaré homology sphere $\overline{\Sigma(2,3,5)}$ does not admit any positive integral Dehn surgery that yields a lens space.
• The non-existence of tight contact structures on $\overline{\Sigma(2,3,5)}$ (as shown by Etnyre and Honda) implies that no positive L-space surgery structure can exist on this manifold.
• For any L-space homology sphere $Y$ that carries a positive proper L-surgery structure, $Y$ must admit a positive tight contact structure.
• The contact invariant $c(\xi)$ does not vanish for the contact structure induced by a fibered knot in an L-space homology sphere, implying tightness.
• The connected sum $\Sigma(2,3,5) \# \overline{\Sigma(2,3,5)}$ does not carry any proper L-surgery structure, positive or negative, due to the absence of tight contact structures.
• The author conjectures that for any Brieskorn homology sphere $\Sigma(p,q,r)$, $\Sigma(p,q,r)$ admits proper L-surgery structure, but $\overline{\Sigma(p,q,r)}$ does not.