Contact surgery numbers of projective spaces

We classify all contact projective spaces with contact surgery number one. In particular, this implies that there exist infinitely many non-isotopic contact structures on the real projective 3-space which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. Large parts of our proofs deal with a detailed analysis of Gompf’s $Γ$-invariant of tangential 2-plane fields on 3-manifolds. From our main result we also deduce that the $Γ$-invariant of a tangential 2-plane field on the real projective 3-space only depends on its $d_3$-invariant.

💡 Research Summary

The paper “Contact surgery numbers of projective spaces” by Marc Kegel and Monika Yadav investigates the minimal number of Legendrian components required to obtain a given contact 3‑manifold from the standard tight contact 3‑sphere (S³, ξ_st) via rational contact surgery. This minimal number is called the contact surgery number cs(M, ξ). The authors focus on the real projective space RP³ and aim to classify all contact structures on RP³ whose contact surgery number equals one.

The starting point is the Ding–Geiges theorem, which guarantees that any closed, co‑oriented, positive contact 3‑manifold can be obtained from (S³, ξ_st) by contact surgery with coefficients ±1. Consequently, cs(M, ξ) measures the smallest number of Legendrian knots needed when one allows arbitrary non‑zero rational coefficients. The authors also consider variants cs_Z, cs_{1/Z}, and cs_{±1} where the coefficients are required to be integers, reciprocals of integers, or exactly ±1, respectively.

A crucial topological observation (Lemma 4.1, a corollary of KMOS07) is that a rational surgery on a knot K ⊂ S³ yields RP³ if and only if K is the unknot and the surgery coefficient is of the form 2·2ⁿ+1 for some integer n. This implies that any contact structure on RP³ with cs = 1 must be produced by a single Legendrian unknot with an appropriate contact surgery coefficient. By varying n over ℤ and using the classification of Legendrian unknots (Etnyre–Furuta, Eliashberg–Fraser), the authors generate a complete list of possible surgery diagrams.

For each diagram they compute two homotopical invariants of the underlying tangential 2‑plane field: the d₃‑invariant and Gompf’s Γ‑invariant. The d₃‑invariant is obtained from a generalized linking matrix Q (Lemma 3.1) together with the Thurston–Bennequin numbers t_i, rotation numbers r_i, and the surgery coefficients. The Γ‑invariant, which lives in H₁(RP³) ≅ ℤ₂, is computed from a characteristic sublink representing a spin structure (Lemma 3.2) and its behavior under Kirby moves (Lemma 3.3).

The main classification (Theorem 1.1) can be summarized as follows:

1. Every contact structure on RP³ satisfies cs_{±1} ≤ 3.
2. The unique tight structure ξ_st has cs_{±1}=cs_{1/Z}=cs_Z=cs=1, and its pair of invariants is (Γ, d₃) = (0, 1 + ¼).
3. An overtwisted structure has cs_{±1}=1 if and only if (Γ, d₃) equals (0, 1 + ¼) or (1, ¾).
4. cs_Z=1 occurs for the two pairs above and additionally for infinitely many families indexed by an integer m ≤ −1, giving four explicit formulas for (Γ, d₃).
5. cs=1 is realized exactly by the eleven cases listed in Table 2 of the paper.

From these results the authors deduce several corollaries. Corollary 1.2 shows that for each value of Γ∈ℤ₂ there are infinitely many overtwisted contact structures with cs = 2, demonstrating that a single rational surgery is not sufficient for many structures. Corollaries 1.3 and 1.4 give a precise description of the Legendrian unknot (its tb and rot) and the surgery coefficient that produce each cs=1 structure. In particular, the unknot with tb = −3, rot = 0 and a +1 surgery yields the overtwisted structure with (Γ, d₃) = (0, 1 + ¼), while tb = −3, rot = ±2 and a +1 surgery gives (Γ, d₃) = (1, ¾).

The most striking consequence is Corollary 1.5: on RP³ the Γ‑invariant is completely determined by the d₃‑invariant. Specifically, d₃ always lies in ℤ + ¼ or ℤ + ¾; the former corresponds to Γ = 0 and the latter to Γ = 1. Moreover, for any admissible pair (Γ, d₃) there exists a unique (up to homotopy) tangential 2‑plane field realizing it. This answers a natural question about the relationship between these two invariants and suggests a broader phenomenon. The authors pose Question 1.6, asking for which other rational homology 3‑spheres the Γ‑invariant is likewise determined by d₃.

Methodologically, the paper combines careful Kirby calculus (blow‑up/down, handle slides, Rolfsen twists) with explicit homological algebra to track spin structures and characteristic sublinks. The authors adopt a normalization of the d₃‑invariant that sets d₃(S³, ξ_st)=0, making the invariant additive under connected sum and integral on homology spheres. This choice aligns with recent literature (e.g., CEK24, EKO23) and simplifies comparisons across different surgery diagrams.

In summary, Kegel and Yadav provide a complete classification of contact structures on RP³ with contact surgery number one, compute all associated homotopical invariants, and reveal a surprising rigidity: the Γ‑invariant is a function of d₃. Their work not only settles the specific case of RP³ but also opens a line of inquiry into the interplay between Γ and d₃ on other rational homology spheres, potentially influencing future studies of contact surgery complexity and the homotopy classification of plane fields.