[Paper Review] Symplectic 4--manifolds with K = 0 and the Lubotzky alternative

Stefan Friedl, Stefano Vidussi|arXiv (Cornell University)|Feb 4, 2011

Geometric and Algebraic Topology20 references3 citations

TL;DR

This paper classifies symplectic 4-manifolds with trivial canonical class (K=0) that admit a free circle action, using the Lubotzky alternative for linear groups. It proves that such manifolds are either torus bundles over tori or finite covers of such bundles, confirming a conjecture for this class of manifolds and extending prior results on product bundles and vanishing Thurston norm.

ABSTRACT

In this paper we use the Lubotzky alternative for finitely generated linear groups to determine which 4-manifolds admitting a free circle action can be endowed with a symplectic structure with trivial canonical class. The content of this paper partly overlaps with the content of the unpublished preprint "Symplectic 4-manifolds with a free circle action" (arXiv:0801.1313 [math.GT]).

Motivation & Objective

• To determine which 4-manifolds admitting a free circle action can support a symplectic structure with trivial canonical class.
• To extend previous results on product bundles (S¹×N) with K=0 to non-trivial circle bundles via new topological tools.
• To use the Lubotzky alternative for finitely generated linear groups to constrain the topology of the orbit space N.
• To prove that for such manifolds, the orbit space N must be a torus bundle over S¹, and M a torus bundle over a torus, except possibly when b₁(N)=1 and e is torsion.
• To provide evidence toward the conjecture that all symplectic 4-manifolds with K=0 are covered by torus bundles over tori.

Proposed method

• Apply the Gysin sequence to relate Betti numbers of M and its orbit space N, using the non-torsion Euler class to derive b₂⁺(M) = b₁(N) - 1.
• Use Taubes' Seiberg-Witten constraints for symplectic 4-manifolds with K=0 to deduce SW_M(0) = 1 and K=0 as the only basic class.
• Employ the Lubotzky alternative to bound the virtual first Betti number vb₁(N; ℤ) ≤ 3 when e is non-torsion.
• For torsion Euler class, construct a regular cover N_Γ → N via the torsion subgroup of H₁(N), showing that the pullback M_Γ is a product S¹ × N_Γ.
• Leverage the result from [FV08a] that S¹ × N_Γ symplectic with K=0 implies N_Γ is a torus bundle over S¹.
• Use the fact that if (N_Γ, φ_Γ) is a torus bundle for all φ_Γ, then (N, φ) is a torus bundle for any non-trivial φ ∈ H¹(N; ℤ).

Experimental results

Research questions

• RQ1 Which 4-manifolds with a free circle action can admit a symplectic structure with trivial canonical class?
• RQ2 Can the refined adjunction inequality approach used for product bundles (S¹×N) be extended to non-trivial circle bundles?
• RQ3 How does the Lubotzky alternative constrain the virtual Betti numbers of 3-manifolds that are orbit spaces of such symplectic 4-manifolds?
• RQ4 What is the topological structure of the orbit space N when the Euler class e is torsion or non-torsion?
• RQ5 Is the list of known symplectic 4-manifolds with K=0 complete for the class of manifolds with free circle actions?

Key findings

• For symplectic 4-manifolds M with K=0 and a free circle action with non-torsion Euler class e, the orbit space N is a torus bundle over S¹, and M is a torus bundle over a torus.
• When the Euler class e is torsion and b₁(N) > 1 or e = 0, the manifold M is again a torus bundle over a torus.
• The virtual first Betti number vb₁(N; ℤ) is bounded by 3 when e is non-torsion, a consequence of the Lubotzky alternative and Seiberg-Witten invariants.
• In the torsion Euler class case, the finite cover M_Γ is a product S¹ × N_Γ, and N_Γ is a torus bundle, implying N itself is a torus bundle over S¹.
• All such manifolds are finitely covered by torus bundles over tori, supporting the conjecture that this class exhausts all symplectic 4-manifolds with K=0.
• The results confirm that the known list of symplectic 4-manifolds with K=0—K3, torus bundles over tori, and S¹-bundles over T²-bundles—is complete for this class.