[Paper Review] Seifert fibered four-manifolds with nonzero Seiberg-Witten invariant

Chen Weimin|arXiv (Cornell University)|Mar 29, 2011

Geometric and Algebraic Topology21 references3 citations

TL;DR

This paper establishes that a Seifert fibered 4-manifold with nonzero Seiberg-Witten invariant must have a regular fiber whose homotopy class generates an infinite cyclic subgroup in the fundamental group. This topological obstruction prevents smooth circle actions unless the fiber class has infinite order, and the authors use knot surgery to destroy such actions while preserving homology and Seiberg-Witten invariants.

ABSTRACT

The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as applications, we show how to destroy smooth circle actions on a 4-manifold by knot surgery, without changing the integral homology, intersection form, and even the Seiberg-Witten invariant. Results concerning classification of Seifert fibered complex surfaces or symplectic 4-manifolds are included. We also show that every smooth circle action on the 4-torus is smoothly conjugate to a linear action.

Motivation & Objective

• To identify topological obstructions to smooth circle actions on 4-manifolds via Seiberg-Witten invariants.
• To determine when a Seifert fibered 4-manifold can support a smooth circle action, particularly in relation to the homotopy class of regular fibers.
• To show that knot surgery can destroy smooth circle actions on 4-manifolds while preserving integral homology, intersection form, and Seiberg-Witten invariants.
• To classify Seifert fibered complex surfaces and symplectic 4-manifolds using Seiberg-Witten invariants and orbifold topology.
• To prove that every smooth circle action on the 4-torus is smoothly conjugate to a linear action.

Proposed method

• Use the de-singularization formula for Seiberg-Witten invariants of 3-orbifolds to relate the nonvanishing of the invariant on the 4-manifold to the base 3-orbifold.
• Construct a finite manifold cover of the base 3-orbifold to lift the Seifert fibration to a circle bundle over a 3-manifold.
• Apply the Hurwitz homomorphism to analyze the image of π₂ in H₂, showing it is finite when b₂⁺ > 1.
• Use the Loop Theorem and compressing disks to derive contradictions when the regular fiber has finite order.
• Perform knot surgery on a 2-torus in the 4-manifold, replacing a tubular neighborhood with (S³ ∖ Nd(K)) × S¹.
• Analyze the resulting fundamental group via amalgamated free product structure, showing the center becomes trivial after surgery.

Experimental results

Research questions

• RQ1 What topological condition must be satisfied by the homotopy class of a regular fiber in a Seifert fibered 4-manifold with nonzero Seiberg-Witten invariant?
• RQ2 Can smooth circle actions on 4-manifolds be destroyed via knot surgery without altering the Seiberg-Witten invariant or integral homology?
• RQ3 Which complex surfaces or symplectic 4-manifolds can admit a Seifert fibration, and what are the obstructions?
• RQ4 Under what conditions is a 4-manifold with infinite center in its fundamental group and nonzero Seiberg-Witten invariant necessarily Seifert fibered?
• RQ5 Is every smooth circle action on the 4-torus smoothly conjugate to a linear action?

Key findings

• The homotopy class of a regular fiber in a Seifert fibered 4-manifold with nonzero Seiberg-Witten invariant must have infinite order.
• If b₂⁺ > 1, the image of the Hurwitz homomorphism π₂(X) → H₂(X) is finite.
• Knot surgery on a 2-torus with non-torus knot produces a 4-manifold with trivial center in the fundamental group, thus no smooth circle action.
• The Seiberg-Witten invariant remains nonzero after knot surgery, provided the original invariant is nonzero and the knot has trivial Alexander polynomial.
• A 4-manifold with z(π₁) = ℤ² and π₂ ≠ 0 is Seifert fibered only if it is diffeomorphic to T² × S²; otherwise, it may not be Seifert fibered.
• Every smooth circle action on the 4-torus is smoothly conjugate to a linear action.