[Paper Review] Dehn surgery, the fundamental group and SU(2)

P. B. Kronheimer, Tomasz Mrowka|ArXiv.org|Dec 17, 2003

Geometric and Algebraic Topology10 references17 citations

TL;DR

This paper proves that for any non-trivial knot in the 3-sphere, Dehn surgery with coefficient |r| ≤ 2 yields a 3-manifold whose fundamental group admits a non-cyclic homomorphism to SU(2). The proof uses holonomy perturbations of the Chern-Simons functional and gauge-theoretic techniques, avoiding instanton Floer homology, to show that such manifolds cannot have cyclic fundamental group—providing an independent verification of Property P for surgeries with |r| ≤ 2.

ABSTRACT

Let K be a non-trivial knot in the 3-sphere and let Y(r) be the 3-manifold obtained by surgery on K with surgery-coefficient a rational number r. We show that there is a homomorphism from the fundamental group of Y(r) to SU(2) with non-cyclic image if r is less than or equal to 2.

Motivation & Objective

• To establish that Dehn surgery on a non-trivial knot with |r| ≤ 2 yields a 3-manifold with non-cyclic fundamental group.
• To demonstrate the existence of a non-cyclic homomorphism from π₁(Y_r) to SU(2) for such surgeries, independent of the cyclic surgery theorem.
• To provide a gauge-theoretic proof of Property P for surgeries with |r| ≤ 2, avoiding reliance on instanton Floer homology.
• To clarify the distinction between cyclic fundamental groups and the absence of non-abelian SU(2) representations in Dehn surgeries.

Proposed method

• Applies holonomy perturbations to the Chern-Simons functional on SU(2) connections over 3-manifolds, using a tubular neighborhood of a knot to define perturbation functions.
• Constructs a perturbed Chern-Simons functional Φ(A) = ∫_D φ(Hol_γ_z(A)) μ(z) using a class function φ on SU(2) and a 2-form μ with integral 1.
• Analyzes critical points of the perturbed functional CS + Φ, showing that flat connections with non-cyclic monodromy correspond to non-trivial SU(2) representations.
• Uses Uhlenbeck compactness and perturbation theory to define Donaldson invariants on 4-manifolds obtained by doubling the surgery manifold.
• Applies a result from [13] that if the representation variety R^w_ι,φ(Y₀) is empty, then the Donaldson invariants vanish.
• Combines this with a symplectic 4-manifold construction from [17] and Witten's conjecture to derive a contradiction if the fundamental group is cyclic.

Experimental results

Research questions

• RQ1 Does Dehn surgery on a non-trivial knot with |r| ≤ 2 produce a 3-manifold with cyclic fundamental group?
• RQ2 Can the fundamental group of such a surgery manifold admit a non-cyclic homomorphism to SU(2)?
• RQ3 Is there a gauge-theoretic proof of Property P for surgeries with |r| ≤ 2 that avoids instanton Floer homology?
• RQ4 What is the relationship between the fundamental group being cyclic and the existence of non-abelian SU(2) representations in Dehn surgeries?

Key findings

• For any non-trivial knot K in S³, Dehn surgery with |r| ≤ 2 yields a 3-manifold Y_r whose fundamental group π₁(Y_r) admits a non-cyclic homomorphism to SU(2).
• The representation variety R^w_ι,φ(Y₀) is empty for zero-surgery Y₀ if and only if the Donaldson invariants of the doubled 4-manifold vanish.
• The proof shows that if π₁(Y_r) were cyclic for |r| ≤ 2, then the Donaldson invariants would vanish, contradicting the non-triviality of Seiberg-Witten invariants in symplectic 4-manifolds.
• The result provides an independent verification of Property P for surgeries with |r| ≤ 2, without relying on the cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen.
• The example of the (−2,3,7) pretzel knot shows that surgeries with |r| > 2 can yield manifolds with non-cyclic fundamental group but no non-abelian SU(2) representations.
• The paper establishes that the absence of non-abelian SU(2) representations is strictly stronger than having a cyclic fundamental group in Dehn surgeries.