[Paper Review] Twisted Alexander polynomials and symplectic structures

Stefan Friedl, Stefano Vidussi|arXiv (Cornell University)|Apr 18, 2006

Geometric and Algebraic Topology4 citations

TL;DR

This paper provides strong evidence for the conjecture that $S^1 \times N$ admits a symplectic structure only if $N$ fibers over the circle. By analyzing twisted Alexander polynomials associated to finite group epimorphisms of $\pi_1(N)$, the authors show that symplectic $S^1 \times N$ forces these polynomials to be monic and achieve maximal degree, a property characteristic of fibered 3-manifolds. The key result confirms that $S^1 \times N(P)$, where $N(P)$ is the 0-surgery on the pretzel knot $(5,-3,5)$, does not admit a symplectic structure.

ABSTRACT

Let N be a closed, oriented 3-manifold. A folklore conjecture states that S^1 x N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing that their behavior is the same as of those of fibered 3-manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S^1 x N. As an application of these results we will show that S^1 x N(P) does not admit a symplectic structure, where N(P) is the 0-surgery along the pretzel knot P = (5,-3,5), answering a question of Peter Kronheimer.

Motivation & Objective

• To provide evidence for the conjecture that $S^1 \times N$ is symplectic only if $N$ fibers over $S^1$.
• To use twisted Alexander polynomials to detect obstructions to symplectic structures on $S^1 \times N$.
• To extend Kronheimer's obstructions based on the ordinary Alexander polynomial to stronger invariants via twisted polynomials.
• To resolve a question of Peter Kronheimer on the symplectic nature of $S^1 \times N(P)$ for the pretzel knot $P = (5,-3,5)$.

Proposed method

• Use twisted Alexander polynomials $\Delta_{N,\phi}^\alpha$ associated to epimorphisms $\alpha: \pi_1(N) \to G$ for finite groups $G$.
• Apply the Meng-Taubes relation between Seiberg-Witten invariants and Alexander polynomials to translate Taubes' constraints on symplectic 4-manifolds into polynomial conditions.
• Establish that for symplectic $S^1 \times N$, the twisted polynomial $\Delta_{N,\phi}^\alpha$ must be monic and achieve degree $|G|\|\phi\|_T + 2\,\text{div}\,\phi_G$, matching the fibered case.
• Use Mayer-Vietoris sequences with $\mathbb{Z}[G]$-coefficients to relate homology of surfaces and manifolds, proving isomorphism of $H_1^\alpha$ groups.
• Apply the universal coefficient theorem and $p$-adic reduction to show that $H_1^\alpha(S;\mathbb{Z}[G]) \cong H_1^\alpha(M;\mathbb{Z}[G])$ under the degree equality.
• Prove that the inclusion-induced map $i_+: H_1^\alpha(S;\mathbb{Z}[G]) \to H_1^\alpha(M;\mathbb{Z}[G])$ is an isomorphism, implying surjectivity of $\pi_1(S) \to \pi_1(M)$ under conjectural conditions.

Experimental results

Research questions

• RQ1 Does $S^1 \times N$ admit a symplectic structure only if $N$ fibers over $S^1$?
• RQ2 Can twisted Alexander polynomials detect symplectic obstructions beyond the ordinary Alexander polynomial?
• RQ3 Is $S^1 \times N(P)$ symplectic for the pretzel knot $P = (5,-3,5)$, where Kronheimer's obstructions do not apply?
• RQ4 Do twisted Alexander polynomials of $S^1 \times N$ satisfy the same degree and monicity conditions as in the fibered case?
• RQ5 Can the isomorphism of $H_1^\alpha$ groups for all finite quotients imply fibration of $N$?

Key findings

• The twisted Alexander polynomial $\Delta_{N,\phi}^\alpha$ is monic and achieves degree $|G|\|\phi\|_T + 2\,\text{div}\,\phi_G$ for any symplectic $S^1 \times N$, matching the fibered case.
• For the pretzel knot $P = (5,-3,5)$, $S^1 \times N(P)$ does not admit a symplectic structure, resolving a question of Kronheimer.
• The degree bound $\deg \Delta_{N,\phi}^\alpha \leq |G|\|\phi\|_T + 2\,\text{div}\,\phi_G$ generalizes McMullen's inequality for the ordinary Alexander polynomial.
• The isomorphism $H_1^\alpha(S;\mathbb{Z}[G]) \cong H_1^\alpha(M;\mathbb{Z}[G])$ holds under the symplectic assumption, implying $\det(i_+) = 1$.
• The conjecture that $i_+: \pi_1(S) \to \pi_1(M)$ is surjective if $H_1^\alpha(S;\mathbb{Z}[G]) \to H_1^\alpha(M;\mathbb{Z}[G])$ is an isomorphism for all finite $G$ would imply the main conjecture.
• For all knots with up to 12 crossings, Conjecture 1.1 holds: $S^1 \times N(K)$ is symplectic only if $K$ is fibered.