Stefano Vidussi

A SURVEY OF TWISTED ALEXANDER POLYNOMIALS

We give a short introduction to the theory of twisted Alexander polynomials of a 3-manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.

A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds

In this paper we show that given any 3-manifold N and any non-fibered class in H^1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. This is obtained by extending earlier work of the authors, together with results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.

Twisted Alexander polynomials and symplectic structures

Let <i>N</i> be a closed, oriented 3-manifold. A folklore conjecture states that <i>S</i>¹ × <i>N</i> admits a symplectic structure only if <i>N</i> admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of <i>N</i>, 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 <i>S</i>¹ × <i>N</i>. As an application of these results we will show that <i>S</i>¹ × <i>N</i>(<i>P</i>) does not admit a symplectic structure, where <i>N</i>(<i>P</i>) is the 0-surgery along the pretzel knot <i>P</i> = (5, -3, 5), answering a question of Peter Kronheimer.

Symplectic

Let be a closed, oriented -manifold. A folklore conjecture states that admits a symplectic structure if and only if admits a fibration over the circle. We will prove this conjecture in the case when is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes -manifolds with vanishing Thurston norm, graph manifolds and -manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic -manifolds). Our result covers, in particular, the case of 0-framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the -manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered.

\mathbf{S}^{1} \times N^3

Let M be a 4-manifold with residually finite fundamental group G having b1(G) > 0. Assume that M carries a symplectic structure with trivial canonical class K =0 ∈ H 2 (M). Using a theorem of Bauer and Li, together with some classical results in 4-manifold topology, we show that for a large class of groups M is determined up to homotopy and, in favorable circumstances, up to homeomorphism by its fundamental group. This is analogous to what was proved by Morgan‐

, subgroup separability, and vanishing Thurston norm

In this note we prove that, for any n2 N, there exist a smooth 4{manifold, homotopic to a K3 surface, dened by applying the link surgery method of Fintushel{Stern to a certain 2{component graph link, which admits n inequivalent symplectic structures.

On the topology of symplectic Calabi–Yau 4-manifolds

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.

Homotopy K3's with several symplectic structures

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1) K , obtained by knot surgery on the rational elliptic surface E(1) = P 2 #9P 2 with the left-handed trefoil knot K. E(1) K has the simplest homotopy type among simply-connected symplectic 4-manifolds known to exhibit such a property.

Homologous non-isotopic symplectic surfaces of higher genus

In this paper we prove that simple type four manifolds with b+2 > 1 which are diffeomorphic outside a point or outside a wedge of circles have the same Seiberg-Witten invariants, excluding the use of these invariants to detect eventual inequivalent smooth structures.

Symplectic tori in homotopy E(1)'S

It follows from earlier work of Silver and Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopflink. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split, and whether it is totally split.