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We classify the metabelian unitary representations of π 1 (M K), where M K is the result of zero-surgery along a knot K ⊂ S 3. We show that certain eta invariants associated to metabelian representations π 1 (M K) → U (k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that this result(More)
In [11], two of us constructed a closed oriented 4-dimensional manifold with fundamental group Z that does not split off S 1 × S 3. In this note we show that this 4-manifold, and various others derived from it, do not admit smooth structures. Moreover, we find an infinite family of 4-manifolds with exactly the same properties. As a corollary, we obtain(More)
Let N be a closed irreducible 3–manifold such that π 1 (N) is virtually RFRS. Using work of Ian Agol we will improve, for all but finitely many S 1 –bundles M over N , the adjunction inequality for the minimal complexity of embedded surfaces. Appealing to a result of Dani Wise, this allows us to completely determine the minimal complexity of embedded(More)
Every element in the first cohomology group of a 3–manifold is dual to embedded surfaces. The Thurston norm measures the minimal 'complexity' of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3–sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm,(More)
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.(More)
Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 × 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–(More)
T T T T T T T T T T T T T T T Abstract In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z(More)