Murasugi found two criteria that must be satisfied by the Alexan-der polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Basic examples demonstrate the application of these new criteria. More delicate examples indicate their applicability to knots with trivial Alexander polynomial, including the two such knots… (More)
Chapter 2. Homology and Duality in Covers 27 2.1. Homology and cohomology with local coefficients 27 2.2. Covers of link exteriors 28 2.3. Some terminology and notation 30 2.4. Poincaré duality and the Blanchfield pairings 30 2.5. The total linking number cover 33 2.6. The maximal abelian cover 35 2.
Given a knot and an SL n C representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL 3 Z representation that is not conjugate to its dual for which the twisted Reidemeister torsion is… (More)
We show that although closed SL × E n-manifolds do not admit met-rics of nonpositive sectional curvature, the arguments of Farrell and Jones can be extended to show that such manifolds are topologically rigid, if n ≥ 2. Smooth manifolds with Riemannian metrics of nonpositive curvature are topologically rigid, by the work of Farrell and Jones . In … (More)
We show that there are two homotopy types of P D 3-complexes with fundamental group S 3 * Z/2Z S 3 , and give explicit constructions for each, which differ only in the attachment of the top cell. In  we showed that π = S 3 * Z/2Z S 3 satisfies the criterion of  and thus is the fundamental group of a P D 3-complex. As π has infinitely many ends but is… (More)
We show that if X is an indecomposable P D 3-complex and π 1 (X) is the fundamental group of a reduced finite graph of finite groups but is neither Z nor Z ⊕ Z/2Z then X is orientable, the underlying graph is a tree, the vertex groups have cohomolog-ical period dividing 4 and all but at most one of the edge groups is Z/2Z. If there are no exceptions then… (More)
This paper is a synthesis and extension of three earlier papers on P D 4-complexes X with fundamental group π such that c.d.π = 2 and π has one end. Our goal is to show that the homotopy types of such complexes are determined by π, the Stiefel-Whitney classes and the equivariant intersection pairing on π 2 (X). We achieve this under further conditions on π.
We state a number of open questions on 3-dimensional Poincaré duality groups and their subgroups, motivated by considerations from 3-manifold topology. The notion of Poincaré duality group of dimension n (or P D n-group, for short) is essentially an algebraic analogue of the notion of aspherical n-manifold. When n = 1 or 2 the analogy is precise; the only… (More)
A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an auto-morphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is… (More)
Let Γ be a finitely presentable prop group with a nontrivial finitely generated closed normal subgroup N of infinite index. Then def(Γ) ≤ 1, and if def(Γ) = 1 then Γ is a prop duality group of dimension 2, N is a free prop group and Γ/N is virtually free. In particular, if the centre of Γ is nontrivial and def(Γ) ≥ 1, then def(Γ) = 1, cd G ≤ 2 and Γ is… (More)