Jonathan A. Hillman

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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)
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 [3]. In [7](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)
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)
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 [3] we showed that π = S 3 * Z/2Z S 3 satisfies the criterion of [5] and thus is the fundamental group of a P D 3-complex. As π has infinitely many ends but is(More)
We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L; ; ]), where L = Z 2 (X) 3 (X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, is composition with the Hopf map and ] is the orbit of the homotopy class of the longitude in 4 (X) under the group of self(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)
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 show that the isotopy type of a 1-simple n-knot K is determined by the Postnikov (n ?1)-stage of its exterior X(K), together with the homotopy class of the longitude K 2 n (X(K)). Moreover any pair (X; j) where X is a 4-dimensional homology circle with 1 (X) = Z and j : S 4 S 1 ! X is a map such that (X; j) = (MCyl(j); S 4 S 1) is an orientable PD 6-pair(More)