Jonathan A. Hillman

Algebraic invariants of links

Abelian Covers: Links Homology and Duality in Covers Determinantal Invariants The Maximal Abelian Cover Sublinks and Other Abelian Covers Twisted Polynomial Invariants Applications: Special Cases and Symmetries: Knot Modules Links with Two Components Symmetries Singularities of Plane Curves Free Covers, Nilpotent Quotients and Completion: Free Covers Nilpotent Quotients Algebraic Closure Disc Links.

Elementary amenable groups of finite hirsch length are locally-finite by virtually-solvable

If G is an elementary amenable group of finite Hirsch length h , then the quotient of G by its maximal locally finite normal subgroup has a maximal solvable normal subgroup, of derived length and index bounded in terms of h . 1991 Mathematics subject classification (Amer. Math. Soc.): Primary 20 F 19; Secondary 20 F 38.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

We extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is aspherical and the group is virtually polyZ of Hirsch length 4. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): primary 57 N 13; secondary 57 M 20, 20 F 99.

Twisted Alexander polynomials of periodic knots

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings. Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartleys criterion does not apply.

On 4-manifolds homotopy equivalent to surface bundles over surfaces

Abstract We show that a closed 4-manifold is homotopy equivalent to the total space of a surface bundle over a surface if the obviously necessary conditions on the fundamental group and Euler characteristic hold. When the base is the 2-sphere we need also conditions on the characteristic classes of the manifold. (Our results are incomplete when the base is the projective plane.) In most cases we can show the manifold is s -cobordant to the total space of the bundle.

On reciprocality of twisted Alexander invariants

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 not reciprocal.

OnL2-homology and asphericity

We useL2 methods to show that if a group with a presentation of deficiency one is an extension ofZ by a finitely generated normal subgroup then the 2-complex corresponding to any presentation of optimal deficiency is aspherical and to prove a converse of the Cheeger-Gromov-Gottlieb theorem relating Euler characteristic and asphericity. These results are applied to the Whitehead conjecture, 4-manifolds and 2-knot groups.

High dimensional knot groups which are not two-knot groups

This paper presents three arguments, one involving orientability, and the others Milnor duality and, respectively, the injectivity of cup product into H 2 for an abelian group and free finite group actions on homotopy 3-spheres to show that there are high dimensional knot groups which are not the groups of knotted 2-spheres in S 4 , thus answering a question of Fox (“Some problems in knot theory”, Topology of 3-manifolds and related topics ”, 168–176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).

Strongly minimal PD4-complexes

Abstract We consider the homotopy types of PD 4 -complexes X with fundamental group π such that c . d . π = 2 and π has one end. Let β = β 2 ( π ; F 2 ) and w = w 1 ( X ) . Our main result is that (modulo two technical conditions on ( π , w ) ) there are at most 2 β orbits of k -invariants determining “strongly minimal” complexes (i.e., those with homotopy intersection pairing λ X trivial). The homotopy type of a PD 4 -complex X with π a PD 2 -group is determined by π , w , λ X and the v 2 -type of X . Our result also implies that Foxs 2-knot with metabelian group is determined up to homeomorphism by its group.

UNKNOTTING ORIENTABLE SURFACES IN THE 4-SPHERE

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an ...