$PD_4$-complexes and 2-dimensional duality groups

Abstract

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.1991 Mathematics Subject Classification. 57P10. Key words and phrases. cohomological dimension, homotopy intersection, k-invariant, P D 4complex.1 2 JONATHAN A. HILLMAN for π a free group, a surface group, a semidirect product F (r) ⋊ Z or a solvable Baumslag-Solitar group Z * m .We shall now outline the paper in more detail. The first two sections are algebraic. In particular, Theorem 3 (in §2) establishes a connection between hermitean pairings and the Whitehead quadratic functor Γ W . Sections 3-8 consider the homotopy classification of P D 4 -complexes, and introduce several notions of minimality. The first main result is Theorem 19 in §7, where it is shown that two P D 4 -complexes with the same strongly minimal model and ±isometric intersection pairings are homotopy equivalent, provided w : π → Z × does not split. Sections 9 and 10 determine the strongly minimal P D 4 -complexes with π 2 = 0 and for which π has finitely many ends. Strongly minimal P D 4 -complexes with π a semidirect product ν ⋊ Z (with ν finitely presentable) are shown to be mapping tori in §11. When ν is a free group the homotopy type of such a mapping torus is determined by π and the Stiefel-Whitney classes, by Theorem 36. The next five sections lead to the second main result, Theorem 45 (in §16), which extends the result of Theorem 36 to the case when π has one end and c.This theorem is modelled on the much simpler case analyzed in §14, in which π is a P D 2 -group. Apart from the notion of minimality, the main technical points are the connection between hermitean pairings and Γ W , the fact that a certain "cup product" defines an isomorphism, and the 2-torsion condition. In [39], we showed that the cup-product condition held for surface groups, torus knot groups and solvable Baumslag-Solitar groups. Here we show that it holds for all finitely presentable groups π with one end and c.d.π = 2 (Theorem 43). The 2-torsion condition is only known for π a P D 2 -group or π a solvable Baumslag-Solitar group (Theorem 49), and does not hold for all the cases covered by Theorem 36. The final section considers the classification up to TOP s-cobordism or homeomorphism of closed 4-manifolds with groups as in Theorem 45. In particular, it is shown that a remarkable 2-knot discovered by Fox is determined up to TOP isotopy and reflection by its knot group.The theme of Hambleton, Kreck and Teichner [28] is close to ours, although their methods are very different. They use Kreck's modified surgery theory to classify up to s-cobordism closed orientable 4-manifolds with fundamental groups of geometric dimension 2 (subject to some K-and L-theoretic hypotheses), and they show also that every automorphism ...

Homotopy classification of $$\textit{PD}_4$$ PD 4 -complexes relative an order relation

“…Remark 5.3. We should point out that our result gives a classification over the complex P , whereas Baues and Bleile [2] give classification result over Bπ and Hillman [6,8] gives a classification result over the strongly minimal model.…”

“…These two notions of minimality coincide whenever the cohomological dimension of the fundamental group is less than or equal to 2 (see for example [8,Theorem 25]). All known examples of strongly minimal models are P D 4 -complexes with such fundamental groups ( [6,7,8]). Therefore one might consider the following natural question: Problem 1.4.…”

“…Hillman [6,8] gives a homotopy classification for P D 4 -complexes over the strongly minimal models subject to a k-invariant constraint. He considers the same obstruction as in the proof of our main result Theorem 5.2.…”

Solvable normal subgroups of 2-knot groups

“…It should extend to all knot groups π with g.d.π = 2, but at present is incomplete; there is a 2-torsion condition which holds for Φ, but is not otherwise easily verified. (See [13] for a much expanded exposition of [12] and earlier work.) When g.d.π = 2 and π ′ is finitely generated there is a quite different argument.…”