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

Abstract

We define an order relation among oriented P D 4 -complexes. We show that with respect to this relation, two P D 4 -complexes over the same complex are homotopy equivalent if and only if there is an isometry between the second homology groups. We also consider minimal objects of this relation.Starting with a P D 4 -complex X, we also define a minimal P D 4 -complex P for X, called X-minimal, which is minimal with respect to the order relation ≻ (see Definition 3.1). Minimal P D 4 -complexes are also considered by Hillman ([6, 7, 8]) with special emphasis on a particular type of minimal P D 4 -complex, called a strongly minimal P D 4 -complex.Recall that for a P D 4 -complex X, the radical of the intersection form λ X , denoted by Rad(λ X ), is isomorphic to the module H 2 (π; Λ). 1 2 FRIEDRICH HEGENBARTH, MEHMETCİK PAMUK AND DUŠAN REPOVŠ Definition 1.2. A P D 4 -complex P is said to be strongly minimal if H 2 (P ; Z[π 1 (P )])/ Rad(λ P ) = 0.Remark 1.3. Obviously, if P is strongly minimal and X ≻ P , then P is X-minimal.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:Find examples of (strongly) minimal P D 4 -complexes whose fundamental group has cohomological dimension greater than 2.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. However, we point out that our method in this paper is different: to see that the obstruction vanishes Hillman realizes it by a self-equivalence, whereas we use a map A ′ to relate the obstruction to intersection forms and cap products. We also remove the hypothesis on the k-invariant.The outline of the paper is as follows: In Section two we list some of the immediate properties of the order relation ≻. In Section three, for a P D 4 -complex X, we define X-minimal P D 4 complexes. We show that if H 2 (X; Λ) is finitely generated, than such minimal complexes exist (Theorem 3.5). Section four is about Postnikov decomposition of the map f : X → P . In section five, we prove our main result: two P D 4 -complexes X and X ′ over the same minimal complex P are homotopy equivalent if and only if there is an isometry Φ : H 2 (X; Λ) → H 2 (X ′ ; Λ) (Theorem 5.2). Acknowledgements. The authors thank the referees for the clarifications of essential points of the paper and for several suggestions which led to a simplification of the proof of Theorem 5.2.In this section we will list some of the immediate properties of the above definition of the order relation ≻.(1) The relation ≻ is transitive and since Id : X → X realizes X ≻ X it is clear that ≻ is also reflexive.(2) The relation ≻ is symmetric in the sense of the following theorem:Theorem 2.1. If ...

$PD_4$-complexes and 2-dimensional duality groups

“…" in v2 of the present article (put on the arXiv on 8 October 2013). The final step is due to Hegenbarth, Pamuk and Repovš, who noted that Poincaré duality in Z may be used to establish an equivalent condition [30]. (This observation has been used in the current version of Lemma 16 above.…”