On minimal Poincaré $4$-complexes

Cavicchioli, Alberto; Hegenbarth, Friedrich; Emilia, Reggio

doi:10.3906/mat-1211-54

Cited by 5 publications

(13 citation statements)

References 16 publications

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“…( Y (3) )), this also implies that H 4 (Y α+β ; Z) ∼ = Z with a generator given by the top cell (see [3,Lemma 4.3]). We define the map…”

mentioning

confidence: 89%

“…Hence, up to homotopy equivalence E 3 can be constructed from X by attaching cells of dimension ≥ 4, so X (3) = (E 3 ) (3) . In fact, this is the way E 3 is constructed.…”

Section: Minimal P D 4 -Complexesmentioning

confidence: 99%

“…Proof. We write P = P (3) ∪ ψ D 4 and we have the following isomorphisms f * : H 4 (X, X (3) ; Λ) ∼ = / / H 4 (P, P (3) ; Λ), f ′ * : H 4 (X ′ , X ′(3) ; Λ) ∼ = / / H 4 (P, P (3) ; Λ) by the degree-1 property of f and f ′ , respectively. Consider the diagram π 4 (X, X (3) ) = H 4 (X, X (3) ; Λ)…”

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confidence: 99%

“…, where ϕ : S 3 → X (3) and 3) are the attaching maps of the 4-cells [12,Theorem 2.4]. For simplicity, we denote ϕ(S 3 ) = ∂D 4 and ϕ ′ (S 3 ) = ∂D ′4 .…”

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confidence: 99%

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

2014

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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 ...

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“…( Y (3) )), this also implies that H 4 (Y α+β ; Z) ∼ = Z with a generator given by the top cell (see [3,Lemma 4.3]). We define the map…”

mentioning

confidence: 89%

“…Hence, up to homotopy equivalence E 3 can be constructed from X by attaching cells of dimension ≥ 4, so X (3) = (E 3 ) (3) . In fact, this is the way E 3 is constructed.…”

Section: Minimal P D 4 -Complexesmentioning

confidence: 99%

mentioning

confidence: 99%

“…, where ϕ : S 3 → X (3) and 3) are the attaching maps of the 4-cells [12,Theorem 2.4]. For simplicity, we denote ϕ(S 3 ) = ∂D 4 and ϕ ′ (S 3 ) = ∂D ′4 .…”

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confidence: 99%

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

2014

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“…Let X be a PD 4 -complex. Then (I) X is homotopy equivalent to K ∪ ϕ D 4 , where K is a 3-complex and ϕ : S 3 → K is the attaching map of the (unique) 4-cell (called disc property);…”

Section: Introductionmentioning

confidence: 99%

$PD_4$-complexes: constructions, cobordisms and signatures

Cavicchioli

Hegenbarth

Spaggiari

2016

Homology, Homotopy and Applications

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The oriented topological cobordism group Ω 4 (P ) of an oriented PD 4 -complex P is isomorphic to Z ⊕ Z. The invariants of an element {f : X → P } ∈ Ω 4 (P ) are the signature of X and the degree of f . We prove an analogous result for the Poincaré duality cobordism group Ω PD 4 (P ): If π 1 (P ) does not contain nontrivial elements of order 2, then Ω PD 4 (P ) is isomorphic to L 0 (Λ) ⊕ Z, where L 0 (Λ) is the Witt group of nondegenerated hermitian forms on finitely generated stably free Λmodules. The component of an element {f : X → P } ∈ Ω PD 4 (P ) in L 0 (Λ) is related to the symmetric signature of X. Then we construct explicitly PD 4 -complexes, define the well-known map L 4 (π 1 (P )) → Ω PD 4 (P ), and characterize the image of the map Ω PD 4 (P ) → Ω N 4 (P ). The results are summarized in Theorems 1.1 and 1.2 stated in the introduction. Prolog: background and motivationWe want to explain how the results fit in with the prior knowledge and the contributions in this area. The paper is occupied with 4-dimensional topology. An outstanding problem in 4-dimensional surgery theory is the following: Given a degree-1 normal map M → P , M a topological 4-manifold, can it be transformed by surgeries to a homotopy equivalence, provided an obstruction in the Wall group L 4 (π 1 (P )) vanishes? The analogous problem is solved if the dimension of M is equal to or greater than 5. It is expressed in the exact surgery sequence (/)where S(P ) is the so-called structure set of P . It consists of equivalence classes of homotopy equivalences N → P , where N is a topological n-manifold and dim M = n 5. Here P has to be a CW-complex satisfying n-Poincaré duality, called shortly a PD n -complex. If there is at least one degree-1 normal map M → P , then the homotopy classes [P, G/T OP ] classify all degree-1 normal maps up to normal cobordism. Ranicki

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