$PD_4$-complexes: constructions, cobordisms and signatures

Abstract

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