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Poincare Duality Pairs of Dimension Three - Homotopy Classification, Realisation and Splitting
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Poincaré duality pairs of dimension n (PDn-pairs) are homotopy generalisations of n-manifolds with boundary. In particular, PD3-pairs represent the homotopy theory of 3-manifolds with boundary. Poincaré duality is an algebraic condition on the chain complex of the universal covering space of a CW-complex which involves the cap product. This book presents properties of the cap product on chain complexes not readily available in the literature. Up to oriented homotopy equivalence, PD3-pairs are classified by their \"fundamental triple\", which consists of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. The derived module category provides the language for the formulation of necessary and sufficient conditions for a given triple to be realised by a PD3-pair. The results on classification and realisation yield splitting theorems for PD3-pairs, that is, conditions under which a given PD3-pair can be decomposed as a connected sum of two PD3-pairs. This book is sufficiently detailed to be accessible after a first course on algebraic topology while containing new results.
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