2010
DOI: 10.1515/forum.2010.016
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Poincaré duality pairs of dimension three

Abstract: We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of PD 4 -complexes. Generalizing Turaev's fundamental triples of PD 3 -complexes we introduce fundamental triples for PD n -complexes and show that two PD n -complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional manifolds.

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Cited by 7 publications
(31 citation statements)
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References 27 publications
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“…Direct verification shows that each G q is a functor from the category of chain complexes of left Λ-modules to the category of left Λ-modules. By Lemma 4.2 in [3], chain homotopic maps…”
Section: Formulation and Necessity Of The Realization Conditionsmentioning
confidence: 90%
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“…Direct verification shows that each G q is a functor from the category of chain complexes of left Λ-modules to the category of left Λ-modules. By Lemma 4.2 in [3], chain homotopic maps…”
Section: Formulation and Necessity Of The Realization Conditionsmentioning
confidence: 90%
“…Let x denote the chain representing the n-cell attached by f 1 + f 2 . Repeated application of Theorem 2.3 of [3] shows that L 1 C( X 1 ) +L 2 C( X 2 ) is a Poincaré duality chain complex. Hence (X, ω X , [1 ⊗ x]) is a Poincaré duality complex.…”
Section: Decomposition As Connected Summentioning
confidence: 97%
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