Poincaré Duality Spaces

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1 Introduction

A Poincaré duality space of dimension $d$${{Authors|Klein}}{{Stub}} ==Introduction== <wikitex>; A Poincaré duality space of dimension $d$ consists of a space $X$ together a pair $(\mathcal{L},[X])$ in which $\mathcal{L}$ is a bundle of local coefficients on $X$ which is free abelian of rank one and $[X] \in H_d(X;\mathcal {L})$ sastifies $$ \cap [X] : H^*(X;M) \to H_{d-*}(X;M \otimes \mathcal{L})$$ is an isomorphism. Here, $M$ is allowed to range over all local coefficient bundles on $X$. </wikitex> ==References== {{#RefList:}} [[Category:Theory]]d$ consists of a space $X$$X$ together a pair $(\mathcal{L},[X])$$(\mathcal{L},[X])$ in which $\mathcal{L}$$\mathcal{L}$ is a bundle of local coefficients on $X$$X$ which is free abelian of rank one and $[X] \in H_d(X;\mathcal {L})$$[X] \in H_d(X;\mathcal {L})$ sastifies

$\displaystyle \cap [X] : H^*(X;M) \to H_{d-*}(X;M \otimes \mathcal{L})$ $$\displaystyle \cap [X] : H^*(X;M) \to H_{d-*}(X;M \otimes \mathcal{L})$$

is an isomorphism. Here,

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$M$ is allowed to range over all local coefficient bundles on $X$$X$.