Poincaré Duality Spaces

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-A Poincaré duality space of dimension $d$ consists of a space $X$ together a pair $(\cal L,[X])$ in which
-$\cal L$ is a bundel local coefficients on $X$ which is free abelian of rank one and $[X] \in H_d(X;\cal L)$ sastifies
-$$
-\cap [X] \: H^*(X;M) \to H_{d-*}(X;M \tensor \cal L)
-$$
-is an isomorphism. Here, $M$ is allowed to range over all local coefficient bundles on $X$.
-To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:
-- For statements like Theorem, Lemma, Definition etc., use e.g.
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 == Introduction ==
 <wikitex>;