Poincaré Duality Spaces

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	{{Authors|Klein}}{{Stub}}		{{Authors|Klein}}{{Stub}}
−	==Introduction==	+	== Introduction ==
	<wikitex>;		<wikitex>;
−		+	</wikitex>
	==Definition==		==Definition==
		+	<wikitex>;
		+	A ''Poincaré pair'' of dimension $d$ consists of a finitely dominated CW pair $(X,\partial X)$ for which there exists $$(\mathcal{L},[X])$$ in which
		+	*$\mathcal{L}$ is a bundle of local coefficients on $X$ which is free abelian of rank one, and
−	A ''Poincaré pair'' of dimension $d$ consists of a pair of spaces $(X,\partial X)$ such that there exists 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 $[X] \in H_d(X,\partial X;\mathcal {L})$ are such that	+	* $[X] \in H_d(X,\partial X;\mathcal {L})$ is a class such that
	$$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$$		$$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$$
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	are isomorphisms.		are isomorphisms.
−	is an isomorphism. Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$.	+	Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$,
−		+	but in fact it is sufficient to check the condition when $\mathcal{B}$ is the local coefficient bundle over $X$
−	If $(X,\partial X)$ is a finitely dominated pair, then it suffices to check the	+	associated with $\Bbb Z[\pi]$, where $\pi$ is the fundamental groupoid of $X$.
−	condition when $\mathcal{B}$ is local coefficient bundle over $X$ defined by	+	</wikitex>
−	$\Bbb Z[\pi]$, with $\pi$ the fundamental groupoid of $X$.	+
−		+
	==Notes==		==Notes==
		+	<wikitex>;
		+	* If $\partial X = \emptyset$, one says that $X$ is a ''Poincaré duality space.'' (In view of this, perhaps better terminology would be to call $(X,\partial X)$ a ''Poincaré duality space with boundary.'')
−	* If $\partial X = \emptyset$, one says that $X$ is a ''Poincaré duality space.'' (In view of this, perhaps better terminology would be to call $(X,\partial X)$ a Poincaré space with boundary.)	+	* $\mathcal L$ is called an ''orientation sheaf'' and $[X]$ is called a ''fundamental class.'' The pair $(\mathcal L,[X])$ is unique up to unique isomorphism.
−	* One typically assumes that $(X,\partial X)$ is finitely dominated.	+	* If $(X,\partial X)$ with respect to $(\mathcal{L},[X])$ a Poincar\'e pair of dimension $d$, then $\partial X$ is a Poincaré space of dimension $d-1$ with respect to $(\mathcal {L}_{|\partial X},\partial [X])$, where $\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$ is the boundary homomorphism.
−		+
−	* $\mathcal L$ is called an ''orientation sheaf'' and $[X]$ is called a fundamental class. The pair $(\mathcal L,[X])$ is unique up to unique isomorphism.	+
−		+
−	* If $(X,\partial X)$ with respect to $(\mathcal{L},[X])$ a Poincar\'e pair of dimension $d$, then $\partial X$ is a of a Poincaré space of dimension $d-1$ with respect to $(\mathcal {L}_{|\partial X},\partial [X])$, where $\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$ is the boundary homomorphism.	+
		+	* A finite CW complex $X$ admits the structure of a Poincaré duality space of dimension $n$ if and only if there exists a framed compact smooth manifold $M$ of dimension $m \ge n+3$ such $M$ is homotopy equivalent to $X$ and the inclusion $\partial M \subset M$ has homotopy fiber homotopy equivalent to $S^{m-n-1}$.
		+	</wikitex>
	==Example==		==Example==
−		+	<wikitex>;
−	A compact (smooth, PL, TOP or homology) manifold $X$ of dimension $d$ is a Poincaré duality space of dimension $d$, where $\mathcal L$ is the orientation sheaf of $X$	+	A compact (smooth, PL, TOP or homology) manifold $(X,\partial X)$ of dimension $d$ is a Poincaré duality pair of dimension $d$, where $\mathcal L$ is the orientation sheaf of $X$ and $[X]$ is the manifold fundamental class.
−	and $[X]$ is the manifold fundamental class.	+
−		+
−		+
	</wikitex>		</wikitex>

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Latest revision as of 17:01, 12 June 2013

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Contents 1 Introduction 2 Definition 3 Notes 4 Example 5 References

1 Introduction

2 Definition

A Poincaré pair of dimension ${{Authors|Klein}}{{Stub}} ==Introduction== <wikitex>; ==Definition== A ''Poincaré pair'' of dimension $d$ consists of a pair of spaces $(X,\partial X)$ such that there exists 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 $[X] \in H_d(X,\partial X;\mathcal {L})$ are such that $$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$$ and $$ \cap [X] : H^*(X,\partial X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})$$ are isomorphisms. is an isomorphism. Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$. If $(X,\partial X)$ is a finitely dominated pair, then it suffices to check the condition when $\mathcal{B}$ is local coefficient bundle over $X$ defined by $\Bbb Z[\pi]$, with $\pi$ the fundamental groupoid of $X$. ==Notes== * If $\partial X = \emptyset$, one says that $X$ is a ''Poincaré duality space.'' (In view of this, perhaps better terminology would be to call $(X,\partial X)$ a Poincaré space with boundary.) * One typically assumes that $(X,\partial X)$ is finitely dominated. * $\mathcal L$ is called an ''orientation sheaf'' and $[X]$ is called a fundamental class. The pair $(\mathcal L,[X])$ is unique up to unique isomorphism. * If $(X,\partial X)$ with respect to $(\mathcal{L},[X])$ a Poincar\'e pair of dimension $d$, then $\partial X$ is a of a Poincaré space of dimension $d-1$ with respect to $(\mathcal {L}_{|\partial X},\partial [X])$, where $\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$ is the boundary homomorphism. ==Example== A compact (smooth, PL, TOP or homology) manifold $X$ of dimension $d$ is a Poincaré duality space of dimension $d$, where $\mathcal L$ is the orientation sheaf of $X$ and $[X]$ is the manifold fundamental class. </wikitex> ==References== {{#RefList:}} [[Category:Theory]]d$ consists of a finitely dominated CW pair $(X,\partial X)$ for which there exists in which

• $\mathcal{L}$ is a bundle of local coefficients on $X$ which is free abelian of rank one, and

• $[X] \in H_d(X,\partial X;\mathcal {L})$ is a class such that

and

are isomorphisms.

Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$, but in fact it is sufficient to check the condition when $\mathcal{B}$ is the local coefficient bundle over $X$ associated with $\Bbb Z[\pi]$, where $\pi$ is the fundamental groupoid of $X$.

3 Notes

• If $\partial X = \emptyset$, one says that $X$ is a Poincaré duality space. (In view of this, perhaps better terminology would be to call $(X,\partial X)$ a Poincaré duality space with boundary.)

• $\mathcal L$ is called an orientation sheaf and $[X]$ is called a fundamental class. The pair $(\mathcal L,[X])$ is unique up to unique isomorphism.

• If $(X,\partial X)$ with respect to $(\mathcal{L},[X])$ a Poincar\'e pair of dimension $d$, then $\partial X$ is a Poincaré space of dimension $d-1$ with respect to $(\mathcal {L}_{|\partial X},\partial [X])$, where $\partial: H_d(X;\mathcal{L}) \to H_{d-1}(\partial X;\mathcal{L}_{|\partial X})$ is the boundary homomorphism.

• A finite CW complex $X$ admits the structure of a Poincaré duality space of dimension $n$ if and only if there exists a framed compact smooth manifold $M$ of dimension $m \ge n+3$ such $M$ is homotopy equivalent to $X$ and the inclusion $\partial M \subset M$ has homotopy fiber homotopy equivalent to $S^{m-n-1}$.

4 Example

A compact (smooth, PL, TOP or homology) manifold $(X,\partial X)$ of dimension $d$ is a Poincaré duality pair of dimension $d$, where $\mathcal L$ is the orientation sheaf of $X$ and $[X]$ is the manifold fundamental class.

5 References