Poincaré Duality Spaces

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1 Definition

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

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

and

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

are isomorphisms.

is an isomorphism. Here, $\mathcal B$$\mathcal B$ is allowed to range over all local coefficient bundles on $X$$X$.

If $(X,\partial X)$$(X,\partial X)$ is a finitely dominated pair, then it suffices to check the condition when $\mathcal{B}$$\mathcal{B}$ is local coefficient bundle over $X$$X$ defined by $\Bbb Z[\pi]$$\Bbb Z[\pi]$, with $\pi$$\pi$ the fundamental groupoid of $X$$X$.

2 Notes

• If $\partial X = \emptyset$$\partial X = \emptyset$, one says that $X$$X$ is a Poincaré duality space.

(Perhaps better terminology would be to call $(X,\partial X)$$(X,\partial X)$ a Poincaré space with boundary.)

• One typically assumes that $(X,\partial X)$$(X,\partial X)$ is finitely dominated.

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

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

3 Example

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