Poincaré Duality Spaces

Revision as of 18:11, 23 March 2011

1 Introduction

1 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 pair of finitely dominated 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

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

and

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

are isomorphisms.

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

It suffices to check the above condition when $\mathcal{B}$ is local coefficient bundle over $X$ defined by $\Bbb Z[\pi]$, with $\pi$ the fundamental groupoid of $X$.

2 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.)

• $\mathcal L$ 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.

3 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.

2 References