Poincaré Duality Spaces
(→Introduction) |
(→Introduction) |
||
Line 6: | Line 6: | ||
− | A Poincaré | + | 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 |
− | $$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})$$ | + | |
+ | $$ \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})$$ | ||
+ | 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 $X$. | is an isomorphism. Here, $\mathcal B$ is allowed to range over all local coefficient bundles on $X$. | ||
− | If $X$ is a finitely dominated | + | 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 | condition when $\mathcal{B}$ is local coefficient bundle over $X$ defined by | ||
− | $\Bbb Z[\pi]$, with $\pi$ the fundamental groupoid of $X$. | + | $\Bbb Z[\pi]$, 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== | ==Example== | ||
− | A | + | 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 fundamental class. | + | and $[X]$ is the manifold fundamental class. |
Revision as of 15:33, 23 March 2011
The user responsible for this page is Klein. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
1 Definition
A Poincaré pair of dimension consists of a pair of spaces such that there exists a pair in which is a bundle of local coefficients on which is free abelian of rank one and are such that
and
are isomorphisms.
is an isomorphism. Here, is allowed to range over all local coefficient bundles on .
If is a finitely dominated pair, then it suffices to check the condition when is local coefficient bundle over defined by , with the fundamental groupoid of .
2 Notes
- If , one says that is a Poincaré duality space.
(Perhaps better terminology would be to call a Poincaré space with boundary.)
- One typically assumes that is finitely dominated.
- is called an orientation sheaf and is called a fundamental class. The pair is unique up to unique isomorphism.
- If with respect to a Poincar\'e pair of dimension , then is a of a Poincaré space of dimension with respect to , where is the boundary homomorphism.
3 Example
A compact (smooth, PL, TOP or homology) manifold of dimension is a Poincaré duality space of dimension , where is the orientation sheaf of and is the manifold fundamental class.