Poincaré Duality Spaces
From Manifold Atlas
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Contents |
1 Introduction
2 Definition
![d](/images/math/2/0/f/20fd65e9c7406034fadc682f06732868.png)
![(X,\partial X)](/images/math/b/c/a/bcae489200a01393164035cce3091da5.png)
![\displaystyle (\mathcal{L},[X])](/images/math/5/2/4/524f4fda4a49a96a3a03636971fc3aec.png)
is a bundle of local coefficients on
which is free abelian of rank one, and
-
is a class such that
![\displaystyle \cap [X] : H^*(X;\mathcal{B}) \to H_{d-*}(X,\partial X;\mathcal{B} \otimes \mathcal{L})](/images/math/5/0/3/503ba21954ed033ab528e17a746c8962.png)
and
![\displaystyle \cap [X] : H^*(X,\partial X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})](/images/math/0/3/5/035b25de612189e48ccd1d1f8779bbfc.png)
are isomorphisms.
Here, is allowed to range over all local coefficient bundles on
,
but in fact it is sufficient to check the condition when
is the local coefficient bundle over
associated with
, where
is the fundamental groupoid of
.
3 Notes
- If
, one says that
is a Poincaré duality space. (In view of this, perhaps better terminology would be to call
a Poincaré duality space with boundary.)
-
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 Poincaré space of dimension
with respect to
, where
is the boundary homomorphism.
- A finite CW complex
admits the structure of a Poincaré duality space of dimension
if and only if there exists a framed compact smooth manifold
of dimension
such
is homotopy equivalent to
and the inclusion
has homotopy fiber homotopy equivalent to
.
4 Example
A compact (smooth, PL, TOP or homology) manifold of dimension
is a Poincaré duality pair of dimension
, where
is the orientation sheaf of
and
is the manifold fundamental class.