Poincaré Duality Spaces
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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
![\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.
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.