Poincaré Duality Spaces

From Manifold Atlas
Jump to: navigation, search

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

2 Definition

A Poincaré pair of dimension d consists of a finitely dominated CW pair (X,\partial X) for which there exists
\displaystyle (\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}) is a class 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,\partial X;\mathcal{B}) \to H_{d-*}(X;\mathcal{B} \otimes \mathcal{L})

are isomorphisms.

Here, \mathcal B is allowed to range over all local coefficient bundles on X, but in fact it is sufficient to check the condition when \mathcal{B} is the local coefficient bundle over X associated with \Bbb Z[\pi], where \pi is the fundamental groupoid of X.

3 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 duality 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.
  • A finite CW complex X admits the structure of a Poincaré duality space of dimension n if and only if there exists a framed compact smooth manifold M of dimension m \ge n+3 such M is homotopy equivalent to X and the inclusion \partial M \subset M has homotopy fiber homotopy equivalent to S^{m-n-1}.

4 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.

5 References

Personal tools