# Poincare duality space

From Topospaces

This article defines a property of topological spaces that depends only on the homology of the topological space, viz it is completely determined by the homology groups. In particular, it is a homotopy-invariant property of topological spacesView all homology-dependent properties of topological spaces OR view all homotopy-invariant properties of topological spaces OR view all properties of topological spaces

## Contents

## Definition

Let a connected space and a commutative ring. We say that is a **Poincare duality space** of formal dimension with respect to if the following hold:

- The homology of with coefficients in is finitely generated
- is a free module of rank over
- Pick a generator for . Then the cap product with this generator induces a map from to . This map is an isomorphism for all .

In particular, has nonvanishing homology groups only for

By default, we take .