Poincaré Complex

A Poincaré complex of formal dimension is a connected CW-complex dominated by a finite CW-complex with an orientation character and a fundamental class that induces Poincaré duality with every local coefficient system. Cap product with identifies cohomology in degree with homology in degree , including the orientation twist on the latter. Formal dimension specifies the degree of the fundamental class.

A closed manifold gives such a homotopy model, but a Poincaré complex need not be a manifold. An aspherical Poincaré complex has a contractible universal cover and a Poincaré duality group as its fundamental group.