Poincaré duality and signature for topological manifolds

The signature of the Poincaré duality of compact topological manifolds with local system of coefficients can be described as a natural invariant of nondegenerate symmetric quadratic forms defined on a category of infinite dimensional linear spaces. The objects of this category are linear spaces of the form W = V ⊕ V ∗ where V is abstract linear space with countable base. The space W is considered with minimal natural topology. The symmetric quadratic form on the space W is generated by the Poincaré duality homomorphism on the abstract chain–cochain groups induced by singular simplices on the topological manifold.