A. S. Mishchenko,  P. S. Popov

On Construction of Signature of Quadratic Forms on Infinite-Dimensional Abstract Spaces

The signature of the Poincare 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 oplus V* where V is an 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 Poincare duality homomorphism on the abstract cochain group induced by nerves of the system of atlases of charts on the topological manifold.