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

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 abstarct 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 cochain group induced by nerves of the system of atlases of charts on the topological manifold. 2000 Mathematics Subject Classification: 55M05, 57N15.