On homological products

With the help of homological algebra and sheaf theory we study the general construction of the $\smallfrown$-product of homology by cohomology. In the most general form we establish a relation between the product and Zeeman filtration in homology. We show that the product on general spaces is defined by products on compact and closed locally compact subspaces. We reveal a new feature of the product in the case of homology and cohomology of pairs. We give a natural interpretation of the $\smallfrown$-product in terms of the diagonal embedding of a space in its Cartesian square. In the case of manifolds (including generalized manifolds) for homology and cohomology classes that are dual under Poincare duality, we establish identity between the general construction of the $\smallfrown$-product and the $\smallsmile$-product and describe the duality itself in terms of the product.