Multiplications on a Complex

In their communications at the First International Topological Conference (Moscow, September 1935), J. W. Alexander and A. Kolmogoroff introduced the notion of a dual cycle1 and defined a product of a dual p-eycle and a dual q-eycle, this product being a dual (p + q)-eyele. A different multiplication of the same sort is considered in this paper. It may be shown that the Alexander-Kolmogoroff product, augmented by the dual boundary of a suitable (p + q - 1)- chain, is equal to the $$ \left( {_{p}^{{p + q}}} \right)th $$ multiple of the product here introduced.2 Moreover, I consider also a product of an ordinary n-cycle and a dual p-eycle (n ≥ p), this product being an ordinary (n — p)-cycle. There is a simple algebraic relationship between the two kinds of multiplication, which I shall explain elsewhere. As an application of the general theory, I give a new approach to the duality and intersection theory of a combinatorial manifold, given in a simplicial subdivision. The theory works exclusively in the given subdivision.