Functor $D$ and strong homotopy

This paper studies functor $D$ and strong homotopy, introduced earlier by the author [1]. A theorem is proven on mappings, and the connection is established between the concepts of strong homotopy of DGA-mapping of coalgebras and functor $D$. As topological applications, in particular, it is shown that continuous mappings of the sphere $f,g:S^{2n-1}\to S^n$ have one and the same Hopf invariant if and only if the induced chain of mappings is strongly homotopic.