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–1Sⁿ have one and the same Hopf invariant if and only if the induced chain of mappings is strongly homotopic.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 557–564, April, 1977.