A study of stable Hopf invariants and Hopf constructions

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Formulas are obtained among box brackets and matrix Toda brackets. Generalizations of classical formulas by Toda are obtained including relations with the Hopf invariants. A class of spaces is defined which is closely connected with the LS category. Relations between the class and covering spaces are clarified which are useful to study examples. Topologies of function spaces which are defined by the exponential topology are studied. Exponential homeomorphisms are proved which hold for any topological spaces and applications of the theorem are obtained.