The present paper is devoted to a study of the following process: take an ~ rbitrary map d: A-~X, use it to attach the cone CA to X and obtain a space B = X u CA, and then lift d in a natural way to a map e: A ~ F , where F is the of the inclusion map X-*B. An important particular case arises when X is a point; the resulting map e is then readily identified to the natural embedding A ~ r , A of A in the loops of its suspension. This case, which is crucial for computing homotopy groups of spheres, has been thoroughly investigated in [21], and most of the results therein extend to the general situation considered here [4]. Next, as shown in [4] and [6], study of the above process yields a satisfactory theory of the dual of Lusternik-Schnirelmann category; in particular, it elucidates the relationship between this dual and the homotopical nilpotency of loop spaces. Finally, the analysis of low-dimensional cases presented in the last section of this paper reveals that some results of homological algebra may also be derived from the general theorems pertinent to the process described. The main problem in this context is to study the map e; specifically, one seeks convenient descriptions of the homotopy types of the fibre E and of the cofibre K of e so as to obtain generalizations of the EHP sequence [21]. Duality and the result in [4; 1. l ] suggest that the homotopy type of K only depends on those of A and B; however, examples (see 1.4 below) disprove this conjecture. Nevertheless, the homotopy type of the suspension 2~K is determined by those of A and B, and this enables us to express the homotopy type of K itself in terms of A and B in a limited range of dimensions. There are several ways of doing that, and the maps which relate K to various functors of the two arguments A and B appear as generalizations of various forms of the Hopf invariant. Description of the homotopy type of E brings the generalized Whitehead product into the picture. Some of those questions were already studied in [4]. The present paper improves and simplifies the results in [4; w 3 and w 4], and is independent of them. In [4], a