Let i: SX-* 93X be the identification map, where 3X is the reduced suspension. G. WV. Whitehead [17] studied the homotopy suspension E: rn (X) +, (SX) by using the map +(i): X -I?eaX. We consider a dual situation: abbreviate 0 (X, x0) by 2, and let j: Q? -> f2 be the identity. Then the map +-'(j): SQ -X induces homomorphisms of the homology groups which are closely related to the homology suspension r: -in(Q) H,+ (X). It is convenient to convert +-l (j) into an equivalent fibre map. The fibre is of the homotopy type of the join Q2 * Q, and the Serre homology sequence of the fibering is essentially the same as G. W. Whitehead's sequence [18] involving a, but contains an extra term. This gives an alternative proof of Whitehead's main result, and also allows us to extend several of his corollaries by one dimension: e. g. a cohomology operation of type (n, q;7r, G), q < 3n, is additive if and only if it is a suspension. As a further application, in Part II we apply the above fibering to the problem of calculating the Postnikov invariants of the suspension of an Eilenberg-MacLane space K (7r, n).