The Suspension of a Loop Space

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).