Title:

On certain problems in homotopy theory

John Milnor has given a construction [J. Milnor. Construction of Universal Bundles, II. Ann. Math. 63 (1956) 430436] which associates with each topological group G a sequence of spaces GP(i) [our notation] which can be regarded as generalizations of the classical projective spaces RP(i) = S^{0}P(i), CP(i) = S^{1}P(i) and QP(i) = S^{3}P(i). If one considers Hspaces instead of topological groups, one finds that additional homotopy structure is necessary to imitate this construction. Intuitively, an A_{n}space X is an Hspace with sufficient additional structure to permit the construction for i ≤ n of spaces XP(i) with properties similar to those of the spaces GP(i). An A_{n}map can be thought of as a map from one A_{n}space to another which up to homotopy respects the A_{n}structure. The principal goals of this study are 1) to investigate the multiplication on an A_{n}space in terms of the geometry of the projective spaces and 2) to analyse the algebra of the homology of an A_{n}space. In the first direction we observe that in the classical case there exist maps RP(r) × RP(s) → RP(r+s) and CP(r) × CP(s) → CP(r+s) which restricted to either factor are the usual imbeddings RP(r) ⊂ RP(r+s) and CP(r) ⊂ CP(r+s). Working with A_{n}spaces we find that the existence of such maps is implied by the multiplication's being itself an A_{r+s}map. In particular, an Hspace (X,m) is homotopy abelian if and only if there is a map of SX × SX → XP(2) which is the inclusion on either factor. The second direction is guided by William Massey's definition of cohomology products. The essential properties of the singular chain complex on an A_{n}space are embodied in the concept of an A(n)algebra. Other algebraic objects called stacks are defined which give rise to spectral sequences of topological significance. Applied to A_{n}spaces, these spectral sequences yield homology operations which serve as criteria for A_{n}maps, and are related to the homology of the protective spaces. The two paths of study are reunited in application to ΩCP(3) the loop space on complex projective 3space. We find that ΩCP(3 is homotopy abelian and that there exists a homotopy equivalence f : S^{1} × ΩS^{7} → ΩCP(3) which is an Hmap but not an A_{4}map.
