On certain problems in homotopy theory - Stasheff, J (1960)
John Milnor has given a construction [J. Milnor. Construction of Universal Bundles, II. Ann. Math. 63 (1956) 430-436] 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) = S0P(i), CP(i) = S1P(i) and QP(i) = S3P(i). If one considers H-spaces instead of topological groups, one finds that additional homotopy structure is necessary to imitate this construction. Intuitively, an An-space X is an H-space 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 An-map can be thought of as a map from one An-space to another which up to homotopy respects the An-structure. The principal goals of this study are 1) to investigate the multiplication on an An-space in terms of the geometry of the projective spaces and 2) to analyse the algebra of the homology of an An-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 An-spaces we find that the existence of such maps is implied by the multiplication's being itself an Ar+s-map. In particular, an H-space (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 An-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 An-spaces, these spectral sequences yield homology operations which serve as criteria for An-maps, and are related to the homology of the protective spaces. The two paths of study are re-united in application to ΩCP(3) the loop space on complex projective 3-space. We find that ΩCP(3 is homotopy abelian and that there exists a homotopy equivalence f : S1 × ΩS7 → ΩCP(3) which is an H-map but not an A4-map.