On principal bundles over spheres

Sphere Bundles Over Spheres and Non-Cancellation Phenomena

“…This paper continues the study of products of manifolds and non-cancellation phenomena for factors in such products initiated in papers [3,4,9]. In these papers (see also [7,8,11,12] for related results) the following phenomenon is observed.…”

“…Thus M x and M 2 are certainly not simply-connected and, indeed, the phenomenon reflects a non-cancellation phenomenon in the category of finitely-presented groups. In the examples constructed in [3], [4], and [9], JV is a compact connected simple Lie group G (only G = S 3 occurs in [3,4]) and M l 9 M 2 are certain principal G-bundles over spheres S" with n ^ 7. Thus in these cases the manifolds M 1} M 2 are at most 2-connected since TT 3 (G) contains a Z-summand.…”

“…These arguments are given in Section 3, where we also show explicitly how one may pass from the hypotheses of Z-fibre-homotopy-equivalence, for all /, and quasiprincipality, to a conclusion generalizing (1.1). This form of argument would thus have been appropriate in [4,9] and would probably have given better insight into some of the examples given in [4].…”

H-spaces of rank two and non-cancellation phenomena

“…We write Xp for the localization of X at p. We may now prove the main theorem of this section. [7]), and hence of all the examples of non-cancellation adduced in [3]. Moreover, we see that all these examples are necessarily concerned with total spaces of principal fibrations of the same genus.…”

“…The proof is given in Section 2, and incorporates ideas involved in the main result of [3] and their generalization by Scheerer [7]. In fact, we prove in Section 2 that all the examples of non-cancellation given in [3] and [7] are necessarily examples of total spaces of principal bundles of the same genus, to which we may then apply the general theorem of [2] already referred to. In Section 4 we study the powers of the H-spaces of rank (3,11) to illustrate further non-cancellation phenomena.…”