On the stable James numbers of Thom complexes

Abstract

of stable cohomotopy groups. Then the stable James number of X*, which we shall denote by d(X, ξ), is defined to be the non-negative generator of image i* (see [7]). Thus d(X, ξ) is the least positive integer r such that a map S^S of degree r can be stably extended to X*, if it exists, or zero otherwise. For a map/: X->S, we shall call the degree offoi the degree of/ simply. Suppose, for example, that X is the projective space FP~ (F=C or H), and ξ is w-fold Whitney sum of the canonical line bundle η over FPk~ιy then X%— ppk+n-ηppn-i a n d d(FP ~ tiη) is the same as F {n, k) in [9]. In that paper, Oshima determined F {n, k} for several small k's (see also [3], [7] and [8] for F{\,k}). Now let X and ξ be as before. Let J(X) denote the group of stable fibre homotopy equivalence classes of real vector bundles over X, and J(ξ) the class of ξ in J(X). Since a stable fibre homotopy equivalence of bundles induces a stable homotopy equivalence of their Thorn complexes, we may regard d(X, —) as a function from J(X) to Z. We shall abuse notations, and not distinguish d(Xy J(ξ)) from d(X> ξ). Our main result is as follows: