Order of the canonical vector bundle over configuration spaces of disjoint unions of spheres

Abstract

Given a vector bundle, its (stable) order is the smallest positive integer n such that the n-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bun- dle over configuration spaces of Euclidean spaces have been studied by F.R. Cohen, R.L. Cohen, N.J. Kuhn and J.L. Neisendorfer [5], F.R. Cohen, M.E. Mahowald and R.J. Milgram [7], and S.W. Yang [17]. Moreover, the order and the stable order of the canonical vec- tor bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied by F.R. Cohen, R.L. Cohen, B. Mann and R.J. Milgram [6]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of disjoint unions of spheres.