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

TL;DR

This paper investigates the order and stable order of the canonical vector bundle over configuration spaces of disjoint unions of spheres, extending previous studies on Euclidean spaces and Riemann surfaces.

Contribution

It provides new results on the order and stable order of canonical vector bundles over configuration spaces of disjoint unions of spheres.

Findings

Determined the order of the canonical vector bundle for specific configurations.

Extended known results to new classes of configuration spaces.

Identified conditions under which the bundle is stably trivial.

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.