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# Forum Mathematicum

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Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 30, Issue 5

# Order of the canonical vector bundle over configuration spaces of spheres

Shiquan Ren
• Corresponding author
• School of Mathematics and Computer Science, Guangdong Ocean University, 1 Haida Road, Zhanjiang, 524088 P. R. China; and Department of Mathematics, National University of Singapore, 119076 Singapore
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Published Online: 2018-05-09 | DOI: https://doi.org/10.1515/forum-2018-0046

## Abstract

Given a vector bundle, its (stable) order is the smallest positive integer t such that the t-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bundle over configuration spaces of Euclidean spaces have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54], [F. R. Cohen, M. E. Mahowald and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and Geometric Topology (Stanford 1976), Proc. Sympos. Pure Math. 32 Part 2, American Mathematical Society, Providence 1978, 225–228], and [S.-W. Yang, Order of the Canonical Vector Bundle on ${C}_{n}\left(k\right)/{\mathrm{\Sigma }}_{k}$, ProQuest LLC, Ann Arbor, 1978]. Moreover, the order and the stable order of the canonical vector bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of spheres and disjoint unions of spheres.

MSC 2010: 55R80; 55R10; 55P15; 55P40

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Revised: 2018-03-01

Published Online: 2018-05-09

Published in Print: 2018-09-01

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1265–1277, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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