TY - JOUR

T1 - A criterion for delooping the fibre of the self-map of a sphere with degree a power of a prime

AU - Cejtin, H.

AU - Kleinerman, S.

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1986

Y1 - 1986

N2 - Fix, once and for all, p to be an odd prime, and n and j to be strictly positive integers. Let F be the homotopy fibre of the self-map of S2n-1 of degree pj is a fibration up to homotopy). Notice that F is its own localization at p. The sphere S2n-1 itself, localized at p, deloops if and only if n divides p -1. In [2], the second author showed that for certain values of p, n and j, the fibre F deloops. The deloopings are of the form BG(Fq)+(p) where G(Fq)is the universal Chevalley group of some exceptional Lie type over the finite field Fq, q a power of a prime different from p. Here "+" denotes Quillen’s "plus construction" (see [6]) and (p) denotes localization at the prime p. In all these cases n divides p- 1.

AB - Fix, once and for all, p to be an odd prime, and n and j to be strictly positive integers. Let F be the homotopy fibre of the self-map of S2n-1 of degree pj is a fibration up to homotopy). Notice that F is its own localization at p. The sphere S2n-1 itself, localized at p, deloops if and only if n divides p -1. In [2], the second author showed that for certain values of p, n and j, the fibre F deloops. The deloopings are of the form BG(Fq)+(p) where G(Fq)is the universal Chevalley group of some exceptional Lie type over the finite field Fq, q a power of a prime different from p. Here "+" denotes Quillen’s "plus construction" (see [6]) and (p) denotes localization at the prime p. In all these cases n divides p- 1.

UR - http://www.scopus.com/inward/record.url?scp=84972564370&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972564370&partnerID=8YFLogxK

U2 - 10.1215/ijm/1256064231

DO - 10.1215/ijm/1256064231

M3 - Article

AN - SCOPUS:84972564370

VL - 30

SP - 566

EP - 573

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

SN - 0019-2082

IS - 4

ER -