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 , 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 ) and (p) denotes localization at the prime p. In all these cases n divides p- 1.
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