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Loop spaces of finite complexes at large primes. (English) Zbl 0594.55006

If X is a finite 1-connected CW-complex with \(\pi_*(X)\otimes {\mathbb{Q}}\) finite dimensional and nonzero, it is shown that for almost all primes p the mod p homotopy type of the loop space \(\Omega\) X is that of a product of odd dimensional spheres and loop spaces of odd dimensional spheres. It follows that the homotopy exponent of X at a prime is finite, at least for all but a finite number of primes, establishing in part a stronger conjecture of J. C. Moore. The main theorem is proved by induction on \(\pi_*(X)\otimes {\mathbb{Q}}\) using different fibrations involving spheres and at one point a result of S. Halperin on finiteness in rational homotopy theory.
Reviewer: J.R.Hubbuck

MSC:

55P35 Loop spaces
55Q05 Homotopy groups, general; sets of homotopy classes
55P62 Rational homotopy theory
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