ABSTRACT. Let X be a finite, simply connected CW complex with only a finite number of nonzero rational homotopy groups. Localized away from a certain finite set of primes, the loop space of X is shown to be homotopy equivalent to a product of spheres and loop spaces of spheres. Applications to the homotopy groups of X and the homological properties of QX are given.Let X be a simply connected CW complex with a finite numer of cells. Define the exponent of X at a prime p to be the least upper bound of the orders of p-primary torsion elements in the homotopy groups, rrtX.Is this exponent finite or not? The following conjecture, due to J. C. Moore [2], asserts that the answer depends only on a certain rational homotopy invariant of X.CONJECTURE. Given X as above, the exponent of X at p is finite if and only if the total rational homotopy rank, J2n rank(7r"X ® Q), is finite. D This conjecture was the starting point for the results in this paper. We prove one direction of this conjecture for all but a finite number of primes. Our main geometric result is the following.
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