Torsion primes in loop space homology

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We construct a finite 1-connected CW complex X such that $H_*(ΩX; Z)$ has p-torsion for the infinitely many primes satisfying p ~ 5,7,17,19 mod 24, but no p-torsion for the infinitely many primes satisfying p ~ 13 or 23 mod 24.

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