Abstract
Let A be the Hopf algebra over Z p for a prime p given by A = Λ(x i, y i ¦ 0 ⩽ i ⩽ p − 2) ⊗ ( Z p [z] (z p) ) (deg x i = 2(p + 1)i + 3, deg y i = 2(p + 1)(i + 1) − 1, deg z = 2(p + 1)) . Kane showed that A is a minimum candidate for the mod p cohomology of a simply connected mod p finite loop space with p-torsion. In fact, if p = 2, 3, 5, then for X = G 2, F 4, E 8, we have H ∗(X; Z p ) ≅ A . We prove that if p ⩾ 7, then there are no such loop spaces X.
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