Finite Complexes with Vanishing Lines of Small Slope

a p–local CW–complex with H∗X a finite dimensional Fp–vector space. Then there is an integer NX (see 2.5) depending on H ∗X as a graded vector space and: (1) For p = 2, if P s t H ∗X 6= 0 for all P s t ∈ A with s < t, then X∧NX has a non–trivial stable summand Y such that H∗Y is A–free. (2) For p 6= 2, if P s t HX 6= 0 for all P s t ∈ A with s < t and QtHX 6= 0 for all Qt ∈ A, then X∧NX has a non–trivial stable summand Y such that H∗Y is A–free. (3) For p = 2, if P s t H ∗X 6= 0 for all P s t ∈ A such that s < t and |P s t | ≤ d, then X∧NX has a non–trivial stable summand Y such that H∗Y has a vanishing line over A of