The transcendence degree of the mod p cohomology of finite Postnikov systems, Fields Inst

Abstract. We examine the transcendence degree of the mod p cohomology of a nite Postnikov system E. We prove that, under mild assumptions on E, the transcendence degree of H(E;Fp) is always positive, and give a com-plete classication of the Postnikov systems where the transcendence degree of H(E;Fp) is nite. More precisely we prove that H(E;Fp) is of nite transcendence degree i E is Fp-equivalent to the classifying space of a p-toral group. To obtain the results we establish a general formula for determining the transcendence degree of an unstable algebra given in terms of the growth of certain `unstable Betti numbers'. As an application of these results we de-rive statements about the n-connected cover Xhni of a nite complex X. We show for instance that, under suitable connectivity assumptions on X, the LS category of Xhni is always innite assuming Xhni 6 = X. Finally we discuss generalizations of the obtained results to polyGEMs. 1