H-spaces analogous to E8mod?3

Abstract

Let p be an odd prime. Let X 0 be a finite, p-local, simply connected homotopy associative H -space. Suppose H * (X 0 ; Z p ) contains the subalgebrasatisfying z 0 = P p x 0 = Q 0 P p P 1 r 0 , P p P 1 r 0 = P 1 y 0 for r 0 ∈ H 3 (X 0 ; Z p ). The only known examples occur for p = 3 and involve the Lie group E 8 . In this note we prove that if X 0 exists, then p must be 3. (2000): 55N22, 55P35, 55P45, 55Q25, 55R05, 55S05, 55S10, 55T10 Mathematics Subject Classification IntroductionIn this note we begin a study of the following questions. Question 1.For p an odd prime and X 0 a finite H -space when is QH even (X 0 ; Z p ) nontrivial?In the late 70's it was shown [21] that QH even (X 0 ; Z p ) could be nontrivial only in degrees of the form 2(p k + p k−1 +p i + · · · + 1). On the other hand, there are examples only for 2p + 2 for any p, and 2p 2 + 2 for p = 3. One of the major questions for finite H -space theory is to determine the possible degrees for QH even (X 0 ; Z p ).If X 0 admits the structure of a p-complete finite loop space, one can construct a generalized maximal torus and Weyl group [7], [8], [26]. Using these ideas, one can classify the mod p cohomology algebras of finite p-complete loop spaces [1]. It is known [1] that QH even (X 0 ; Z p ) can only be nontrivial in degrees 2p + 2 and

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