mod 3 cohomology algebras of finite \(H\)-spaces (Q697335)

Let \(X\) be a \(3\)-local, finite, simply connected \(H\)-space whose homology is associative. For example, \(X\) could be a localized odd dimensional sphere, the Lie group \(E_{8}\) or the \(H\)-space \(X(3)\) constructed by \textit{J. R. Harper} [\(H\)-spaces with torsion, Mem. Am. Math. Soc. 223 (1979; Zbl 0421.55006)]. The main theorem of this paper states that, up to isomorphism of mod \(3\) cohomology algebras, these examples, and products of them, are the only ones. Concretely, the mod \(3\) cohomology of \(X\) must be isomorphic, as an algebra, to the cohomology of a product of copies of \(X(3)\), copies of \(E_{8}\) and odd dimensional spheres. There are also partial results to suggest that an isomorphism as algebras over the Steenrod algebra may also be possible, as demonstrated by \textit{R. Kane} [J. Pure Appl. Algebra 41, 213-232 (1986; Zbl 0635.57027)] in the case where there is exactly one copy of \(E_{8}\).