Rank \(p - 1\) mod-\(p\) \(H\)-spaces
For any odd prime \(p\), two constructions of torsion free finite \(p\)-local \(H\)-spaces are known: one by \textit{G. Cooke, J. R. Harper} and \textit{A. Zabrodsky} [Topology 18, 349--359 (1979; Zbl 0426.55009)] and the other by \textit{F. Cohen} and \textit{J. Neisendorfer} [Algebraic topology, Proc. Conf., Aarhus 1982, Lect. Notes Math. 1051, 351--359 (1984; Zbl 0582.55010)]. In both constructions, the resulting \(H\)-spaces have rank less than \(p-1\). On the other hand, the Cooke-Harper-Zabrodsky construction goes through when the rank is \(p-1\) but the space produced need not be an \(H\)-space. In the paper under review, the authors first show that the same is true using the Cohen-Neisendorfer method: they show that for any \(CW\)-complex \(A\) with \(p-1\) cells, all in odd dimensions, there is a spherically resolved space \(B\) such that \(H_*(B)\cong \Lambda(\tilde H_*(A))\) as coalgebras and a map \(A \to B\) which induces the inclusion of the exterior algebra generators in homology. Then they give a criterion for the space \(B\) to be an \(H\)-space. In particular, they study closely the case \(p=3\), in which the space \(B\) is a sphere bundle over a sphere. Finally, they give some criteria for a spherically resolved space of rank less than \(p-1\) to be an \(H\)-space.
scientific article; zbMATH DE number 6176736
scientific article