Generalized Whitehead spaces with few cells (Q1328889)

A topological space \(E\) is called a GW-space if every generalised Whitehead product on \(E\) is trivial. Every H-space is a GW-space and for a suspended space the two notions are equivalent. In particular \(S^ n\) is a GW-space (at 2) if and only if \(n = 1\), 3 or 7. GW-spaces with two or three cells (other than the base point) were studied by \textit{H. Kachi} [Hiroshima Math. J. 20, No. 2, 365-384 (1990; Zbl 0711.55007)], who showed that the only GW-spaces with two cells are contractible spaces. Kachi also considered complexes of type \((q,n,m)\), i.e. CW complexes with cells only in dimensions \(0,q,n\) and \(m\) with \(0 < q \leq n \leq m\). The authors now prove the following. If a complex \(E\) of type \((q,n,m)\) is a GW-space (at 2), then \(E\) has the homotopy type of either a sphere of dimension 1, 3 or 7, or a Poincaré complex of type \((q,n,q + n)\), where \(\{q,n\} \subseteq \{1,3,7\}\) or \((q,n) = (1,2) \), (2,4), (3,4), (3,5) or (3, 7). In the latter case, \(E\) has the homotopy type (at 2) of one of the spaces \(S^ q \times S^ n\) for \(\{q,n\} \subseteq \{1,3,7\}\), \(L^ 3 (p,l)\) for \((q,n) = (1,2)\), \(CP (3)\) for \((q,n) = (2,4)\), \(S^ 7\) for \((q,n) = (3,4)\), \(SU (3)\) for \((q,n) = (3,5)\), \(E_{k \omega}\) with \(k \neq 2 \bmod 4\) for \((q,n) = (3,7)\).