Torsion in finite \(H\)-spaces and the homotopy of the three-sphere (Q985321)

The central theorem of this paper is that if \(X\) is an \(H\)-space (localized at an odd prime \(p\), as all spaces in this paper are assumed to be) and \(i:S^3 \to X\) a map, then \(\Omega i \langle 3 \rangle\), the loop of the 3-connected cover of \(i\), is null-homotopic if, and only if, \(i \circ \bar{\alpha}\) is null-homotopic, where \(\bar{\alpha}: P^{2p+1}(p) \to S^3\) is the unique extension of a representative \(\alpha : S^{2p} \to S^3\) of a generator of \(\pi_{2p}(S^3) \cong {\mathbb Z}/p\). This theorem is built on in a couple of ways. If \(X\) is a \(2\)-connected finite \(H\)-space with a single cell in dimension~\(3\), and \(i\) is the inclusion of this bottom cell, then these two null-homotopy conditions are equivalent to the non-vanishing of \(\beta {\mathcal P}^1(x)\), \(x \in H^3(X)\) being such that \(i^*(x)\) is a generator of \(H^3(S^3)\). If \(X\) has multiple \(3\)-cells then we lose the equivalence but there is still some relation between the conditions. Moreover, if \(X\) is a simple, simply-connected, compact Lie group, then these condition are equivalent to \(G\) having \(p\)-torsion in its integral cohomology.