Torsion in finite $H$-spaces and the homotopy of the three-sphere

Abstract

Let $X$ be a 2-connected $p$-local finite $H$-space with a single cell in dimension three. We give a simple cohomological criterion which distinguishes when the inclusion i: $S^3 \underset {\longrightarrow}{i} X$ has the property that the loop of its three-connected cover is null homotopic. In particular, such a null homotopy implies that $\pi_m(i )= 0$ for $m \geq 4$. Applications are made to Harper's rank 2 finite $H$-space and simple, simply-connected, compact Lie groups.