The number of roots in a simply-connected $H$-manifold

An $H$-manifold is a triple $(M,m,e)$ where $M$ is a compact connected triangulable manifold without boundary, $e \in M$, and $m:M \times M \to M$ is a map such that $m(x,e) = m(e,x) = x$ for all $x \in M$. Define ${m_1}:M \to M$ to be the identity map and, for $k \geqslant 2$, define ${m_k}:M \to M$ by ${m_k}(x) = m(x,{m_{k - 1}}(x))$. It is proven that if $(M,m,e)$ is an $H$-manifold, then $M$ is simply-connected if and only if given $k \geqslant 1$ there exists a multiplication $m'$ on $M$ homotopic to $m$ such that ${m'_j}(x) = e$ implies $x = e$ for all $j \leqslant k$.