Divisible $H$-spaces

Let $X$ be an $H$-space with multiplication $m$. Define, for $x \in X,{m_2}(x) = m(x,x)$ and ${m_k}(x) = m(x,{m_{k - 1}}(x))$, for all $k > 2$. If ${m_k}(x) = y$, then $x$ is called a $k$th root of $y$. The $H$-space $(X,m)$ is divisible if every $y$ in $X$ has a $k$th root for each $k \geqq 2$. We prove that if $X$ is a compact connected topological manifold without boundary, then $(X,m)$ is divisible and, in fact, that every $y$ in $X$ has at least ${k^\beta }k$th roots for each $k \geqq 2$, where $\beta$ is the first Betti number of $X$.