Published online by Cambridge University Press: 20 November 2018
The fiber ${{W}_{n}}$ of the double suspension
${{S}^{2n-1}}\,\to \,{{\Omega }^{2}}{{S}^{2n+1}}$ is known to have a classifying space
$B{{W}_{n}}$. An important conjecture linking the
$EPH$ sequence to the homotopy theory of Moore spaces is that
$B{{W}_{n}}\,\simeq \,\Omega {{T}^{2np+1}}(p)$, where
${{T}^{2np+1}}(p)$ is Anick's space. This is known if
$n\,=\,1$. We prove the
$n\,=\,p$ case and establish some related properties.