  Cohen conjectured that for r>=2 there is a space T^2n+1(2^r) and a homotopy
fibration sequence Loop^2 S^2n+1 --> S^2n-1 --> T^2n+1(2^r) --> Loop S^2n+1
with the property that the left map composed with the double suspension, Loop^2
S^2n+1 --> S^2n-1 --> Loop^2 S^2n+1, is homotopic to the 2^r-power map. We
positively resolve this conjecture when r>=3. Several preliminary results are
also proved which are of interest in their own right.

Algebraic Topology