  We study co-H-maps from a suspension to the suspension of the projective
plane and provide examples of non-suspension 3-cell co-H-spaces. These
(infinitely many) examples are related to the homotopy groups of the 3-sphere.
For each element of order 2 in $\pi_n(S^3)$, there is a corresponding
non-suspension co-H-space of cells in dimensions 2, 3 and n+2.
  Our ideas are to study Hopf invariants in combinatorial way by using the
Cohen groups.
