  For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle
x,y\,|\,x^{k+1}yxy \rangle$, with $k\geq1$ odd, we describe representatives for
all free and based homotopy classes of maps from $K$ into the real projective
plane and we classify the homotopy classes containing only surjective maps.
With this approach we get an answer, for maps into the real projective plane,
for a classical question in topological root theory, which is known so far, in
dimension two, only for maps into the sphere, the torus and the Klein bottle.
The answer follows by proving that for all $k\geq1$ odd, $H^2(K;mathbb{Z})=0$
and, for $k\geq3$ odd, there exist maps from $K$ into the real projective plane
which are strongly surjective. For $k=1$, there is no such a strongly
surjective map. 55N25, 57M20.
