Homotopy in non metrizable omega-bounded surfaces

We investigate the problem of describing the homotopy classes $[X,Y]$ of continuous functions between $\omega$-bounded non metrizable manifolds $X,Y$. We define a family of surfaces $X$ built with the first octant $C$ in $L^2$ ($L$ is the longline and $R$ the longray), and show that $[X,R]$ is in bijection with so called `adapted' subsets of a partially ordered set. We also show that $[M,R]$ can be computed for some surfaces $M$ that, unlike $C$, do not contain $R$. This indicates that when $X,Y$ are $\omega$-bounded non metrizable surfaces, there might be a link between $[X,Y]$ and the concept of $Y$-directions in X$. Many pictures are used and the proofs are quite detailed.