  In this work we study homeomorphisms of closed orientable surfaces homotopic
to the identity, focusing on the existence of non-contractible periodic orbits.
We show that, if $g$ is such a homeomorphism, and if $\hat g$ is its lift to
the universal covering of $S$ that commutes with the deck transformations, then
one of the following three conditions must be satisfied: (1) The set of fixed
points for $\hat g$ projects to a closed subset $F$ which contains an essential
continuum, (2) $g$ has non-contratible periodic points of every sufficiently
large period, or (3) there exists an uniform bound $M$ such that, if $\hat x$
projects to a contractible periodic point then the $\hat g$ orbit of $\hat x$
has diameter less or equal to $M$. Some consequences for homeomorphisms of
surfaces whose rotation set is a singleton are derived.
