  We consider in this paper the minimally twisted chain link with 5 components
in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn
fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic
one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or
fewer (including 0) cusps. In view of Thurston's hyperbolic Dehn filling
theorem it is then natural to face the problem of classifying all the
exceptional fillings on M5, namely those yielding non-hyperbolic 3-manifolds.
Here we completely solve this problem, also showing that, thanks to the
symmetries of M5 and of some hyperbolic manifolds resulting from fillings of
M5, the set of exceptional fillings on M5 is described by a very small amount
of information.
