  The non-perturbative effect of interaction can sometimes make interacting
bosons behave as though they were free fermions. The system of neutral bosons
in a rapidly rotating atomic trap is equivalent to charged bosons coupled to a
magnetic field, which has opened up the possibility of fractional quantum Hall
effect for bosons interacting with a short range interaction. Motivated by the
composite fermion theory of the fractional Hall effect of electrons, we test
the idea that the interacting bosons map into non-interacting spinless fermions
carrying one vortex each, by comparing wave functions incorporating this
physics with exact wave functions available for systems containing up to 12
bosons. We study here the analogy between interacting bosons at filling factors
$\nu=n/(n+1)$ with non-interacting fermions at $\nu^*=n$ for the ground state
as well as the low-energy excited states and find that it provides a good
account of the behavior for small $n$, but interactions between fermions become
increasingly important with $n$. At $\nu=1$, which is obtained in the limit
$n\rightarrow \infty$, the fermionization appears to overcompensate for the
repulsive interaction between bosons, producing an {\em attractive}
interactions between fermions, as evidenced by a pairing of fermions here.
