  Two-component equal-mass Fermi gases, in which unlike atoms interact through
a short-range two-body potential and like atoms do not interact, are stable
even when the interspecies s-wave scattering length becomes infinitely large.
Solving the many-body Schroedinger equation within a hyperspherical framework
and by Monte Carlo techniques, this paper investigates how the properties of
trapped two-component gases change if a third or fourth component are added. If
all interspecies scattering lengths are equal and negative, our calculations
suggest that both three- and four-component Fermi gases become unstable for a
certain critical set of parameters. The relevant length scale associated with
the collapse is set by the interspecies scattering length and we argue that the
collapse is, similar to the collapse of an attractive trapped Bose gas, a
many-body phenomenon. Furthermore, we consider a three-component Fermi gas in
which two interspecies scattering lengths are negative while the other
interspecies scattering length is zero. In this case, the stability of the
Fermi system is predicted to depend appreciably on the range of the underlying
two-body potential. We find parameter combinations for which the system appears
to become unstable for a finite negative scattering length and parameter
combinations for which the system appears to be made up of weakly-bound trimers
that consist of one fermion of each species.
