  Observations of the centers of galaxies continue to evolve, and it is useful
to take a fresh look at the constraints that exist on alternatives to
supermassive black holes at their centers. We discuss constraints complementary
to those of Maoz (1998) and demonstrate that an extremely wide range of other
possibilities can be excluded. In particular, we present the new argument that
for the velocity dispersions inferred for many galactic nuclei, even binaries
made of point masses cannot stave off core collapse because hard binaries are
so tight that they merge via emission of gravitational radiation before they
can engage in three-body or four-body interactions. We also show that under
these conditions core collapse leads inevitably to runaway growth of a central
black hole with a significant fraction of the initial mass, regardless of the
masses of the individual stars. For clusters of noninteracting low-mass objects
(from low-mass stars to elementary particles), relaxation of stars and compact
objects that pass inside the dark region will be accelerated by interactions
with the dark mass. If the dark region is instead a self-supported object such
as a fermion ball, then if stellar-mass black holes exist they will collide
with the object, settle, and consume it. The net result is that the keyhole
through which alternatives to supermassive black holes must pass is
substantially smaller and more contrived than it was even a few years ago.
