  WFPC-2 images are used to study the central structure of M31, M32, and M33.
The dimmer peak, P2, of the M31 double nucleus is centered on the bulge to
0.1", implying that it is the dynamical center of M31. P2 contains a compact
source discovered by King et al. (1995) at 1700 A. This source is resolved,
with r_{1/2} approx0.2 pc. It dominates the nucleus at 3000 A, and is
consistent with late B-early A stars. This probable cluster may consist of
young stars and be an older version of the cluster of hot stars at the center
of the Milky Way, or it may consist of heavier stars built up from collisions
in a possible cold disk of stars orbiting P2. In M32, the central cusp rises
into the HST limit with gamma approx0.5, and the central density
rho_0>10^7M_sol pc^-3. The V-I and U-V color profiles are flat, and there is no
sign of an inner disk, dust, or any other structure. This total lack of
features seems at variance with a nominal stellar collision time of 2 X 10^10
yr, which implies that a significant fraction of the light in the central pixel
should come from blue stragglers. InM33, the nucleus has an extremely steep
gamma=1.49 power-law profile for 0.05"<r<0.2" that becomes shallower as the HST
resolution limit is approached. The profile for r<0.04" has either a gamma
approx 0.8 cusp or a small core with r_c ~<0.13 pc. The central density is
rho_0 > 2 10^6M_sol pc^-3, and the implied relaxation time is only ~3 X 10^6
yr, indicating that the nucleus is highly relaxed. The accompanying short
collision time of 7 X 10^9 yr predicts a central blue straggler component
quantitatively consistent with the strong V-I and B-R color gradients seen with
HST and from the ground.
