  The number density of galaxy clusters provides tight statistical constraints
on the matter fluctuation power spectrum normalization, traditionally phrased
in terms of sigma_8, the root mean square mass fluctuation in spheres with
radius 8 h^-1 Mpc. We present constraints on sigma_8 and the total matter
density Omega_m0 from local cluster counts as a function of X-ray temperature,
taking care to incorporate and minimize systematic errors that plagued previous
work with this method. In particular, we present new determinations of the
cluster luminosity - temperature and mass - temperature relations, including
their intrinsic scatter, and a determination of the Jenkins mass function
parameters for the same mass definition as the mass - temperature calibration.
Marginalizing over the 12 uninteresting parameters associated with this method,
we find that the local cluster temperature function implies sigma_8
(Omega_m0/0.32)^alpha = 0.86+/-0.04 with alpha = 0.30 (0.41) for Omega_m0 <
0.32 (Omega_mo > 0.32) (68% confidence for two parameters). This result agrees
with a wide range of recent independent determinations, and we find no evidence
of any additional sources of systematic error for the X-ray cluster temperature
function determination of the matter power spectrum normalization. The joint
WMAP5 + cluster constraints are: Omega_m0 = 0.30+0.03/-0.02 and sigma_8 =
0.85+0.04/-0.02 (68% confidence for two parameters).
