From the complete sample and the individually determined cluster masses the galaxy cluster mass function is determined and used to constrain the mean cosmic matter density and the amplitude of mass fluctuations. Comparison to Press--Schechter type model mass functions in the framework of Cold Dark Matter cosmological models and a Harrison--Zeldovich initial density fluctuation spectrum yields the constraints OmegaM = 0.12^{+0.06}_{-0.04} and sigma8 = 0.96^{+0.15}_{-0.12} (90% c.l.). The degeneracy between OmegaM and sigma8 previously encountered for local cluster samples therefore has been broken mainly due to the large covered mass range. Various possible systematic uncertainties are quantified. Adding all identified systematic uncertainties to the statistical uncertainty in a worst case fashion results in an upper limit OmegaM < 0.31. For comparison to previous results a relation sigma8 = 0.43 OmegaM^{-0.38} is derived.
Two further constraints on OmegaM obtained from the HIFLUGCS clusters agree well with the above results. The mean intracluster gas fraction combined with independent estimates of the baryon density yields the upper limit OmegaM ~< 0.34. Calculation of the median mass-to-light ratio for 18 clusters in common to the sample of Girardi et al. (2000) combined with estimates of the total luminosity density in the Universe yields OmegaM ~ 0.15.
The mass function is integrated to show that the contribution of mass bound within virialized cluster regions to the total matter density is small, i.e., OmegaCluster = 0.012^{+0.003}_{-0.004} for cluster masses larger than 6.4^{+0.7}_{-0.6} * 10^{13} h_{50}^{-1} Msun. If light traces mass this also implies that most galaxies sit outside clusters.
The more distant future of cluster cosmology with eROSITA.