  This is the fourth of a series of papers in which we derive simultaneous
constraints on cosmological parameters and X-ray scaling relations using
observations of the growth of massive, X-ray flux-selected galaxy clusters.
Here we examine the constraints on neutrino properties that are enabled by the
precise and robust constraint on the amplitude of the matter power spectrum at
low redshift that is available from our data. In combination with cluster
gas-mass fraction, cosmic microwave background, supernova and baryon acoustic
oscillation data, and incorporating conservative allowances for systematic
uncertainties, we limit the species-summed neutrino mass, M_nu, to <0.33 eV at
95.4 per cent confidence in a spatially flat, cosmological constant (LambdaCDM)
model. In a flat LambdaCDM model where the effective number of neutrino
species, N_eff, is allowed to vary, we find N_eff = 3.4 -0.5 +0.6 (68.3 per
cent confidence, incorporating a direct constraint on the Hubble parameter from
Cepheid and supernova data). We also obtain results with additional degrees of
freedom in the cosmological model, in the form of global spatial curvature
(Omega_k) and a primordial spectrum of tensor perturbations (r and n_t). The
results are not immune to these generalizations; however, in the most general
case we consider, in which M_nu, N_eff, curvature and tensors are all free, we
still obtain M_nu < 0.70 eV and N_eff = 3.7 +- 0.7 (at respectively the same
confidence levels as above). These results agree well with recent work using
independent data, and highlight the importance of measuring cosmic structure
and expansion at low as well as high redshifts. Although our cluster data
extend to redshift z=0.5, the effect of neutrino mass on the growth of
structure at late times is not yet detected at a significant level.
