  We examine whether future, nearly all-sky galaxy redshift surveys, in
combination with CMB priors, will be able to detect the signature of the cosmic
neutrino background and determine the absolute neutrino mass scale. We also
consider what constraints can be imposed on the effective number of neutrino
species. In particular we consider two spectroscopic strategies in the near-IR,
the so-called "slitless" and "multi-slit" approaches, whose examples are given
by future space-based galaxy surveys, as EUCLID for the slitless case, or
SPACE, JEDI, and possibly WFIRST in the future, for the multi-slit case. We
find that, in combination with Planck, these galaxy probes will be able to
detect at better than 3--sigma level and measure the mass of cosmic neutrinos:
a) in a cosmology-independent way, if the sum of neutrino masses is above 0.1
eV; b) assuming spatial flatness and that dark energy is a cosmological
constant, otherwise. We find that the sensitivity of such surveys is well
suited to span the entire range of neutrino masses allowed by neutrino
oscillation experiments, and to yield a clear detection of non-zero neutrino
mass. The detection of the cosmic relic neutrino background with cosmological
experiments will be a spectacular confirmation of our model for the early
Universe and a window into one of the oldest relic components of our Universe.
