  LSST will open new vistas for cosmology in the next decade, but it cannot
reach its full potential without data from other telescopes. Cosmological
constraints can be greatly enhanced using wide-field ($>20$ deg$^2$ total
survey area), highly-multiplexed optical and near-infrared multi-object
spectroscopy (MOS) on 4-15m telescopes. This could come in the form of
suitably-designed large surveys and/or community access to add new targets to
existing projects. First, photometric redshifts can be calibrated with high
precision using cross-correlations of photometric samples against spectroscopic
samples at $0 < z < 3$ that span thousands of sq. deg. Cross-correlations of
faint LSST objects and lensing maps with these spectroscopic samples can also
improve weak lensing cosmology by constraining intrinsic alignment systematics,
and will also provide new tests of modified gravity theories. Large samples of
LSST strong lens systems and supernovae can be studied most efficiently by
piggybacking on spectroscopic surveys covering as much of the LSST
extragalactic footprint as possible (up to $\sim20,000$ square degrees).
Finally, redshifts can be measured efficiently for a high fraction of the
supernovae in the LSST Deep Drilling Fields (DDFs) by targeting their hosts
with wide-field spectrographs. Targeting distant galaxies, supernovae, and
strong lens systems over wide areas in extended surveys with (e.g.) DESI or MSE
in the northern portion of the LSST footprint or 4MOST in the south could
realize many of these gains; DESI, 4MOST, Subaru/PFS, or MSE would all be
well-suited for DDF surveys. The most efficient solution would be a new
wide-field, highly-multiplexed spectroscopic instrument in the southern
hemisphere with $>6$m aperture. In two companion white papers we present gains
from deep, small-area MOS and from single-target imaging and spectroscopy.
