  We measure rotational broadening in spectra taken by the Apache Point
Observatory Galactic Evolution Experiment (APOGEE) survey to characterise the
relationship between stellar multiplicity and rotation. We create a sample of
2786 giants and 24 496 dwarfs with stellar parameters and multiple radial
velocities from the APOGEE pipeline, projected rotation speeds \vsini\
determined from our own pipeline, and distances, masses, and ages measured by
Sanders \& Das. We use the statistical distribution of the maximum shift in the
radial velocities, \drvm, as a proxy for the close binary fraction to explore
the interplay between stellar evolution, rotation, and multiplicity. Assuming
that the minimum orbital period allowed is the critical period for Roche Lobe
overflow and rotational synchronization, we calculate theoretical upper limits
on expected \vsini\ and \drvm\ values. These expectations agree with the
positive correlation between the maximum \drvm\ and \vsini\ values observed in
our sample as a function of \logg. We find that the fast rotators in our sample
have a high occurrence of short-period ($\log(P/\text{d})\lesssim 4$)
companions. We also find that old, rapidly-rotating main sequence stars have
larger completeness-corrected close binary fractions than their younger peers.
Furthermore, rapidly-rotating stars with large \drvm\ consistently show
differences of 1-10 Gyr between the predicted gyrochronological and measured
isochronal ages. These results point towards a link between rapid rotation and
close binarity through tidal interactions. We conclude that stellar rotation is
strongly correlated with stellar multiplicity in the field, and caution should
be taken in the application of gyrochronology relations to cool stars.
