  GJ 1132b, which orbits an M dwarf, is one of the few known Earth-sized
planets, and at 12 pc away it is one of the closest known transiting planets.
Receiving roughly 19x Earth's insolation, this planet is too hot to be
habitable but can inform us about the volatile content of rocky planet
atmospheres around cool stars. Using Hubble STIS spectra, we search for a
transit in the Lyman-alpha line of neutral hydrogen (Ly-alpha). If we were to
observe a deep Ly-alpha absorption signature, that would indicate the presence
of a neutral hydrogen envelope flowing from GJ 1132b. On the other hand, ruling
out deep absorption from neutral hydrogen may indicate that this planet does
not have a detectable amount of hydrogen loss, is not losing hydrogen, or lost
hydrogen and other volatiles early in the star's life. We do not detect a
transit and determine a 2-sigma upper limit on the effective envelope radius of
0.36 R* in the red wing of the Ly-alpha line, which is the only portion of the
spectrum we detect after absorption by the ISM. We analyze the Ly-alpha
spectrum and stellar variability of GJ1132, which is a slowly-rotating 0.18
solar mass M dwarf with previously uncharacterized UV activity. Our data show
stellar variabilities of 5-22%, which is consistent with the M dwarf UV
variabilities of up to 41% found by \citet{Loyd2014}. Understanding the role
that UV variability plays in planetary atmospheres is crucial to assess
atmospheric evolution and the habitability of cooler rocky exoplanets.
