  The temperature of the diffuse, photo-heated intergalactic medium (IGM)
depends on its reionization history because the thermal time scales are long.
The widths of the hydrogen Lyman-alpha absorption lines seen in the spectra of
distant quasars that arise in the IGM can be used to determine its temperature.
We use a wavelet analysis of the Lyman-alpha forest region of quasar spectra to
demonstrate that there is a relatively sudden increase in the line widths
between redshifts z=3.5 and 3.0, which we associate with entropy injection
resulting from the reionization of HeII. The subsequent fall-off in temperature
after z~3.5, is consistent with a thermal evolution dominated by adiabatic
expansion. If, as expected, the temperature also drops rapidly after hydrogen
reionization, then the high temperatures inferred from the line widths before
HeII reionization imply that hydrogen reionization occurred below redshift z=9.
