  Quantum reactive scattering calculations for the hydrogen exchange reaction H
+ H$_2$($v=4$, $j=0$) $\to$ H + H$_2$($v'$, $j'$) and its isotopic analogues
are reported for ultracold collision energies. Due to the unique properties
associated with ultracold collisions, it is shown that the geometric phase
effectively controls the reactivity. The rotationally resolved rate
coefficients computed with and without the geometric phase are shown to differ
by up to four orders of magnitude. The effect is also significant in the
vibrationally resolved and total rate coefficients. The dynamical origin of the
effect is discussed and the large geometric phase effect reported here might be
exploited to control the reactivity through the application of external fields
or by the selection of a particular nuclear spin state.
