  We present an exact mapping between two simple spin models: the
Fredrickson-Andersen (FA) model and a model of annihilating random walks with
spontaneous creation from the vacuum, A+A <-> 0. We discuss the geometric
structure of the mapping and its consequences for symmetries of the models.
Hence we are able to show that the upper critical dimension of the FA model is
two, and that critical exponents are known exactly in all dimensions. These
conclusions also generalise to a mapping between A+A <-> 0 and the
reaction-diffusion system in which the reactions are branching and coagulation,
A+A <-> A. We discuss the relation of our analysis to earlier work, and explain
why the models considered do not fall into the directed percolation
universality class.
