  A comparison of the quantum Heisenberg antiferromagnetic model (QHAM) on the
pyrochlore lattice, the checkerboard lattice and the square lattice with
crossing interactions is performed. The three lattices are constructed with the
same tetrahedral unit cell and this property is used to describe the low energy
spectrum by means of an effective hamiltonian restricted to the singlet sector.
We analyze the structure of the effective hamiltonian and solve it within mean
field approximation for the three lattices.
