  We study possible quantum phases of the Heisenberg antiferromagnet on the
planar pyrochlore lattice, also known as the checkerboard lattice or the square
lattice with crossings. It is assumed that the exchange coupling on the
square-lattice-links is not necessarily the same as those along the crossing
links. When all the couplings are the same, this model may be regarded as a two
dimensional analog of the pyrochlore antiferromagnet. The large-N limit of the
Sp($N$) generalized model is considered and the phase diagram is obtained by
analyzing the fluctuation effects about the $N \to \infty$ limit. We find a
topologically-ordered $Z_2$-spin-liquid phase in a narrow region of the phase
diagram as well as the plaquette-ordered and staggered spin-Peierls phases as
possible quantum-disordered paramagnetic phases.
