  We investigate ground states of $s$=1/2 Heisenberg antiferromagnets on the
eleven two-dimensional (2D) Archimedian lattices by using the coupled cluster
method. Magnetic interactions and quantum fluctuations play against each other
subtly in 2D quantum magnets and the magnetic ordering is thus sensitive to the
features of lattice topology. Archimedean lattices are those lattices that have
2D arrangements of regular polygons and they often build the underlying
magnetic lattices of insulating quasi-two-dimensional quantum magnetic
materials. Hence they allow a systematic study of the relationship between
lattice topology and magnetic ordering. We find that the Archimedian lattices
fall into three groups: those with semiclassical magnetic ground-state
long-range order, those with a magnetically disordered (cooperative quantum
paramagnetic) ground state, and those with a fragile magnetic order. The most
relevant parameters affecting the magnetic ordering are the coordination number
and the degree of frustration present.
