  We study the spin-1/2 Heisenberg model in a lattice that interpolates between
the triangular and the kagome lattices. The exchange interaction along the
bonds of the kagome lattice is J, and the one along the bonds connecting kagome
and non-kagome sites is J', so that J'=J corresponds to the triangular limit
and J'=0 to the kagome one. We use variational and exact diagonalization
techniques. We analyze the behavior of the order parameter for the
antiferromagnetic phase of the triangular lattice, the spin gap, and the
structure of the spin excitations as functions of J'/J. Our results indicate
that the antiferromagnetic order is not affected by the reduction of J' down to
J'/J ~ 0.2. Below this value, antiferromagnetic correlations grow weaker, a
description of the ground state in terms of a Neel phase renormalized by
quantum fluctuations becomes inadequate, and the finite-size spectra develop
features that are not compatible with antiferromagnetic ordering. However, this
phase does not appear to be connected to the kagome phase as well, as the
low-energy spectra do not evolve with continuity for J'-> 0 to the kagome
limit. In particular, for any non-zero value of J', the latter interaction sets
the energy scale for the low-lying spin excitations, and a gapless triplet
spectrum, destabilizing the kagome phase, is expected.
