The spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain with ferromagnetic next-nearestneighbour (NNN) interaction is investigated. The ground state is the Haldane phase for weak NNN interaction, and is the ferromagnetic phase for weak antiferromagnetic interaction. We find a series of topologically distinct spin-gap phases with various magnitudes of edge spins for strong NNN interaction. The phase boundaries between these phases are determined on the basis of the DMRG calculation with additional spins that compensate the edge spins. It is found that each of the exact solutions with short-range antiferromagnetic correlation on the ferromagnetic-nonmagnetic phase boundary is representative of each spin gap phase. Fig. 2. Ground-state phase diagram. The ferromagnetic phase, the Haldane phase, and the intermediate spin-gap phases with edge spin S e in the open chain Hamiltonian (18) are indicated by F, H, and I Se , respectively. The stability limit of the ferromagnetic phase (17) is shown by the dotted line. The open squares are data for the boundary between different spin-gap phases determined numerically in Sect. 4.3. The filled circles correspond to the ''special points'' defined by Dmitriev et al. 18,19) The solid curves are guides for the eye.Fig. 4. J F -dependence of the scaled gap LÁE with the periodic boundary condition for J L ¼ À1:5.