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A Heisenberg spin-$s$ chain with alternating ferromagnetic ($-J_1^F<0$) andantiferromagnetic ($J_1^A>0$) nearest-neighbor (NN) interactions, exhibits theDimer and spin-$2s$ Haldane phases in the limits $J_1^F/J_1^A \rightarrow 0$and $J_1^F/J_1^A \rightarrow \infty$ respectively. These two phases areunderstood to be topologically equivalent. Induction of the frustration throughthe next nearest-neighbor ferromagnetic interaction ($-J_2^F<0$) produces avery rich quantum phase diagram. With frustration, the whole phase diagram isdivided into a ferromagnetic (FM) and a nonmagnetic (NM) phase. For $s=1/2$,the full NM phase is seen to be of Haldane-Dimer type, but for $s>1/2$, aspiral phase comes between the FM and the Haldane-Dimer phases. The study of asuitably defined string-order parameter and spin-gap at the phase boundaryindicates that the Haldane-Dimer and spiral phases have different topologicalcharacters. We also find that, along the $J_2^F=\frac 12 J_1^F$ line in the NMphase, an NN dimer state is the {\it exact} groundstate, provided$J_1^A>J_C=\kappa J_1^F$ where $\kappa \le s + h$ for applied magnetic field$h$. Without magnetic field, the position of $J_C$ is on the FM-NM phaseboundary when $s=1/2$, but for $s>1/2$, the location of $J_C$ is on the phaseseparation line between the Haldane-Dimer and spiral phases.

Haldane and Dimer phases in a frustrated spin chain: an exact groundstate and associated topological phase transition