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Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifoldequipped with a semi-free Hamiltonian $S^1$ action such that the fixed pointset consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in\cite{L}, we defined a certain invariant of such spaces which consists of fixedpoint data and twist type, and we divided the possible values of theseinvariants into six ``types''. In this paper, we construct such manifolds withthese ``types''. As a consequence, we have a precise list of the values ofthese invariants.

On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions