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The aim of this article is to answer a question posed by J. Oprea in his talk at the Workshop “Homotopy and Geometry”. In equivariant symplectic geometry it is often important to give a sufficient condition for the action to be Hamiltonian. For example McDuff [M] showed that, if the circle group acts on a compact symplectic 4-manifold with non-empty fixed point set, then the action is necessarily Hamiltonian. This phenomenon does not occur in higher dimensions. Let S act on a symplectic manifold (M,ω). If X denotes the vector field on M generated by the action, then the action is called Hamiltonian if the 1-form i(X)ω is exact. This condition is also equivalent to the existence of an equivariant cohomology class in H S1(M) descending to [ω] ∈ H(M) [A-B]. This cohomological condition can also be expressed in several apparently different ways, e.g. [H-Y, G]. Here we utilize the following one [L-O]. Let φ : S ×M → M denote the given action and φ∗ : H∗(M) → H∗(S1 ×M) = H∗(S1) ⊗H∗(M) the induced homomorphism. Here the coefficients are taken in the reals. For x ∈ H(M) let λ(x) ∈ Hq−1(M) be defined by φ∗(x) = 1⊗ x+ u1 ⊗ λ(x)

4-dimensional c-symplectic $S^1$-manifolds with non-empty fixed point set need not be c-Hamiltonian