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4-dimensional $$c$$-symplectic $$S^1$$-manifolds with non-empty fixed point set need not be $$c$$-Hamiltonian. (English) Zbl 0994.57032
Oprea, John (ed.) et al., Homotopy and geometry. Proceedings of the workshop, Banach Center, Warsaw, Poland, June 9-13, 1997. Warsaw: Polish Academy of Sciences, Banach Cent. Publ. 45, 91-93 (1998).
It is known that if the circle group acts on a compact symplectic 4-manifold with non-empty fixed point set, then the action is necessarily Hamiltonian and this phenomenon does not occur in higher dimensions. Let $$(M,x)$$ be a cohomologically symplectic manifold, which means $$M$$ is $$2n$$-dimensional and $$x$$ is a cohomology class in $$H^2(M)$$ such that $$x^n\neq 0$$ in $$H^{2n}(M)$$. Suppose that the group $$S^1$$ acts on $$M$$. The author addresses the following questions posed by J. Oprea: Whether $$(M,x)$$ is $$c$$-Hamiltonian when $$\dim M= Y$$ and the fixed point set $$M^{S^1}$$ is non-empty? Note that by definition $$(M,x)$$ is called $$c$$-Hamiltonian of $$\lambda(x)= 0$$, where for $$x\in H^q(M)$$, $$\lambda(x)\in H^{q-1}(M)$$ defined by $$\varphi^*(x)= 1\otimes x+ u^*_1\otimes \lambda(x)$$, where $$u^*_1\in H^1(S^1)$$ is the standard generator. The author shows that there are examples in which $$\dim M= 4$$, $$M^{S_1}\neq\emptyset$$ but $$(M,x)$$ is not $$c$$-Hamiltonian.
For the entire collection see [Zbl 0906.00019].
##### MSC:
 57S25 Groups acting on specific manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension 53D35 Global theory of symplectic and contact manifolds
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