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

The aim of this article is to answer a question posed by J. Oprea in his talk at the Workshop "Homotopy and Geometry".

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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".

It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family ${w}_{\alpha}^{H}\left(X\right)$ of elements of the groups ${K}_{0}\left(\mathbb{Z}\left[{\pi}_{0}{\left(WH\left(X\right)\right)}_{\alpha}^{*}\right]\right)$. We prove that every family ${w}_{\alpha}^{H}$ of elements of the groups ${K}_{0}\left(\mathbb{Z}\left[{\pi}_{0}{\left(WH\left(X\right)\right)}_{\alpha}^{*}\right]\right)$ can be realized as the family of equivariant finiteness obstructions ${w}_{\alpha}^{H}\left(X\right)$ of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall’s obstruction...

Sullivan associated a uniquely determined $DGA{|}_{\mathbf{Q}}$ to any simply connected simplicial complex $E$. This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space $E$. In case $E$ is the total space of a principal $G$-bundle, ($G$ is a compact connected Lie-group) we associate a $G$-equivariant model ${U}_{G}\left[E\right]$, which is a collection of “$G$-homotopic” $DGA$’s${|}_{\mathbf{R}}$ with $G$-action. ${U}_{G}\left[E\right]$ will, in general, be different from the Sullivan’s minimal model of the space $E$. ${U}_{G}\left[E\right]$ contains the total rational...