### A complement to the theory of equivariant finiteness obstructions

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...