### 15-vertex triangulations of an 8-manifold.

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A n-knot group is the fundamental group of the complement of an n-sphere smoothly embedded in Sn+2. Artin gave in 1925 ([A]) an algebraic characterization of 1-knot groups. M. Kervaire gave in 1965 ([K]) an algebraic characterization of n-knot groups for n ≥ 3. The problem of characterizing algebraically 2-knot groups has been posed several times (see for example [Su, Problem 4.7]). Ribbon 2-knot groups have been characterized algebraically by Yajima [Y].We will give here a characterization of 2-knot...

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...