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...

An algorithmic computation of the set of unpointed stable homotopy classes of equivariant fibrewise maps was described in a recent paper [4] of the author and his collaborators. In the present paper, we describe a simplification of this computation that uses an abelian heap structure on this set that was observed in another paper [5] of the author. A heap is essentially a group without a choice of its neutral element; in addition, we allow it to be empty.

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.