On the link of a stratum in a real algebraic set (Q1196970)

Recall, an Euler space is a triangulable space such that the link of every point on it has an even Euler characteristic. Sullivan proved that real algebraic sets are Euler spaces. Later the reviewer and King and also Benedetti and Dedo showed that the converse of this is also true in dimension two. In dimension greater than two the reviewer and King gave examples of Euler spaces that can not be homeomorphic to algebraic sets. They proved that algebraic sets have to satisfy further topological-combinatorial restrictions. In dimension three they gave necessary and sufficient (topological) conditions for a set to be homeomorphic (or isomorphic) to a real algebraic set. The authors give some generalizations of these results. As an application they give an example of an algebraic set, which topologically imbeds into \(\mathbb{R}^ 4\), but cannot be homeomorphic to an algebraic subset of \(\mathbb{R}^ 4\) [see also the discussion of this result by the reviewer and \textit{A. King} in Real algebraic geometry, Proc. Conf., Rennes/Fr. 1991, Lect. Notes Math. 1524, 120-127 (1992; Zbl 0771.14016)].