The topology of real algebraic sets
This paper is a survey of the author's work on the question: ''Which topological spaces are homeomorphic to real algebraic sets?''. In the non-singular case it is known [due to \textit{J. Nash}, Ann. Math. (2) 56, 405--421 (1952; Zbl 0048.38501) and \textit{A. Tognoli}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 27, 167--185 (1973; Zbl 0263.57011)] that any compact smooth manifold is diffeomorphic to a real algebraic set. In the singular case, the idea emphasized by this paper is the following: starting with a topological space, first try to have some topological resolution of singularities; then use the preceding result for the non- singular case, and end on the algebraic side by algebraic ''blowing-downs''. The authors give their results in the case of isolated singularities, in the case of P. L. manifolds. They describe a class \({\mathcal R}\) of topological spaces which admit a ''topological resolution tower'', a serious candidate to be the class of spaces homeomorphic to real algebraic sets (and, in any case, containing this class). Still, there would be the problem of finding combinatorial conditions which characterize real algebraic sets. Sullivan's theorem on local Euler characteristic [\textit{D. Sullivan}, Proc. Liverpool singularities - Sympos. I, Dep. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. 192, 165--168 (1971; Zbl 0227.32005)] gives such a necessary condition. The authors prove that it is sufficient for compact polyhedra of dimension \(\leq 2\), but not for higher dimension: they give an example of compact polyhedron of dimension 3 which satisfies the local Euler characteristic condition but which is not in \({\mathcal R}\). [This work was published twice, the second time in the book announced in this Zbl 0527.00017.]
scientific article; zbMATH DE number 3859290