King, H. The topology of real algebraic sets. (English) Zbl 0537.14018 Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 641-654 (1983). [For the entire collection see Zbl 0509.00008.] This paper is a preliminary description of joint work of S. Akbulut and the author on the topology of real algebraic sets. This work is still in progress and is directed toward a characterization in topological terms of topological spaces which are homeomorphic to real algebraic sets. Some partial results in this direction are presented, including the solution of the characterization problem in dimensions \(\leq 6\). The solution is based on a notion of topological resolution of topological spaces, modelled on the notion of resolution of singularities in algebraic geometry. It is not easy to formulate a good notion of topological resolution, and the author express the hope that the current version (”resolution towers”) is the final one. The big part of the methods, including the concept of topological resolution, applies to all dimensions, providing a solution of the characterization problem under the assumption that the following conjecture is true: every compact smooth manifold is diffeomorphic to a nonsingular real algebraic set such that all mod 2 homology classes can be represented by a compact real algebraic subset. The author suspects that this solution does not really depend on the conjecture. [Reviewer’s note: This conjecture is, indeed, false. This was recently proved by R. Benedetti and M. Dedo, cf. ”Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism” (preprint).] This short paper presents a good introduction to the work of Akbulut and the author. But the reader should be aware that not all their results are discussed in the paper. For example, the beautiful theorem about the existence of real algebraic structures on all PL-manifolds is not touched here, see S. Akbulut and the author, Publ. Math., Inst. Hautes Étud. Sci. 53, 79-162 (1981). [In addition to methods discussed in the paper, some pure topological results of S. Akbulut and L. Taylor, Publ. Math., Inst. Hautes Étud. Sci. 53, 163-195 (1981; Zbl 0476.57008) are needed to prove this theorem.] Reviewer: N.V.Ivanov Cited in 3 Documents MSC: 14Pxx Real algebraic and real-analytic geometry 14F45 Topological properties in algebraic geometry 57R95 Realizing cycles by submanifolds 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 57Q99 PL-topology Keywords:topology of real algebraic sets; topological resolution of topological spaces; resolution towers Citations:Zbl 0509.00008; Zbl 0476.57008 PDFBibTeX XML