Deformation retracts of Stein spaces
Let \(X\) be an \(n\)-dimensional Stein space. It was proved by \textit{H. Hamm} [J. Reine Angew. Math. 338, 121-135 (1983; Zbl 0491.32010); J. Reine Angew. Math. 364, 1-9 (1986; Zbl 0567.32005)], \textit{M.Goresky} and \textit{R. MacPherson} [Stratified Morse Theory. (Springer, Berlin, Heidelberg) (1988; Zbl 0639.14012)] that \(X\) has the homotopy type of a CW-complex of dimension \(\leqq n;\) in the algebraic case this was proved by \textit{K. Karchyauskas} [Homotopy properties of complex algebraic sets, Studies in Topology, Steklov Institute (1979)] with the additional conclusion that the CW-complex is finite. It is therefore natural to ask if there exists a subset \(\mathcal Q\) of \(X\) with the same topological properties like \(X,\) for instance \(\mathcal Q\) is a strong definition retract of \(X,\) and \(\mathcal Q\) is a CW-complex of dimension \(\leqq n.\) The main result of the paper is: There exists a closed subanalytic subset \(\mathcal Q\) of \(X,\) dim\(\mathcal Q\leqq n,\) such that \(\mathcal Q\) is a strong deformation retract of \(X.\) If \(X\) is affine algebraic then \(\mathcal Q\) can be chosen to be compact.
scientific article; zbMATH DE number 1057242