An end theorem for stratified spaces
A fundamental theorem of geometric topology characterizes those topological manifolds which are the interiors of compact manifolds with boundary. The general case of this theorem was proved by \textit{L. C. Siebenmann} (1965) in his Ph.D. Thesis and it says that a tame ended topological \(n\)-manifold \(X\) (\(n \geq 6\)) is the interior of a compact manifold with boundary if and only if the Siebenmann obstruction \(\sigma(X) \in \widetilde {K}_{0}({\mathbb Z}\pi)\) vanishes, where \(\pi\) is the fundamental group of the end of \(X\). \textit{F. Quinn} [Ann. Math., II. Ser. 110, 275-331 (1979; Zbl 0394.57022); Invent. Math. 68, 353-424 (1982; Zbl 0533.57008)] generalizes Siebenmann's result greatly using his controlled \(K\)-theory. For any locally compact pair \((X,A)\), where \(A\) is a closed and tame ANR in \(X\) and \(X-A\) is a topological \(n\)-manifold (\(n \geq 6\)), he defines a \(K\)-theoretic obstruction, which vanishes if and only if \(A\) has a mapping cylinder neighborhood in \(X\). In the paper under review, the authors show, using localized control methods of Quinn, that the Siebenmann theorem generalizes to the context of manifold stratified spaces in the sense of \textit{F. Quinn} [J. Am. Math. Soc. 1, No. 2, 441-499 (1988; Zbl 0655.57010)]. Roughly, a manifold stratified space is a locally compact Hausdorff space \(X\) together with a finite filtration by closed \(m\)-skeleta \(X^m\), \(X = X^n \supset X^{n-1} \supset \ldots \supset X^{-1} = \emptyset\), such that each stratum \(X_m = X^m - X^{m-1}\) is a separable \(m\)-manifold with neighborhoods in \(X_m \cup X_k\) (for \(k>m\)) which have the local homotopy properties of mapping cylinders of fibrations. \(X\) is tame ended if the one point compactification of \(X\) is again a manifold stratified space. The boundaries of all strata form the boundary \(\partial X\) of \(X\). For a manifold stratified space \(X\) with empty boundary, a completion of \(X\) is a compact stratified space \(\overline {X}\) such that \(X = \overline {X} - \partial\overline {X}\), and \(\partial\overline {X}\) has a collar neighborhood in \(\overline {X}\). A completion may not always exist. The authors prove that if \(X\) is a manifold stratified space, with empty boundary, which admits a completion \(\overline {X}\), then a single \(K\)-theoretic obstruction \(\gamma_{\ast}(X)\) vanishes. The obstruction \(\gamma_{\ast}(X)\) is a localization of Quinn's mapping cylinder neighborhood obstruction, and it reduces to the Siebenmann obstruction if \(X\) is a manifold. Moreover, they prove that if \(\gamma_{\ast}(X)=0\), where \(X\) is a tame ended manifold stratified space with empty boundary, then for any closed subset \(A\) of \(X\) containing \(X^5\), which is the union of components of strata and admits a completion \(\overline {A}\), \(X\) also admits a completion \(\overline {X}\) and additionally \(Cl_{\overline {X}}(A)=\overline {A}\). Finally, these results are applied to stratified \(G\)-manifolds in the sense of \textit{S. Weinberger} [The topological classification of stratified spaces (1994; Zbl 0826.57001)].
scientific article; zbMATH DE number 1337382
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