Repovš, Dušan The recognition problem for topological manifolds: A survey. (English) Zbl 0859.57023 Kodai Math. J. 17, No. 3, 538-548 (1994). This is a survey of recent work on the problem of recognizing topological manifolds. Since J. W. Cannon’s paper on the recognition problem [Bull. Am. Math. Soc. 84, 832-866 (1978; 418.57005)], much progress has been made. A locally compact Hausdorff space \(X\) is said to be a generalized \(n\)-manifold if it is a Euclidean neighborhood retract and if it has the local homology of Euclidean \(n\)-space. The recognition problem is to recognize when a generalized \(n\)-manifold is actually an \(n\)-manifold. Edwards showed that resolvable \(n\)-manifolds for \(n \geq 5\) with a minimal amount of general position properties are indeed manifolds. Quinn showed that there is an integral invariant \(I(X)\) so that \(I(X) = 1\) if and only if \(X\) has a resolution. The recent results of Bryant, Ferry, Mio and Weinberger show that there are generalized \(n\)-manifolds for \(n \geq 6\) that have no resolution. In fact they are not even homotopy equivalent to any topological manifold. In dimension three there are positive results in recognizing when certain generalized homology 3-manifolds are 3-manifolds. These results involve more subtle general position properties than those used in higher dimensions. Reviewer: D.G.Wright (Provo) Cited in 1 ReviewCited in 4 Documents MSC: 57P99 Generalized manifolds Keywords:recognition of manifolds; generalized \(n\)-manifold; general position; resolution Citations:Zbl 0418.57005 PDFBibTeX XMLCite \textit{D. Repovš}, Kodai Math. J. 17, No. 3, 538--548 (1994; Zbl 0859.57023) Full Text: DOI References: [1] J. W. Alexander, An example of a simply connected surface bounding a region which is not simply connected, Proc. Nat. Acad. Sci. U. S. A. 10 (1924), 8-10. · JFM 50.0661.02 [2] M. Bestvina and J. J. 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