A mapping theorem for topological sigma-compact manifolds. (English) Zbl 0626.57009

It is the purpose of this paper to prove a generalization to \(\sigma\)- compact manifolds of a well known result due to M. Brown [Topology of 3-manifolds and related topics, Proc. Univ. Georgia Inst. 1961, 92-94 (1962; Zbl 0132.203)], which asserts the existence of a special kind of continuous, “non-pathological” surjections from the unit n-dimensional cube onto a given compact connected manifold \(M^ n.\)
In the more general setting when \(M^ n\) is \(\sigma\)-compact, the space \({\mathcal E}(M)\) of ends of \(M^ n\) plays an important role: Since \({\mathcal E}(M)\) is a totally disconnected, compact, metrizable space, a set E contained in the boundary of the unit cube \(I^ n\) can be constructed in such a way that E is homeomorphic to \({\mathcal E}(M)\). Now \(I^ n\setminus E\) and \(M^ n\) are two manifolds with the same set of ends. Broadly speaking, our result states that \(M^ n\) is the identification space obtained from \(I^ n\setminus E\) by identifying points within the boundary of \(I^ n\setminus E\) alone.


57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)


Zbl 0132.203
Full Text: Numdam EuDML


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