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**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.

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.

### MSC:

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

### Citations:

Zbl 0132.203### References:

[1] | L.V. Ahlfors and L. Sario : Riemann Surfaces . Princeton University Press (1960). · Zbl 0196.33801 |

[2] | R. Berlanga and D.B.A. Epstein : Measures on sigma-compact manifolds and their equivalence under homeomorphism . J. London Math. Soc. (2) 27(1983) 63-74. · Zbl 0523.28013 · doi:10.1112/jlms/s2-27.1.63 |

[3] | R. Berlanga : Homeomorphisms preserving a good measure in a manifold . Ph.D. Warwick (1983). |

[4] | M. Brown : A mapping theorem for untriangulated manifolds . In: M.K. Fort (editor): Topology of 3-manifolds and related topics . Prentice Hall (1963) 92-94. · Zbl 1246.57052 |

[5] | W. Hurewicz and H. Wallman : Dimension Theory . Princeton University Press (1948). · Zbl 0036.12501 |

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