Celebratio Mathematica — Neumann — Complete Bibliography

Complete Bibliography

Works connected to Ulrich Hirsch

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U. Hirsch and W. D. Neumann: “On cyclic branched coverings of spheres,” Math. Ann. 215 (October 1975), pp. 289–291. MR 375321 Zbl 0289.57003 article

In the \( PL \) category every orientable closed \( n \)-manifold \( M^n \) can be exhibited as a branched covering of the \( n \)-sphere \( S^n \) [Alexander 1920]. If \( M^n \) is given one can therefore ask for the minimal integer \( d \geq 1 \) for which there exists such a map whose degree is equal to \( d \). If \( n = 2 \) then always \( d\leq 2 \). If \( n = 3 \) then \( d \leq 3 \), ([Hilden 1974; Hirsch 1974, §7.3; Montesinos 1974]). In general these maps are irregular.

In this note we are concerned however with a class of regular branched coverings of \( S^n \). With reference to papers by Fox [1972], Montesinos [1975], and Birman and Hilden [1973] it is shown that there is a big class of orientable closed manifolds which cannot be realized as cyclic, in particular as 2-fold branched coverings of the sphere.

D. Eisenbud, U. Hirsch, and W. Neumann: “Transverse foliations of Seifert bundles and self-homeomorphism of the circle,” Comment. Math. Helv. 56 : 4 (1981), pp. 638–660. MR 656217 Zbl 0516.57015 article

David Eisenbud	Related
Ulrich Hirsch	Related