Projective space as a branched covering of the sphere with orientable branch set (Q801331)

A classical theorem of Alexander states that every closed, orientable PL n-manifold M is a piecewise linear branched covering of the n-sphere, \(S^ n\), i.e. there is a finite-to-one open PL map \(f: M\to S^ n\). The set of points in \(S^ n\), which are images of points in M where f fails to be a local homeomorphism, is called the branch set. \textit{N. Brand} [Indiana Univ. Math. J. 29, 229-248 (1980; Zbl 0438.57002)] has shown that if the real projective n-space, \({\mathbb{R}}P^ n\), is a branched covering of \(S^ n\) with branch set a locally flat submanifold of \(S^ n\), then \(n=2^ t\pm 1.\) In the present paper the author shows that the values of n can be further limited if the branch set is orientable. Main theorem: If \({\mathbb{R}}P^ n\) is a branched covering of \(S^ n\) with locally flat, orientable branch set, then \(n=1,3\), or 7.