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

If RP " is a branched covering of S " with locally flat, orientable branch set, then n — 1,3, or 7. 1. Introduction. Let M be a closed, orientable PL ^-manifold. A theorem of Alexander [2] states that every such manifold is a piecewise linear branched covering of the /i-sphere, Sn, i.e. there is a finite-to-one open PL map /: M-> Sn. The subset of M where / fails to be a local homeomorphism is called the singular set and the image of the singular se