Canceling branch points on projections of surfaces in 4-space

Carter, J. Scott; Saito, Masahico

doi:10.1090/S0002-9939-1992-1126191-0

Canceling branch points on projections of surfaces in $4$-space
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by J. Scott Carter and Masahico Saito PDF

Proc. Amer. Math. Soc. 116 (1992), 229-237 Request permission

Abstract:

A surface embedded in $4$-space projects to a generic map in $3$-space that may have branch points—each contributing $\pm 1$ to the normal Euler class of the surface. The sign depends on crossing information near the branch point. A pair of oppositely signed branch points are geometrically canceled by an isotopy of the surface in $4$-space. In particular, any orientable manifold is isotopic to one that projects without branch points. This last result was originally obtained by Giller. Our methods apply to give a proof of Whitney’s theorem.

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