Canceling branch points on projections of surfaces in $4$-space

Authors:
J. Scott Carter and Masahico Saito

Journal:
Proc. Amer. Math. Soc. **116** (1992), 229-237

MSC:
Primary 57Q35; Secondary 57Q45

DOI:
https://doi.org/10.1090/S0002-9939-1992-1126191-0

MathSciNet review:
1126191

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Abstract | References | Similar Articles | Additional Information

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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*The normal Euler class of a polyhedral surface in*$4$

*-space*, unpublished notes and letters. T. F. Banchoff and Ockle Johnson (to appear).

*Reidemeister-type moves for surfaces in four dimensional space*, preprint.

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Keywords:
Embedded surfaces in <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$4$">-space,
projections,
branch points,
normal Euler number

Article copyright:
© Copyright 1992
American Mathematical Society