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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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: 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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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