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
HTML articles powered by AMS MathViewer

by J. Scott Carter and Masahico Saito

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

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

PDF | 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.

References

Thomas F. Banchoff, Double tangency theorems for pairs of submanifolds, Geometry Symposium, Utrecht 1980 (Utrecht, 1980) Lecture Notes in Math., vol. 894, Springer, Berlin-New York, 1981, pp. 26–48. MR 655418
Thomas F. Banchoff, Normal curvatures and Euler classes for polyhedral surfaces in $4$-space, Proc. Amer. Math. Soc. 92 (1984), no. 4, 593–596. MR 760950, DOI 10.1090/S0002-9939-1984-0760950-4

The normal Euler class of a polyhedral surface in

-space

Sylvain E. Cappell and Julius L. Shaneson, An introduction to embeddings, immersions and singularities in codimension two, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 129–149. MR 520529
Cole A. Giller, Towards a classical knot theory for surfaces in $\textbf {R}^{4}$, Illinois J. Math. 26 (1982), no. 4, 591–631. MR 674227
Seiichi Kamada, Nonorientable surfaces in $4$-space, Osaka J. Math. 26 (1989), no. 2, 367–385. MR 1017592
Dennis Roseman, Projections of codimension two embeddings, Knots in Hellas ’98 (Delphi), Ser. Knots Everything, vol. 24, World Sci. Publ., River Edge, NJ, 2000, pp. 380–410. MR 1865719, DOI 10.1142/9789812792679_{0}024

Reidemeister-type moves for surfaces in four dimensional space

Bruce Trace, A general position theorem for surfaces in Euclidean $4$-space, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 123–137. MR 813108, DOI 10.1090/conm/044/813108
Hassler Whitney, On the topology of differentiable manifolds, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, pp. 101–141. MR 0005300