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ArticleOriginal scientific text

Title

Surfaces in 3-space that do not lift to embeddings in 4-space

Authors 1, 2

Affiliations

  1. University of South Alabama, Mobile, Alabama 36688, U.S.A.
  2. University of South Florida, Tampa, Florida 33620, U.S.A.

Abstract

A necessary and sufficient condition for an immersed surface in 3-space to be lifted to an embedding in 4-space is given in terms of colorings of the preimage of the double point set. Giller's example and two new examples of non-liftable generic surfaces in 3-space are presented. One of these examples has branch points. The other is based on a construction similar to the construction of Giller's example in which the orientation double cover of a surface with odd Euler characteristic is immersed in general position. A similar example is shown to be liftable with an explicit lifting given. The problem of lifting is discussed in relation to the theory of surface braids. Finally, the orientations of the double point sets are studied in relation to the lifting problem.

Keywords

ENG
orientations, triple points, Boy's surface, knotted surfaces

Bibliography

  1. F. Apery, 'Models of the Real Projective Plane,' Vieweg (Braunschweig 1987).
  2. T. F. Banchoff, Triple Points and Surgery of Immersed Surfaces, Proc. AMS 46, No.3 (Dec. 1974), 403-413.
  3. J. S. Carter, 'How Surfaces Intersect in Space: an Introduction to Topology,' World Scientific Publishing, 2nd edition (Singapore 1995).
  4. J. S. Carter and M Saito, Reidemeister Moves for Surface Isotopies and Their Interpretation as Moves to Movies, J. of Knot Theory and its Ram., vol. 2, no. 3, (1993), 251-284.
  5. J. S. Carter and M. Saito, Braids and Movies, J. of Knot Theory and its Ram. 5, no. 5 (1996), 589-608.
  6. J. S. Carter and M. Saito, Knotted Surfaces, Braid Movies and Beyond, in Baez, J., 'Knots and Quantum Gravity,' Oxford Science Publishing (Oxford 1994), 191-229.
  7. J. S. Carter and M. Saito, Knot Diagrams and Braid Theories in Dimension 4, Real and Complex Singularities, W. L. Marar (ed.), Pitman Res. Notes Math. Ser. 333, Longman Sci. Tech., Harlow, 1995, 112-147.
  8. J. S. Carter, D. E. Flath and M. Saito, 'The Classical and Quantum 6j-symbols,' Princeton University Press Lecture Notes in Math Series (1995).
  9. G. F. Francis, 'A Topological Picturebook,' Springer-Verlag (New York 1987).
  10. A. Hansen, Knot^4, a video presented at SIGGRAPH '93. His software Meshview was used to produce this video.
  11. C. Giller, Towards a Classical Knot Theory for Surfaces in R4, Illinois Journal of Mathematics 26, No. 4, (Winter 1982), 591-631.
  12. S. Kamada, Surfaces in R4 of braid index three are ribbon, Journal of Knot Theory and its Ramifications 1 (1992), 137-160.
  13. S. Kamada, 2-dimensional braids and chart descriptions, 'Topics in Knot Theory', Proceedings of the NATO Advanced Study Institute on Topics in Knot Theory, Turkey, (1992), 277-287.
  14. S. Kamada, A characterization of groups of closed orientable surfaces in 4-space, Topology 33 (1994), 113-122.
  15. S. Kamada, Generalized Alexander's and Markov's theorems in dimension four, Preprint.
  16. U. Koschorke, Multiple points of Immersions and the Kahn-Priddy Theorem, Math Z. 169 (1979), 223-236.
  17. G. Mikhalkin and M. Polyak, Whitney formula in higher dimensions, J. Differential Geom. 44 (1996), 583-594.
  18. D. Roseman, Reidemeister-type moves for surfaces in four dimensional space, this volume.
  19. D. Roseman, Twisting and Turning in Four Dimensions, A video made at the Geometry Center (1993).
Banach Center Publications
1998/42/1
Export to PBN, Opens in new tab
Pages:
29-47
Main language of publication
English
Published
1998
Exact and natural sciences