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

J. Carter; Masahico Saito

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 29-47
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Carter, J., and Saito, Masahico. "Surfaces in 3-space that do not lift to embeddings in 4-space." Banach Center Publications 42.1 (1998): 29-47. <http://eudml.org/doc/208814>.

@article{Carter1998,
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.},
author = {Carter, J., Saito, Masahico},
journal = {Banach Center Publications},
keywords = {orientations; triple points; Boy's surface; knotted surfaces; surface; immersion; embedding; lifting},
language = {eng},
number = {1},
pages = {29-47},
title = {Surfaces in 3-space that do not lift to embeddings in 4-space},
url = {http://eudml.org/doc/208814},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Carter, J.
AU - Saito, Masahico
TI - Surfaces in 3-space that do not lift to embeddings in 4-space
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 29
EP - 47
AB - 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.
LA - eng
KW - orientations; triple points; Boy's surface; knotted surfaces; surface; immersion; embedding; lifting
UR - http://eudml.org/doc/208814
ER -

References

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  1. [1] F. Apery, 'Models of the Real Projective Plane,' Vieweg (Braunschweig 1987). Zbl0623.57001
  2. [2] T. F. Banchoff, Triple Points and Surgery of Immersed Surfaces, Proc. AMS 46, No.3 (Dec. 1974), 403-413. Zbl0309.57017
  3. [3] J. S. Carter, 'How Surfaces Intersect in Space: an Introduction to Topology,' World Scientific Publishing, 2nd edition (Singapore 1995). Zbl0855.57001
  4. [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. Zbl0808.57020
  5. [5] J. S. Carter and M. Saito, Braids and Movies, J. of Knot Theory and its Ram. 5, no. 5 (1996), 589-608. Zbl0889.57034
  6. [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. Zbl0859.57027
  7. [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. Zbl0849.57022
  8. [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). Zbl0851.17001
  9. [9] G. F. Francis, 'A Topological Picturebook,' Springer-Verlag (New York 1987). Zbl0612.57001
  10. [10] A. Hansen, Knot 4 , a video presented at SIGGRAPH ’93. His software Meshview was used to produce this video. 
  11. [11] C. Giller, Towards a Classical Knot Theory for Surfaces in R 4 , Illinois Journal of Mathematics 26, No. 4, (Winter 1982), 591-631. Zbl0476.57009
  12. [12] S. Kamada, Surfaces in R 4 of braid index three are ribbon, Journal of Knot Theory and its Ramifications 1 (1992), 137-160. Zbl0763.57013
  13. [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. [14] S. Kamada, A characterization of groups of closed orientable surfaces in 4-space, Topology 33 (1994), 113-122. Zbl0820.57017
  15. [15] S. Kamada, Generalized Alexander's and Markov's theorems in dimension four, Preprint. Zbl0831.57013
  16. [16] U. Koschorke, Multiple points of Immersions and the Kahn-Priddy Theorem, Math Z. 169 (1979), 223-236. Zbl0406.57017
  17. [17] G. Mikhalkin and M. Polyak, Whitney formula in higher dimensions, J. Differential Geom. 44 (1996), 583-594. Zbl0937.53008
  18. [18] D. Roseman, Reidemeister-type moves for surfaces in four dimensional space, this volume. Zbl0906.57010
  19. [19] D. Roseman, Twisting and Turning in Four Dimensions, A video made at the Geometry Center (1993). 

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