Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-29T09:05:18.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic maps of the projective plane with a single triple point

Published online by Cambridge University Press:  22 February 2012

GREG HOWARD  and
SUE GOODMAN
GREG HOWARD
Affiliation:
Federal Reserve Board of Governors and University of North Carolina, Chapel Hill e-mail: greg.l.howard@gmail.com
SUE GOODMAN
Affiliation:
University of North Carolina, Chapel Hill e-mail: seg@email.unc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Cromwell and Marar present an analysis of semi-regular (generic) surfaces with a single triple point and connected self-intersection set. Six of their surfaces are the projective plane, including Boy's surface and Steiner's surface. We build on their work by incorporating twists similar to that of Apery's immersion of the projective plane and show that with a few additional surfaces, all such generic maps of the projective plane are now identified.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 3 , May 2012 , pp. 455 - 472
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[A]Apery, F.Models of the Real Projective Plane (Braunschweig: Vieweg, 1987).CrossRefGoogle Scholar
[B]Banchoff, T. F. Triple points and surgery of immersed surfaces. Proc. Amer. Math. Soc. (1974), 407–413.CrossRefGoogle Scholar
[C]Carter, J. S.How Surfaces Intersect in Space (World Scientific, 1993).CrossRefGoogle Scholar
[CS]Carter, J. C. and Saito, M.Knotted Surfaces and Their Diagrams. Math. Surveys and Monographs, vol. 55 (1998).CrossRefGoogle Scholar
[CM]Cromwell, P. R., and Marar, W. L. Semiregular surfaces with a single triple point. Amer. Math. Soc. Geom. Dedicata (1994), 143–153.CrossRefGoogle Scholar
[FT]Fenn, R. and Taylor, P. On the number of triple points of an immersed surface, (unpublished) preprint (1977).Google Scholar
[GK]Goodman, S. and Kossowski, M.Immersions of the projective plane with one triple point. Differential Geom. Appl. 27 (2009).CrossRefGoogle Scholar
[HH]Hass, J. and Hughes, J.Immersions of surfaces in 3-manifolds. Topology 24 (1985), 97–112.CrossRefGoogle Scholar
[H]Hirsch, M. W. Immersions of manifolds. Trans. Amer. Math. Soc. (1959), 242–272.CrossRefGoogle Scholar
[IM]Izumiya, S. and Marar, W. L. The Euler characteristic of a generic wavefront in a 3-manifold. Proc. Amer. Math. Soc. (1993), 1347–1350.CrossRefGoogle Scholar
[JT]James, I. and Thomas, E.Note on the classification of cross-sections. Topology 4 (1966), 351–359.CrossRefGoogle Scholar
[J]Juhász, A.Regular homotopy classes of local generic mappings. Topology Appl. 138 (2004), 45–59.CrossRefGoogle Scholar
[NS]Ballesteros, J. J. N. and Saeki, O. Euler characteristic formulas for simplicial maps. Math. Proc. Camb. Phil. Soc. (2001), 307–331.CrossRefGoogle Scholar
[P]Pinkall, U.Regular Homotopy classes of immersed surfaces. Topology 24 (1985), 421–434.CrossRefGoogle Scholar
[S]Samelson, H. Orientability of hypersurfaces in Rn. Proc. Amer. Math. Soc. (1969), 301–302.CrossRefGoogle Scholar
[W]Whitney, H. The singularities of a smooth n-manifold in (2n − 1) space. Ann. of Math. (1944), 247–293.CrossRefGoogle Scholar