Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve
- 38 Downloads
- 2 Citations
Abstract
Let γ be a smooth generic curve in ℝP3. Denote by C the number of its flattening points, and by T the number of planes tangent to γ at three distinct points. Consider the osculating planes to γ at the flattening points. Let N denote the total number of points where γ intersects these osculating plane transversally. Then T ≡ [N + θ(γ)C]/2 (mod 2), where θ(γ) is the number of noncontractible components of γ. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in ℝ3 has no flattening points, then the number of its triple tangent planes is even. We also give multidimensional analogs of this formula and show that these results follow from certain general facts about the topology of codimension 1 singularities of stable maps between manifolds having the same dimension.
Key words
smooth space curve stable map general map corank 1 singularity flattening point osculating plane real projective space smooth manifoldPreview
Unable to display preview. Download preview PDF.
REFERENCES
- 1.Th. Banchoff, T. Gaffney, and C. McCrory, “Counting tritangent planes of space curves,” Topology, 24 (1985), no. 1, 15–23.Google Scholar
- 2.M. H. Freedman, “Planes triply tangent to curves with nonvanishing torsion,” Topology, 19 (1980), no. 1, 1–8.CrossRefGoogle Scholar
- 3.B. Morin, “Formes canoniques des singularites d'une application differentiable,” C. R. Acad. Sci. Paris, 260 (1965), 5662–5665.Google Scholar
- 4.V. I. Arnol'd, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps. vol. I. The Classification of Critical Points, Caustics and Wave Fronts, Birkhauser Boston, Inc., Boston, MA, 1985.Google Scholar
- 5.V. A. Vasil'ev, Lagrange and Legendre Characteristic Classes, Gordon and Breach Science Publishers, New York, 1988.Google Scholar
- 6.M. E. Kazarian, “Thom polynomials for Lagrange, Legendre, and critical point function singularities,” Proc. London Math. Soc., 86 (2003), no. 3, 707–734.CrossRefGoogle Scholar
- 7.V. D. Sedykh, “Resolution of singularities of corank 1 of a front in general position,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 37 (2003), no. 2, 52–64.Google Scholar
- 8.C. McCrory and A. Parusinski, “Algebraically constructible functions,” Ann. Sci. Ecole Norm. Sup., 30 (1997), no. 4, 527–552.Google Scholar
- 9.J. N. Mather, “Stratifications and mappings,” Uspekhi Mat. Nauk [Russian Math. Surveys], 27 (1972), no. 5, 85–118.Google Scholar
- 10.Fuster M. C. Romero, “Sphere stratifications and the Gauss map,” Proc. Roy. Soc. Edinburgh. Sect. A, 95 (1983), 115–136.Google Scholar
- 11.V. D. Sedykh, “On the topology of the image of a stable smooth mapping with singularities of corank 1,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 395 (2004), no. 4, 459–463.Google Scholar
- 12.V. D. Sedykh, “Structure of the convex hull of a space curve,” Trudy Sem. Petrovsk., 6 (1981), 239–256.Google Scholar
- 13.V. D. Sedykh, “Double tangent planes to a space curve,” Sibirsk. Mat. Zh. [Siberian Math. J.], 30 (1989), no. 1, 209–211.Google Scholar