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 Mat. Zametki, 2005, Volume 78, Issue 3, Pages 413–427 (Mi mz2598)

Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve

V. D. Sedykh

Gubkin Russian State University of Oil and Gas

Abstract: Let $\gamma$ be a smooth generic curve in $\mathbb RP^3$. Denote by $C$ the number of its flattening points, and by $T$ the number of planes tangent to $\gamma$ at three distinct points. Consider the osculating planes to $\gamma$ at the flattening points. Let $N$ denote the total number of points where $\gamma$ intersects these osculating plane transversally. Then $T\equiv[N+\theta(\gamma)C]/2\pmod2$, where $\theta(\gamma)$ is the number of noncontractible components of $\gamma$. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in $\mathbb R^3$ 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.

DOI: https://doi.org/10.4213/mzm2598

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English version:
Mathematical Notes, 2005, 78:3, 378–390

Bibliographic databases:

UDC: 515.164.15+514.755.24
Revised: 08.02.2005

Citation: V. D. Sedykh, “Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve”, Mat. Zametki, 78:3 (2005), 413–427; Math. Notes, 78:3 (2005), 378–390

Citation in format AMSBIB
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• https://doi.org/10.4213/mzm2598
• http://mi.mathnet.ru/eng/mz/v78/i3/p413

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This publication is cited in the following articles:
1. V. D. Sedykh, “On the Coexistence of Corank 1 Multisingularities of a Stable Smooth Mapping of Equidimensional Manifolds”, Proc. Steklov Inst. Math., 258 (2007), 194–217
2. V. D. Sedykh, “The Topology of Adjacencies of Type $A$ and $D$ Lagrangian Singularities”, Funct. Anal. Appl., 48:4 (2014), 304–308
3. V. D. Sedykh, “On the topology of stable Lagrangian maps with singularities of types $A$ and $D$”, Izv. Math., 79:3 (2015), 581–622
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