A complete system of linear relations between the Euler characteristics of manifolds of corank 1 singularities of a generic front

Sedykh, V. D.

doi:10.1007/s10688-005-0007-7

Cited by 7 publications

(15 citation statements)

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“…Remark 2.13. As in [8], it can be proved that if a linear relation with real coefficients holds for the Euler characteristics of the manifolds of singularities for all generic compact wave fronts in all spaces of dimension n 6, then this relation is a real linear combination of the relations in Theorem 2.10. § 3.…”

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confidence: 86%

On the topology of wave fronts in spaces of low dimension

Sedykh¹

2012

Izv. Math.

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We calculate the adjacency indices of the singularities of generic wave fronts in spaces of dimension n 6. As a corollary, we find new conditions for the coexistence of singularities of wave fronts.A generic wave front F in R n , n 6, may have only finitely many types of singularities up to diffeomorphisms of the ambient space. Let A F be the set of points x ∈ R n such that F has a singularity of type A at x. Then A F is a smooth (possibly non-closed) submanifold of R n .Suppose that F has a singularity of type B at a point y. Put c = n − dim B F and let H be a c-dimensional affine subspace of R n such that y ∈ H and H is transversal to B F at y. We consider the (c − 1)-dimensional sphere S ε ⊂ H of radius ε > 0 centred at y.If ε is sufficiently small, then all the connected components of the intersection S ε ∩ A F are contractible for every singularity of F of type A. The number of these components depends only on the types A and B of the singularities. It is denoted by J A (B) and called the index of adjacency of singularities of type B to singularities of type A. For example, we see from Figures 1, 2 thatIn this paper we calculate the adjacency indices for all singularities of generic wave fronts in spaces of dimension n 6. Using these calculations, we find all linear relations with real coefficients between the Euler characteristics of manifolds of singularities of a compact generic wave front in any smooth n-dimensional manifold. § 2. Main resultsWe first recall the basic notions and facts from the theory of singularities of Legendrian maps and wave fronts (see [1] for details).

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Section: )mentioning

confidence: 86%

On the topology of wave fronts in spaces of low dimension

Sedykh¹

2012

Izv. Math.

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“…The formulas (2) for n 13 are given in Table 3. Theorem 1.3 follows directly from the results of the paper [10]. The formal deduction is given in the next section.…”

Section: Relations Between the Euler Numbersmentioning

confidence: 87%

“…Recently, in [10,11], we found new restrictions on the coexistence of corank 1 singularities of generic fronts. They imply new topological relations between manifolds of singular tangent hyperplanes.…”

Section: Introductionmentioning

confidence: 99%

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On the Topology of Singularities of the Set of Supporting Hyperplanes of a Smooth Submanifold in an Affine Space

Sedykh

2005

J. London Math. Soc.

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Many new universal relations are obtained between the Euler numbers of manifolds of singular supporting hyperplanes of an arbitrary generic smooth closed k-dimensional submanifold in R n where n 7 or k = 1. These relations are applied to Barner-convex curves in an odd-dimensional space R n . A universal (nontrivial) linear relation is established between the numbers of singular supporting hyperplanes of various types but of the same total multiplicity n of tangency with a given generic smooth closed connected Barner-convex curve in R n . The coefficients of this relation are defined by Catalan numbers.

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“…It can be shown by analogy with [4] that any linear relation with real coefficients between the Euler characteristics of manifolds of singularities of a wave front which is valid for all generic compact fronts in all spaces of dimension n 6 is a real linear combination of the relations given in Theorem 1.…”

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confidence: 99%

On Euler characteristics of manifolds of singularities of wave fronts

Sedykh

2012

Funct Anal Its Appl

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All linear relations with real coefficients between the Euler characteristics of manifolds of singularities of wave fronts are found for all generic compact fronts in all spaces of dimension at most 6.A brief synopsis of all notions and facts of the singularity theory of Legendre mappings [1] which we use in this paper is contained in the author's previous paper [5].Suppose given a smooth (of class C ∞ ) n-manifold B, a smooth (2n − 1)-manifold E endowed with a contact structure, and a Legendre bundle ρ : E → B. Consider a smooth compact (without boundary) Legendre submanifold L in E and the Legendre mapping f = ρ • i : L → B, where i : L → E is the identity embedding. The image F = f (L) of the mapping f is called a (wave) front. We say that a statement about Legendre mappings or their fronts holds for a generic mapping f or a generic front F if it is true for all embeddings i belonging to an open dense subset of the space of all embeddings i : L → E (in the C ∞ -topology).For n 6, each singularity of a generic mapping f is Legendre equivalent to a singularity at zero of the mappingwheret = (t 1 , . . . , t k ),q = (q k+1 , . . . , q n−1 ), and S = S(t,q) is a smooth function of one of the following types (μ is an integer):+ · · · + q 3 t 2 2 , 4 μ n, E 6 : S = S(t 1 , t 2 ,q) = t 3 1 + t 4 2 + q 5 t 1 t 2 2 + q 4 t 1 t 2 + q 3 t 2 2 , μ= 6 n (see [1]). The number μ is called the codimension of the singularity. If μ is odd, then singularities of types D + µ and D − µ are Legendre equivalent (and their type is denoted by D µ ). Legendre singularities of other types are pairwise Legendre nonequivalent.Let S be the free Abelian multiplicative semigroup whose generators are the symbols A µ (μ = 1, 2, . . . ), D − 2+2k , D + 2+2k , D 3+2k (k = 1, 2, . . . ), and E 6 . Consider any nonidentity element A = X 1 · · · X p ∈ S, where X 1 , . . . , X p is any set of generators in the semigroup S. A front F has a singularity of type A at a point y ∈ B if (1) f −1 (y) consists of p pairwise different points and (2) the points of f −1 (y) can be ordered as x 1 , . . . , x p so that f has singularities of types X 1 , . . . , X p , respectively, at these points. The sum of the codimensions of the singularities of types X 1 , . . . , X p is called the codimension of the given singularity of type A and denoted by codim A .If n 6, then any generic front F has a singularity of type A ∈ S with codim A n at each of its points. The set A F of points at which F has singularities of type A is a smooth submanifold of codimension codim A in B. In what follows, we denote the Euler characteristic χ(A F ) of the * This work was supported by RFBR grant no. 08-01-00388.

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A complete system of linear relations between the Euler characteristics of manifolds of corank 1 singularities of a generic front

Cited by 7 publications

References 3 publications

On the topology of wave fronts in spaces of low dimension

On the topology of wave fronts in spaces of low dimension

On the Topology of Singularities of the Set of Supporting Hyperplanes of a Smooth Submanifold in an Affine Space

On Euler characteristics of manifolds of singularities of wave fronts

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