On the topology of stable Lagrangian maps with singularities of types $ A$ and $ D$

Sedykh, V. D.

doi:10.1070/im2015v079n03abeh002754

Cited by 7 publications

(3 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Theorem 3 is a variation of similar statements on the adjacency indices of corank 1 Legendrian multisingularities (see [4], [5], and [8]). In particular, for any X = l i=1 A ± μ i ∈ S + such that μ i > 1 for all i, the sum of the numbers J Y (A + μ ) over all those Y ∈ S + which differ from X only in the number of occurrences of the generator A 1 and in the signs in the superscripts in the notations of multisingularity types equals the adjacency index of a singularity of type A μ−1 to a singularity of type l i=1 A μ i −1 of a stable (noncooriented) wave front.…”

Section: Adjacency Of a Lagrangian Monosingularity Of Type A δ μ To Amentioning

confidence: 95%

The topology of adjacencies of type A and D Lagrangian singularities

Sedykh

2014

Funct Anal Its Appl

View full text Add to dashboard Cite

Adjacencies of multisingularities of stable Lagrangian maps with singularities of types A ± µ and D ± µ are studied.We calculate the homotopy type of adjacencies of multisingularities in the image of a stable Lagrangian map with singularities of types A ± μ and D ± μ . Decompositions of real singularities have also been studied by other authors; see, e.g., [2] and [6].1. Singularities of Lagrangian maps. First, we recall the basic facts of the theory of Lagrangian singularities; see [1] for details.Let E be a smooth 2n-manifold with a symplectic structure ω. A smooth n-dimensional submanifold L in E is said to be Lagrangian if ω| L = 0. A smooth fibration ρ : E → V over a smooth n-manifold V is said to be Lagrangian if all of its fibers are Lagrangian submanifolds in E.Let ρ be a Lagrangian fibration, and let L be a Lagrangian submanifold in E. Then the map f = ρ| L : L → V is said to be Lagrangian. We consider only proper Lagrangian maps.According to Arnold's theorem, all simple stable singularities of Lagrangian maps are of typesTo be more precise, any simple stable germ of a Lagrangian map is Lagrange equivalent to the germ at zero of a Lagrangian map R n → R n , (t, q) → (−∂S(t, q)/∂t, q), where t = (t 1 , . . . , t k ), q = (q k+1 , . . . , q n ), and S = S(t, q) is a smooth function of one of the following forms (μ n + 1 is an integer):+ · · · + q 3 t 2 2 , μ 4, E ± 6 : S = 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, E 7 : S = t 3 1 + t 1 t 3 2 + q 6 t 2 1 t 2 + q 5 t 2 1 + q 4 t 1 t 2 + q 3 t 2 2 , μ= 7,The number μ is called the degree of the singularity, and μ − 1 is its codimension. If μ is even or μ = 1, then all singularities of types A + μ and A − μ are Lagrange equivalent (and we denote these types by A μ ).Let S + be the free Abelian multiplicative semigroup generated by the symbols E 7 , and E 8 . We denote the identity element of this semigroup by 1.The elements of S + are used to index the types of multisingularities in the image a stable Lagrangian map with simple singularities. Namely, consider any element A = X 1 . . . X p ∈ S + \ {1}, where X 1 , . . . , X p is any set of generators of S + . A Lagrangian map f : L → V has a multisingularity of type A at a point y ∈ V if (i) f −1 (y) consists of p pairwise different points; and (ii) the points in 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. At the points of the complement V \ f (L), the map f has multisingularities of type 1.

show abstract

Section: Adjacency Of a Lagrangian Monosingularity Of Type A δ μ To Amentioning

confidence: 95%

The topology of adjacencies of type A and D Lagrangian singularities

Sedykh

2014

Funct Anal Its Appl

View full text Add to dashboard Cite

show abstract

“…In [2] we studied the topology of the manifolds of multisingularities for Lagrangian germs of types A ± µ and D ± µ (for all n). In particular, it follows from Theorem 7.8 in that paper that the total number of connected components of the complement to the caustic of the map (1) with singularity of type D δ µ at the origin is equal to: In [3], [4], and [6] we studied the manifolds of multisingularities for a Lagrangian germ of type E ± 6 at points of its caustic as well as the complement to the image.…”

mentioning

confidence: 99%

The topology of the complement to the caustic of a Lagrangian germ of type $E_6^\pm$

Sedykh

2023

Russian Math. Surveys

View full text Add to dashboard Cite

“…Теорема 3 является вариацией аналогичных утверждений об индексах примыканий лежандровых мультиособенностей коранга 1 (см. [4], [5], [8]). В частности, для любого X =…”

unclassified

Топология Примыканий Лагранжевых Особенностей Типов $A$ И $D$

Седых¹,

Sedykh²

2014

Funktsional. Anal. i Prilozhen.

View full text Add to dashboard Cite

On the topology of stable Lagrangian maps with singularities of types $ A$ and $ D$

Cited by 7 publications

References 14 publications

The topology of adjacencies of type A and D Lagrangian singularities

The topology of adjacencies of type A and D Lagrangian singularities

The topology of the complement to the caustic of a Lagrangian germ of type $E_6^\pm$

Топология Примыканий Лагранжевых Особенностей Типов $A$ И $D$

Contact Info

Product

Resources

About