## Thom polynomials for Lagrange, Legendre, and critical point function singularities (2003)

Venue: | Proc. London Math. Soc |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Kazarian03thompolynomials,

author = {Maxim Kazarian},

title = {Thom polynomials for Lagrange, Legendre, and critical point function singularities},

journal = {Proc. London Math. Soc},

year = {2003},

volume = {3},

pages = {86}

}

### OpenURL

### Abstract

The classi®cation of isolated hypersurface singularities is known to be highly irregular and there is no hope of getting the complete classi®cation. Nevertheless one can make the following observation. Consider the hierarchy of singularities of

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Citation Context ...autological bundle; the generators a i satisfy the relations a 2 i ˆ 0. A similar description exists for the (integer) cohomology ring of the complex Lagrange Grassmannian L C N > Sp…N†=U…N†. Theorem =-=[7, 15, 13]-=-. The ring of Lagrange characteristic classes is isomorphic to the quotient of the polynomial ring in variables a 1; a 2; ... of 725s726 maxim kazarian degrees 2; 4; ... by the ideal generated by elem... |

61 |
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Citation Context ... the classes a ˆ PQ and b ˆ 1. For instance, since i …1† ˆ‰MŠˆcn…Hom…V; I†† ˆ u n u n 1 c1 ‡ ...6 cn; Sometimes in the literature this equivalence is called contact equivalence. We prefer to follow =-=[4, 2]-=- to keep the notion of contact equivalence for contact diffeomorphisms of contact manifolds.sthom polynomials Table 1. Thom polynomials of isolated hypersurface singularities of codimension less than ... |

54 |
Curves and Singularities
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Citation Context ...of ®nding extra relations between Thom polynomials. The coef®cients entering these relations are so-called adjacency exponents which are an analogue of the incidence coef®cients in the real case; see =-=[8, 22]-=-. They are computed from the detailed study of adjacency of singularities of successive complex codimensions. These relations are suf®cient to complete the computation of all Thom polynomials of Theor... |

48 |
Algebro-geometric applications of Schur S- and Q-polynomials
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Citation Context ...autological bundle; the generators a i satisfy the relations a 2 i ˆ 0. A similar description exists for the (integer) cohomology ring of the complex Lagrange Grassmannian L C N > Sp…N†=U…N†. Theorem =-=[7, 15, 13]-=-. The ring of Lagrange characteristic classes is isomorphic to the quotient of the polynomial ring in variables a 1; a 2; ... of 725s726 maxim kazarian degrees 2; 4; ... by the ideal generated by elem... |

32 |
L.: On symmetric and skew-symmetric determinantal varieties
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Citation Context ... k instead of ak, where u ˆ c1…I† and c 1 1 1‡u = 2 1 ‡ ba 1 ‡ ba 2 ‡ ...ˆ c 1 1 ‡ 1 u = 2 ‡ c2 …1‡u = 2† 2 ... ‡ c2 …1 u = 2† 2 ‡ ... ; c i ˆ c i…V †: Proof. We use the following trick borrowed from =-=[10]-=-. Consider ®rst the case when I ˆ J 2 , where J is another line bundle with c 1…J† ˆ1 2 c1…I† ˆ1 2 u. Then S 2 V I ˆ S 2 …V J † ; so f…2† can be treated as a self-adjoint bundle map V J !…V J † and we... |

19 |
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Citation Context ...cal information about adjacencies of singularities is translated into the properties of the spectral sequence constructed by the ®ltration on the classifying space by the codimension of singularities =-=[11]-=-. Let us describe this spectral sequence for the classi®cation of (complex) Lagrange singularities [12]. The initial term of this sequence is E ; 2 ˆ L S H …BGS†, where GS is the symmetry group of the... |

17 |
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Citation Context ...0 2 Y w invariantly de®nes a nowhere-vanishing section of Y j Q… f †. To prove this we must show that the symmetry group of the singularity g acts trivially on the element s w. It is known (see [20], =-=[23]-=-, and the Russian translation of [22]) that, for any isolated critical point singularity, the symmetry group with respect to the right equivalence is ®nite if the number of variables is equal to the c... |

15 |
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Citation Context ...contractible ®bres. The ring of Lagrange characteristic classes is the limit cohomology ring lim N ! 1 H …L C N†. The topology of the real Lagrange Grassmannian L R N > U…N†=O…N† is well studied (see =-=[6, 9]-=-). Its Z 2-cohomology ring H …L R N ; Z 2† is generated by the Stiefel±Whitney classes a i of the tautological bundle; the generators a i satisfy the relations a 2 i ˆ 0. A similar description exists ... |

3 |
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Citation Context ...homomorphism Some of the Thom polynomials of Theorem 1 are computed using the method of resolution of singularities. The computations of this method use a formula for the Gysin homomorphism proved in =-=[13]-=-. Consider vector bundles V; I ! M of ranks n, 1 respectively and a quadratic bundle map f…2†: V ! I. The map f…2† is considered as a section of the bundle S 2 V I or as a linear self-adjoint bundle m... |

2 |
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Citation Context ...siliev has computed the cohomology of this complex in the codimension not exceeding 6 and found the expressions for all these classes (except A7) in terms of the Stiefel±Whitney classes. In the paper =-=[12]-=- we suggested an approach to this problem based on the study of classifying space of Lagrange singularities. This has led to understanding the geometrical meaning of the Vassiliev complex and to intro... |

2 |
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Citation Context ...rank of the critical point. As Wall noticed in [23], these groups are always ®nite for any critical point singularity of ®nite multiplicity. For the case of real functions this assertion is proved in =-=[20]-=-. An independent proof valid also for the complex case is given by Vassiliev in the Russian translation of [22]. For singularities of small codimensions these groups are well known (some extension of ... |

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Citation Context ...llyrequires a lot of computations. One should ®nd all possible adjacencies of classes of successive codimensions and compute the adjacencyexponents. In these computations the methods and results from =-=[1, 22, 19]-=- are used. 721s722 maxim kazarian Lemma. The lists in Table 3 and the notes below exhaust all possible adjacencies of singularity classes of codimension no greater than 6. The functions of the family ... |

1 |
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Citation Context ... the classes a ˆ PQ and b ˆ 1. For instance, since i …1† ˆ‰MŠˆcn…Hom…V; I†† ˆ u n u n 1 c1 ‡ ...6 cn; Sometimes in the literature this equivalence is called contact equivalence. We prefer to follow =-=[4, 2]-=- to keep the notion of contact equivalence for contact diffeomorphisms of contact manifolds.sthom polynomials Table 1. Thom polynomials of isolated hypersurface singularities of codimension less than ... |

1 |
Classes caracteÂristique d'immersions lagrangiennes deÂ®nies par des varieÂteÂs de caustiques (d'apreÁs V. A. Vassiliev), SeÂminaire Sud-Rhodanien de GeÂomeÂtry, travaux en cours 1
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Citation Context ...ectic form. Then the relative Chern classes of the quadruple …E; I; L 1; L 2† are de®ned as u ˆ c 1…I† and a i ˆ c i…L 2 I L 1†. The equalities c…L 1 ‡ L 1 I† ˆc…L 2 ‡ L 2 I† ˆc…E† imply the identity =-=(5)-=- for these classes. The situations considered above ®t into this pattern. In the case of a ®bre bundle map f : V ! I we take E ˆ V V I and L 1 ˆ V 0, and L 2 > L 1 is the bundle of the tangent planes ... |

1 |
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Citation Context ...contractible ®bres. The ring of Lagrange characteristic classes is the limit cohomology ring lim N ! 1 H …L C N†. The topology of the real Lagrange Grassmannian L R N > U…N†=O…N† is well studied (see =-=[6, 9]-=-). Its Z 2-cohomology ring H …L R N ; Z 2† is generated by the Stiefel±Whitney classes a i of the tautological bundle; the generators a i satisfy the relations a 2 i ˆ 0. A similar description exists ... |

1 |
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Citation Context ...ere-degenerate section of L 2 E ). Let L 1; L 2 Ì E be two Lagrange subbundles (in the sense that the ®bres of L 1 and L 2 are Lagrange planes in the ®bres of E). Then the relative Chern classes (cf. =-=[14]-=-) of the triple …E; L 1; L 2† are de®ned as a i ˆ c i…L 2 L 1†. The equalities c…L 1 ‡ L 1†ˆc…L 2 ‡ L 2†ˆc…E† imply the identity (4) for these classes. The situations considered above ®t into this pat... |

1 |
Thom polynomials, symmetries and incidence of singularities
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Citation Context ... very hard to understand the topology of the classifying space from their construction. It should be noticed nevertheless that their construction works as well for the case of multisingularities; see =-=[21, 16, 17]-=- for some applications. It is an interesting problem to ®nd an a priori construction for the classifying space of multisingularities and to describe its topology (the work [18] implies that it should ... |

1 |
Multiple point formulas ± a new point of view
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(Show Context)
Citation Context ... very hard to understand the topology of the classifying space from their construction. It should be noticed nevertheless that their construction works as well for the case of multisingularities; see =-=[21, 16, 17]-=- for some applications. It is an interesting problem to ®nd an a priori construction for the classifying space of multisingularities and to describe its topology (the work [18] implies that it should ... |

1 |
cs, `Generalized Pontrjagin±Thom construction for maps with singularities', Topology 37
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(Show Context)
Citation Context ...ˆ S ´ G EG > BGS is the classifying space of the `symmetry group' GS of the singularity S (the stabiliser of any point x 2 S). A similar description of BS exists even if S consists of many orbits. In =-=[18]-=- SzuÈcs and RimaÂnyi used an alternative approach to the de®nition of the classifying space of singularities, based on SzuÈcs's idea of gluing the classifying spaces of symmetry groups of singularitie... |

1 |
Classi®cation and deformation of singularities', Doctoral dissertation
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(Show Context)
Citation Context ...llyrequires a lot of computations. One should ®nd all possible adjacencies of classes of successive codimensions and compute the adjacencyexponents. In these computations the methods and results from =-=[1, 22, 19]-=- are used. 721s722 maxim kazarian Lemma. The lists in Table 3 and the notes below exhaust all possible adjacencies of singularity classes of codimension no greater than 6. The functions of the family ... |

1 |
cs, `Multiple points of singular maps
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(Show Context)
Citation Context ... very hard to understand the topology of the classifying space from their construction. It should be noticed nevertheless that their construction works as well for the case of multisingularities; see =-=[21, 16, 17]-=- for some applications. It is an interesting problem to ®nd an a priori construction for the classifying space of multisingularities and to describe its topology (the work [18] implies that it should ... |

1 |
and Legendre characteristic classes, 2nd edn (Gordon and
- Vassiliev, Lagrange
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(Show Context)
Citation Context ...nold±Maslov class which is dual to the total critical set of the projection. The theory of characteristic classes related to the real Lagrange singularities was developed by V. Vassiliev. In his book =-=[22]-=- a cochain complex (the so-called Vassiliev universal complex of singularity classes) was constructed whose generators correspond to the singularity classes. The cohomology groups of this complex are ... |