Multiplicity of Boardman strata and deformations of map germs

Abstract

Abstract. We define algebraically for each map germ f:K",0^>K'\ 0 and for each Boardman symbol i = (/,,...,i k ) a number c-,(f) which is ^-invariant. If / is finitely determined, this number is the generalization of the Milnor number of/when p = 1, the number of cusps of/when n =p = 2, or the number of cross caps when n = 2,p = 3. We study some properties of this number and prove that, in some particular cases, this number can be interpreted geometrically as the number of S ! points that appear in a generic deformation of / . In the last part, we compute this number in the case that the map germ is a projection and give some applications to catastrophe map germs.

Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs

“…Note that g| U : U → Int D 2 is a proper smooth map. Let g : U → Int D 2 ⊂ R 2 be a proper C ∞ stable perturbation of g| U in a sense similar to that in [3,4,15,17].…”

Perturbations of caustics and fronts

“…See also [30], [28], [29], [19], [20] for the related studies. After [3], [8], there appears a sequence of investigations on the problem of counting general isolated ThomBoardman singularities [26], [4], [5], [6]. Then we observe that there are two main points for obtaining right formulae: one is seeking the appropriate defining ideals of treating singularities on jet spaces and second is seeing their Cohen-Macaulay property.…”

“…The counting of isolated singularities of caustics in the four space has several difficulties: The ideals associated to Thom-Boardman singularities are actually defined in general ( [25], [26], [4], [5]): They are called Morin ideals of Thom-Boardman singularities. However, for A 5 -singularities or Σ r,1,1,1,1 -singularities, the Morin ideal does not define a Cohen-Macaulay variety (Theorem 3.1 of [5]).…”

The deformation multiplicity of a map germ with respect to a Boardman symbol

“…This is a generalization of the invariant de ned in [17], which is nite only when the Jacobian extensions de ne isolated singularities. This is a generalization of the invariant de ned in [17], which is nite only when the Jacobian extensions de ne isolated singularities.…”

“…It can be expressed using the limit formula for the multiplicity (see [11, p. 107]) as [17] for details). For each map germ f : (C n ; 0) !…”