Brasselet number and function-germs with a one-dimensional critical set

Abstract

The Brasselet number of a function f with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, using the Brasselet number, we present several formulas for germs f : (X, 0) → (C, 0) and g : (X, 0) → (C, 0) in the case where g has a one-dimensional critical locus. We also give applications when f has isolated singularities and when it is a generic linear form.

Local topology of a deformation of a function-germ with a one-dimensional critical set

“…Proof. By [17], g is tractable at the origin with respect to the good stratification V of X induced by a generic linear form l. Hence, the formula follows directly from Proposition 3.9, using that B l,X g (0) = Eu X g (0) and that B l,X g (0) = Eu X g (0).…”

“…. , i k(j) }, B g,X∩f −1 (δ) (x l ) is constant on b j ∩B ǫ (see Remark 4.5 of [17]). Then we denote B g,…”

“…In order to compare the Brasselet numbers B f,X g (0) and B f,X g (0) we need to understand the stratified critical set of g. We use the description presented in [17]. Consider a decomposition of Σ W g into branches b j ,…”

“…Proof. By [17], V ′ = {C n \{f = 0}∪Σ W g, {f = 0}\{0}, Σ W g, {0}} is a good stratification of C n relative to f. Also by [17], V ′′ , given by…”

“…Proof. By [17], g is tractable at te origin with respect to the good stratification V induced by l. Without loss of generality, we can suppose that l = z 0 . Let F g,0 be the Milnor fibre of g at the origin and F g,0 be the Milnor fibre of g at the origin.…”