Combinatorial Models in the Topological Classification of Singularities of Mappings

Nuño–Ballesteros, J. J.

doi:10.1007/978-3-319-73639-6_1

Cited by 3 publications

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“…Finally, let us remark that the techniques used in this paper of obtaining the topological classification of real analytic map germs by means of its associated link has been used by both authors and the second author with several collaborators in other cases [1,2,[15][16][17]18].…”

Section: Introductionmentioning

confidence: 99%

The dual tree of a fold map germ from to

Pérez¹,

Nuño–Ballesteros²

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let $f\colon (\mathbb {R}^{3},0)\to (\mathbb {R}^{4},0)$ be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of $f$ with a small enough sphere $S^{3}_\epsilon$ centred at the origin in $\mathbb {R}^{4}$ . If $f$ is of fold type, we define a tree, that we call dual tree, that contains all the topological information of the link and we prove that in this case it is a complete topological invariant. As an application we give a procedure to obtain normal forms for any topological class of fold type.

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Section: Introductionmentioning

confidence: 99%

The dual tree of a fold map germ from to

Pérez¹,

Nuño–Ballesteros²

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A weak version of the Mond conjecture

Conejero

Nuño–Ballesteros

2023

Collect. Math.

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We prove that a map germ $$f:(\mathbb {C}^n,S)\rightarrow (\mathbb {C}^{n+1},0)$$ f : ( C n , S ) → ( C n + 1 , 0 ) with isolated instability is stable if and only if $$\mu _I(f)=0$$ μ I ( f ) = 0 , where $$\mu _I(f)$$ μ I ( f ) is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that f has corank one. The proof here is also valid for corank $$\ge 2$$ ≥ 2 , provided that $$(n,n+1)$$ ( n , n + 1 ) are nice dimensions in Mather’s sense (so $$\mu _I(f)$$ μ I ( f ) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $$\mathscr {A}_e$$ A e -codimension of f is $$\le \mu _I(f)$$ ≤ μ I ( f ) , with equality if f is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of f is a hypersurface.

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