Singularities of Mappings

David Mond () and Juan José Nuño-Ballesteros ()
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David Mond: University of Warwick
Juan José Nuño-Ballesteros: Universitat de València, Departament de Matemàtiques

Chapter Chapter 2 in Handbook of Geometry and Topology of Singularities III, 2022, pp 81-144 from Springer

Abstract: Abstract We summarise some of the basic theory of A $$\mathscr {A}$$ -equivalence (right-left equivalence) of germs of maps due to John Mather and others, and then go on to explain how to calculate some of the key invariants: the A e $$\mathscr {A}_e$$ -codimensionCodimension , the determinacy degree, and a minimal versal unfolding, introducing a new technique which lead to easily implementable computer algorithms. We then describe the topology of stable perturbations of A $$\mathscr {A}$$ -finite germs, and in particular the image and discriminant Milnor number for germs ( ℂ n , 0 ) → ( ℂ p , 0 ) $$({\mathbb {C}}^n,0)\to ({\mathbb {C}}^p,0)$$ with n + 1 ≥ p. We give a new formula for the discriminant Milnor number Discriminant Milnor number which, once again, can be implemented in a computer algorithm. The survey continues with a brief introduction to multiple point spaces and the image computing spectral sequence, and to the study of the Fitting ideal Fitting ideal s. We end with a number of open problems.

Date: 2022
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DOI: 10.1007/978-3-030-95760-5_2

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