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arXiv · 1804.09414

The extra-nice dimensions

Raúl Oset Sinha·Maria Aparecida Soares Ruas·Roberta Wik Atique
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Abstract

We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $C^{\infty}(N\times[0,1],P)$, also known as pseudo-isotopies, is dense if and only if the pair of dimensions $(\dim N, \dim P)$ is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs $(n,p)$ for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $\mathcal A_e$-codimension 1 hyperplane sections. They are also related to the simplicity of $\mathcal A_e$-codimension 2 germs. We give a sufficient condition for any $\mathscr A_e$-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.

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Raúl Oset Sinha, Maria Aparecida Soares Ruas, Roberta Wik Atique. 2018-04-25. The extra-nice dimensions. https://arxiv.org/abs/1804.09414

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