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The dual tree of a fold map germ from $\mathbb {R}^{3}$
to $\mathbb {R}^{4}$
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 3 / June 2023
- Published online by Cambridge University Press:
- 06 May 2022, pp. 958-977
- Print publication:
- June 2023
-
- Article
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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.
