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Topological triviality of plane-to-plane singularities. (English) Zbl 1005.32020
Nagano, Tadashi (ed.) et al., Proceedings of the workshop on geometry and its applications in honor of Morio Obata, Yokohama, Japan, November 19-21, 1991. Singapore: World Scientific. 29-37 (1993).
The deformation of an analytic map germ \(f:\mathbb{R}^n, 0\to \mathbb{R}^p,0\) is any analytic germ
\(f_t:\mathbb{R}^n\times [a,b],0\times [a,b] \to \mathbb{R}^p,0\) such that \(f_0=f\). Topological triviality of \(f_t\) means that every \(f_t\) is the same as \(f\), modulo homeomorphisms of \(\mathbb{R}^n,0\) and \(\mathbb{R}^p,0\).
The goal is to find computable sufficient conditions for topological triviality. The holomorphic version of this problem is better known, e.g. if \(p=1\) then constancy of the Milnor number implies topological triviality (the Milnor number is the dimension of a certain algebra associated with the germ). In the complex case \(n=p=2\) is also well understood using results of Gaffney and Damon.
In the paper under review it is shown that for real analytic finitely determined germs, if \(n=p=2\) and the dimensions of two algebras associated with \(f_t\) are constant in \(t\) then \(f_t\) is topologically trivial.
For the entire collection see [Zbl 0981.00018].

32S15 Equisingularity (topological and analytic)
58K15 Topological properties of mappings on manifolds