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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].

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