### Finite determinacy and topological triviality. II : sufficient conditions and topological stability

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We will extend the infinitesimal criteria for the equisingularity (i.e. topological triviality) of deformations $f$ of germs of mappings ${f}_{0}:{k}^{s}$, $0\to {k}^{t}$, $0$ to non-finitely determined germs (these occur generically outside the “nice dimensions” for Mather, even among topologically stable mappings). The failure of finite determinacy is described geometrically by the “versality discriminant”, which is the set of points where ${f}_{0}$ is not stable (i.e. viewed as an unfolding it is not versal). The criterion asserts that...

We obtain algebraic and geometric conditions for the topological triviality of versal unfoldings of weighted homogeneous complete intersections along subspaces corresponding to deformations of maximal weight. These results are applied: to infinite families of surface singularities in ${C}^{4}$ which begin with the exceptional unimodular singularities, to the intersection of pairs of generic quadrics, and to certain curve singularities. The algebraic conditions are related to the operation of...

We introduce a skeletal structure $(M,U)$ in ${\mathbb{R}}^{n+1}$, which is an $n$- dimensional Whitney stratified set $M$ on which is defined a multivalued “radial vector field” $U$. This is an extension of notion of the Blum medial axis of a region in ${\mathbb{R}}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” $\mathcal{B}$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to $\mathcal{B}$. Together these allow us to provide sufficient numerical conditions for...

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