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.

We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$. We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$. As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.

We consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

Let f,g:(ℝn, 0) → (ℝ, 0) be Cr+1 functions, r ∈ ℕ. We will show that if ∇f(0)=0 and there exist a neighbourhood U of 0 ∈ ℝn and a constant C > 0 such that

$$\begin{equation*} \left|\partial^m(g-f)(x)\right| ≤ C \left|\nabla f(x)\right|^{r+2-|m|} \quad \textrm{ for } x\in U, \end{equation*} $$

We study orthogonal projections of generic embedded hypersurfaces in ℝ4 with boundary to 2-spaces. Therefore, we classify simple map germs from ℝ3 to the plane of codimension less than or equal to 4 with the source containing a distinguished plane which is preserved by coordinate changes. We also go into some detail on their geometrical properties in order to recognize the cases of codimension less than or equal to 1.

The center symmetry set (CSS) of a smooth hypersurface $S$ in an affine space $\mathbf{R}^n$ is the envelope of lines joining pairs of points where $S$ has parallel tangent hyperplanes. The idea stems from a definition of Janeczko, in an alternative version due to Giblin and Holtom. For $n = 2$ the envelope is always real, while for $n \ge 3$ the existence of a real envelope depends on the geometry of the hypersurface. In this paper we make a local study of the CSS, some results applying to $n \le 5$ and others to the cases $n = 2,3$. The method is to construct a generating function whose bifurcation set contains the CSS and possibly some other redundant components. Focal sets of smooth hypersurfaces are a special case of the construction, but the CSS is an affine and not a euclidean invariant. Besides the familiar local forms of focal sets there are other local forms corresponding to boundary singularities, and yet others which do not appear to have arisen elsewhere in a geometrical context. There are connections with Finsler geometry. This paper concentrates on the theory and the proof of the local normal forms for the CSS.

In this paper we classify families of square matrices up to the following natural equivalence. Thinking of these families as germs of smooth mappings from a manifold to the space of square matrices, we allow arbitrary smooth changes of co-ordinates in the source and pre- and post- multiply our family of matrices by (generally distinct) families of invertible matrices, all dependent on the same variables. We obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli). This is a non-linear generalisation of the classical notion of linear systems of matrices. We also make a start on an understanding of the associated geometry.

We present in this paper sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety V. The main result is an infinitesimal criterion based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu in [8]. When V is a weighted homogeneous variety, we obtain as a corollary, the topological triviality of deformations by terms of non negative weights of a weighted homogeneous germ consistent with V. Application of the results to deformations of Newton non-degenerate germs with respect to a given variety is also given.