Real deformations and invariants of map-germs
Volume 82 / 2008
Abstract
A stable deformation $f^t$ of a real map-germ $f:{\mathbb R} ^n,0\to{\mathbb R} ^p,0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{\mathbb C}^t$ are real. A related notion is that of a good real perturbation $f^t$ of $f$ (studied e.g. by Mond and his coworkers) for which the homology of the image (for $n< p$) or discriminant (for $n\ge p$) of $f^t$ coincides with that of $f_{\bC}^t$. The class of map germs having an M-deformation is, in some sense, much larger than the one having a good real perturbation. We show that all singular map-germs of minimal corank (i.e. of corank $\max (n-p+1,1)$) and ${\cal A} _e$-codimension 1 have an M-deformation. More generally, there is the question whether all ${\cal A}$-simple singular map-germs of minimal corank have an M-deformation. The answer is “yes” for the following three dimension ranges $(n,p)$: $n\ge p$, $p\ge 2n$ and $p=n+1$, $n\neq 4$. We describe some new techniques for obtaining these results, which lead to simpler proofs and also to new results in the dimension range $n+2\le p\le 2n-1$.
