# Publishing house / Banach Center Publications / All volumes

## Real deformations and invariants of map-germs

### Volume 82 / 2008

Banach Center Publications 82 (2008), 183-199 MSC: Primary 58K60, 58K65; Secondary 32S30. DOI: 10.4064/bc82-0-13

#### 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 $n+2\le p\le 2n-1$.

#### Authors

• J. H. RiegerInstitut für Mathematik
Universität Halle
D-06099 Halle (Saale), Germany
e-mail
• M. A. S. RuasICMC, Universidade de São Paulo
13560-970 São Carlos, SP, Brazil
e-mail
• R. Wik AtiqueICMC, Universidade de São Paulo
13560-970 São Carlos, SP, Brazil
e-mail

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