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2008 | 82 | 1 | 183-199
Tytuł artykułu

Real deformations and invariants of map-germs

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A stable deformation $f^t$ of a real map-germ $f:ℝⁿ,0 → ℝ^p,0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{ℂ}^{t}$ 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 ≥ p) of $f^t$ coincides with that of $f_{C}^{t}$. 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 $𝓐_e$-codimension 1 have an M-deformation. More generally, there is the question whether all 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 ≥ p, p ≥ 2n and p = n + 1, n ≠ 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 ≤ p ≤ 2n - 1.
Słowa kluczowe
  • Institut für Mathematik, Universität Halle, D-06099 Halle (Saale), Germany
  • ICMC, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil
  • ICMC, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil
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