# Real deformations and invariants of map-germs

J. H. Rieger; M. A. S. Ruas; R. Wik Atique

Banach Center Publications (2008)

- Volume: 82, Issue: 1, page 183-199
- ISSN: 0137-6934

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topJ. H. Rieger, M. A. S. Ruas, and R. Wik Atique. "Real deformations and invariants of map-germs." Banach Center Publications 82.1 (2008): 183-199. <http://eudml.org/doc/281999>.

@article{J2008,

abstract = {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 $_e$-codimension 1 have an M-deformation. More generally, there is the question whether all -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.},

author = {J. H. Rieger, M. A. S. Ruas, R. Wik Atique},

journal = {Banach Center Publications},

keywords = {stable invariant; deformation; germ},

language = {eng},

number = {1},

pages = {183-199},

title = {Real deformations and invariants of map-germs},

url = {http://eudml.org/doc/281999},

volume = {82},

year = {2008},

}

TY - JOUR

AU - J. H. Rieger

AU - M. A. S. Ruas

AU - R. Wik Atique

TI - Real deformations and invariants of map-germs

JO - Banach Center Publications

PY - 2008

VL - 82

IS - 1

SP - 183

EP - 199

AB - 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 $_e$-codimension 1 have an M-deformation. More generally, there is the question whether all -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.

LA - eng

KW - stable invariant; deformation; germ

UR - http://eudml.org/doc/281999

ER -

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