Real deformations and invariants of map-germs

J. 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 -