The versality discriminant and local topological equivalence of mappings

James Damon

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 4, page 965-1004
  • ISSN: 0373-0956

Abstract

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We will extend the infinitesimal criteria for the equisingularity (i.e. topological triviality) of deformations f of germs of mappings f 0 : k s , 0 → k t , 0 to non-finitely determined germs (these occur generically outside the “nice dimensions” for Mather, even among topologically stable mappings). The failure of finite determinacy is described geometrically by the “versality discriminant”, which is the set of points where f 0 is not stable (i.e. viewed as an unfolding it is not versal). The criterion asserts that algebraic filtration conditions on the infinitesimal deformations together with topological triviality of f in a “conical neighborhood” of the versality discriminant imply topological triviality of f itself.

How to cite

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Damon, James. "The versality discriminant and local topological equivalence of mappings." Annales de l'institut Fourier 40.4 (1990): 965-1004. <http://eudml.org/doc/74908>.

@article{Damon1990,
abstract = {We will extend the infinitesimal criteria for the equisingularity (i.e. topological triviality) of deformations $f$ of germs of mappings $f_0 : k^s$, $0\rightarrow k^t$, $0$ to non-finitely determined germs (these occur generically outside the “nice dimensions” for Mather, even among topologically stable mappings). The failure of finite determinacy is described geometrically by the “versality discriminant”, which is the set of points where $f_0$ is not stable (i.e. viewed as an unfolding it is not versal). The criterion asserts that algebraic filtration conditions on the infinitesimal deformations together with topological triviality of $f$ in a “conical neighborhood” of the versality discriminant imply topological triviality of $f$ itself.},
author = {Damon, James},
journal = {Annales de l'institut Fourier},
keywords = {stratified vector fields; conical neighborhoods; finite determinacy; versality discriminant; infinitesimal deformations; topological triviality},
language = {eng},
number = {4},
pages = {965-1004},
publisher = {Association des Annales de l'Institut Fourier},
title = {The versality discriminant and local topological equivalence of mappings},
url = {http://eudml.org/doc/74908},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Damon, James
TI - The versality discriminant and local topological equivalence of mappings
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 4
SP - 965
EP - 1004
AB - We will extend the infinitesimal criteria for the equisingularity (i.e. topological triviality) of deformations $f$ of germs of mappings $f_0 : k^s$, $0\rightarrow k^t$, $0$ to non-finitely determined germs (these occur generically outside the “nice dimensions” for Mather, even among topologically stable mappings). The failure of finite determinacy is described geometrically by the “versality discriminant”, which is the set of points where $f_0$ is not stable (i.e. viewed as an unfolding it is not versal). The criterion asserts that algebraic filtration conditions on the infinitesimal deformations together with topological triviality of $f$ in a “conical neighborhood” of the versality discriminant imply topological triviality of $f$ itself.
LA - eng
KW - stratified vector fields; conical neighborhoods; finite determinacy; versality discriminant; infinitesimal deformations; topological triviality
UR - http://eudml.org/doc/74908
ER -

References

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  1. [D1] J. DAMON, Finite Determinacy and Topological Triviality I. Invent. Math., 62 (1980), 299-324. Zbl0489.58003MR82f:58018
  2. II. Sufficient Conditions and Topological Stability, Compositio Math., 47 (1982), 101-132. Zbl0523.58005
  3. [D2] J. DAMON, Topological Triviality and Versality for Subgroups of A and K, Memoirs of A.M.S., 389 (1988). Zbl0665.58005MR90a:58012
  4. [D3] J. DAMON, Topological invariants of µ-constant deformations of complete intersection singularities, Quart. J. Math., 40 (1989), 139-160. Zbl0724.32019MR90j:32012
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  7. [Gel] T. GAFFNEY, Properties of Finitely Determined Germs, Thesis, Brandeis Univ., 1975. 
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  9. [LeR] D. T. LE and C. P. RAMANUJAM, Invariance of Minor's number implies the invariance of topological type, Amer. J. Math., 98 (1976), 67-78. Zbl0351.32009
  10. [Lo] E. LOOIJENGA, Semi-universal Deformation of a Simple Elliptic Hypersurface Singularity: I. Unimodularity, Topology, 16 (1977), 257-262. Zbl0373.32004MR56 #8565
  11. [M1] J. MATHER, Stability of C∞ Mappings V: Transversality, Advances in Math., 4 (1970), 301-336. Zbl0207.54303MR43 #1215c
  12. [M2] J. MATHER, Generic projections, Ann. of Math., (2) 98 (1973), 226-245. Zbl0242.58001MR50 #14835
  13. [T] B. TEISSIER, Cycles Évanescents, Sections Planes, et Conditions de Whitney, Singularités à Cargèse, Asterisque 7, 8 (1973), 285-362. Zbl0295.14003
  14. [V] A. N. VARCHENKO, A lower bound for the codimension of the stratum µ-constant in terms of the mixed Hodge structure, Vest. Mosk. Univ. Mat., 37 (1982), 29-31. Zbl0517.32004MR83j:10058
  15. [Wi] K. WIRTHMÜLLER, Universell Topologisch Triviale Deformationen, thesis, Univ. of Regensburg. 
  16. [Wa] C. T. C. WALL, private communication. 

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