# The versality discriminant and local topological equivalence of mappings

Annales de l'institut Fourier (1990)

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

## Access Full Article

top## Abstract

top## How to cite

topDamon, 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

top- [D1] J. DAMON, Finite Determinacy and Topological Triviality I. Invent. Math., 62 (1980), 299-324. Zbl0489.58003MR82f:58018
- II. Sufficient Conditions and Topological Stability, Compositio Math., 47 (1982), 101-132. Zbl0523.58005
- [D2] J. DAMON, Topological Triviality and Versality for Subgroups of A and K, Memoirs of A.M.S., 389 (1988). Zbl0665.58005MR90a:58012
- [D3] J. DAMON, Topological invariants of µ-constant deformations of complete intersection singularities, Quart. J. Math., 40 (1989), 139-160. Zbl0724.32019MR90j:32012
- [DGaf] J. DAMON and T. GAFFNEY, Topological Triviality of Deformation of Functions and Newton Filtrations, Invent. Math., 72 (1983), 335-358. Zbl0519.58021MR85c:58017
- [DGal] J. DAMON and A. GALLIGO, Universal Topological Stratification for the Pham Example, preprint.
- [Gel] T. GAFFNEY, Properties of Finitely Determined Germs, Thesis, Brandeis Univ., 1975.
- [K] A. G. KOUCHNIRENKO, Polyèdres de Newton et Nombres de Milnor, Invent. Math., 32 (1976), 1-31. Zbl0328.32007MR54 #7454
- [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
- [Lo] E. LOOIJENGA, Semi-universal Deformation of a Simple Elliptic Hypersurface Singularity: I. Unimodularity, Topology, 16 (1977), 257-262. Zbl0373.32004MR56 #8565
- [M1] J. MATHER, Stability of C∞ Mappings V: Transversality, Advances in Math., 4 (1970), 301-336. Zbl0207.54303MR43 #1215c
- [M2] J. MATHER, Generic projections, Ann. of Math., (2) 98 (1973), 226-245. Zbl0242.58001MR50 #14835
- [T] B. TEISSIER, Cycles Évanescents, Sections Planes, et Conditions de Whitney, Singularités à Cargèse, Asterisque 7, 8 (1973), 285-362. Zbl0295.14003
- [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
- [Wi] K. WIRTHMÜLLER, Universell Topologisch Triviale Deformationen, thesis, Univ. of Regensburg.
- [Wa] C. T. C. WALL, private communication.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.