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M-deformations of -simple germs from to

Published online by Cambridge University Press:  01 January 2008

J. H. RIEGER
Affiliation:
Institut für Mathematik, Universität Halle, D-06099 Halle (Saale), Germany. e-mail: [email protected]
M. A. S. RUAS
Affiliation:
ICMC, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil. e-mail: [email protected]; [email protected]
R. WIK ATIQUE
Affiliation:
ICMC, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil. e-mail: [email protected]; [email protected]

Abstract

All -simple corank-1 germs from to , where n ≠ 4, have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the image.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]A'Campo, N.Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I. Math. Ann 213 (1975), 132.Google Scholar
[2]Arnol'd, V. I.. Springer numbers and Morsification spaces. J. Algebraic Geom 1 (1992), 197214.Google Scholar
[3]Bruce, J. W., Plessis, A. A. du and Wall, C. T. C.. Determinacy and unipotency. Invent. Math 88 (1987), 521554.CrossRefGoogle Scholar
[4]Bruce, J. W., Kirk, N. P. and Plessis, A. A. du. Complete transversals and the classification of singularities. Nonlinearit 10 (1997), 253275.CrossRefGoogle Scholar
[5]Cooper, T., Mond, D. and Atique, R. Wik. Vanishing topology of codimension 1 multi-germs over bR and bC. Compositio Math 131 (2002), 121160.Google Scholar
[6]Damon, J.. Topological triviality and versality of subgroups of cA and cK. Mem. Amer. Math. Soc 389 (1988).Google Scholar
[7]Damon, J.. Topological triviality and versality of subgroups of cA and cK: II. Sufficient conditions and applications. Nonlinearit 5 (1992), 373412.Google Scholar
[8]Entov, M. R.. On real Morsifications of Dmu singularities. Russ. Acad. Sci. Dokl. Math 46 (1993), 2529.Google Scholar
[9]Giusti, M., Classification des singularités %isolées simples d'intersections complètes, 457-494, in Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence RI, 1983.CrossRefGoogle Scholar
[10]Goryunov, V. and Mond, D.. Vanishing cohomology of singularities of mappings. Compositio Math 89 (1993), 4580.Google Scholar
[11]Goryunov, V. V.. Semi-simplicial resolutions and homology of images and discriminants of mappings. Proc. London Math. Soc 70 (1995), 363385.CrossRefGoogle Scholar
[12]Gusein-Zade, S. M.. Dynkin diagrams of singularities of functions of two variables. Funct. Anal. Appl 8 (1974), 295300.Google Scholar
[13]Hamm, H. A.. Lokale topologische Eigenschaften komplexer Räume. Math. Ann 191 (1971), 235252.CrossRefGoogle Scholar
[14]Houston, K.. Local topology of images of finite complex analytic maps. Topolog 36 (1997), 10771121.Google Scholar
[15]Houston, K.. On the topology of augmentations and concatenations of singularities. Manuscripta Math 117 (2005), 383405.Google Scholar
[16]Houston, K. and Kirk, N. P.. On the classification and geometry of corank 1 map-germs from three-space to four-space. In Singularity Theory. London Math. Soc. Lecture Note Series 263, (Cambridge University Press, 1999), 325351.CrossRefGoogle Scholar
[17]Kharlamov, V.M., Orevkov, S.Yu. and Shustin, E.I., Singularity which has no %M-smoothing, in The Arnoldfest, Eds. E. Bierstone et al., Fields Institute Communications Vol. 24 (1999), 273-309.Google Scholar
[18]Klotz, C., Pop, O. and Rieger, J. H.. Real double-points of deformations of cA-simple map-germs from bR n to bR 2n. Math. Proc. Camb. Phil. Soc 142 (2007), 341363.CrossRefGoogle Scholar
[19]Marar, W. L.. The Euler characteristic of the disentanglement of the image of a corank 1 map-germ. In Singularity Theory and Applications, Warwick 1989. Mond, D. and Montaldi, J. (eds.), Lecture Notes in Math. 1462 (Springer Verlag, 1991).Google Scholar
[20]Marar, W. L. and Mond, D.. Multiple point schemes for corank 1 maps. J. London Math. Soc 39 (1989), 553567.CrossRefGoogle Scholar
[21]Marar, W. L. and Mond, D.. Real map-germs with good perturbations. Topolog 35 (1996), 157165.Google Scholar
[22]Marar, W.L., Montaldi, J.A. and Ruas, M.A.S., Multiplicities of zero-schemes in quasihomogeneous corank-1 singularities Cn to Cn, in Singularity Theory, 353-367, London Math. Soc. Lecture Note Series 263, Cambridge University Press (1999).CrossRefGoogle Scholar
[23]Mond, D.. Some remarks on the geometry and classification of germs of maps from surfaces to 3-space. Topolog 26 (1987), 361383.CrossRefGoogle Scholar
[24]Mond, D.. How good are real pictures? Algebraic geometry and singularities (La Rábida, 1991). Progr. Math 134 (Birkhäuser, 1996), 259276.CrossRefGoogle Scholar
[25]Mond, D. and Atique, R. Wik. Not all codimension 1 germs have good real pictures. In Real and complex singularities. Lecture Notes in Pure and Appl. Math. 232 (Dekker, 2003), 189200.Google Scholar
[26]Plessis, A. A. du and Wall, C. T. C.. The Geometry of Topological Stability (Clarendon Press, 1995).CrossRefGoogle Scholar
[27]Plessis, A. du, On the determinacy of smooth map-germs, Inventiones Math. 58 (1980), 107-160.Google Scholar
[28]Rieger, J. H.. Families of maps from the plane to the plane. J. London Math. Soc 36 (1987), 351369.Google Scholar
[29]Rieger, J. H.Versal topological stratification and bifurcation geometry of map-germs of the plane. Math. Proc. Camb. Phil. Soc 107 (1990), 127147.CrossRefGoogle Scholar
[30]Rieger, J. H.. Recognizing unstable equidimensional maps, and the number of stable projections of algebraic hypersurfaces. Manuscripta Math 99 (1999), 7391.Google Scholar
[31]Rieger, J. H.. Invariants of equidimensional corank-1 maps, in Geometry and Topology of Caustics – Caustics'02. Banach Center Publ. 62 (2004), 239248.Google Scholar
[32]Rieger, J. H. and Ruas, M. A. S.. Classification of cA-simple germs from kn to k2. Compositio Math 79 (1991), 99108.Google Scholar
[33]Rieger, J. H. and Ruas, M. A. S.. M-deformations of cA-simple Σ n-p+1-germs from bR n to bR p, n ≥ p. Math. Proc. Cambridge Phil. Soc 139 (2005), 333349.CrossRefGoogle Scholar
[34]Rieger, J. H., Ruas, M. A. S. and Atique, R. Wik. Real deformations and invariants of map-germs, preprint 2006.Google Scholar
[35]Ruas, M. A. S.. Cl- Determina finita e aplica oes. Thesis, ICMC, USP- S Carlos (1983).Google Scholar
[36]Wall, C. T. C.. Finite determinacy of smooth map-germs. Bull. London Math. Soc 13 (1981), 481539.Google Scholar
[37]Zhitomirskii, M.. Fully simple singularities of plane and space curves, preprint 2005.Google Scholar