Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-08T19:23:05.408Z Has data issue: false hasContentIssue false
  • English
  • Français

M-deformations of -simple germs from to

Published online by Cambridge University Press:  01 January 2008

J. H. RIEGER ,
M. A. S. RUAS  and
R. WIK ATIQUE
J. H. RIEGER
Affiliation:
Institut für Mathematik, Universität Halle, D-06099 Halle (Saale), Germany. e-mail: rieger@mathematik.uni-halle.de
M. A. S. RUAS
Affiliation:
ICMC, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil. e-mail: masruas@icmc.usp.br; rwik@icmc.usp.br
R. WIK ATIQUE
Affiliation:
ICMC, Universidade de São Paulo, 13560-970 São Carlos, SP, Brazil. e-mail: masruas@icmc.usp.br; rwik@icmc.usp.br
Get access Rights & Permissions [Opens in a new window]

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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 1 , January 2008 , pp. 181 - 195
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]A'Campo, N.Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I. Math. Ann 213 (1975), 1–32.Google Scholar
[2]Arnol'd, V. I.. Springer numbers and Morsification spaces. J. Algebraic Geom 1 (1992), 197–214.Google Scholar
[3]Bruce, J. W., Plessis, A. A. du and Wall, C. T. C.. Determinacy and unipotency. Invent. Math 88 (1987), 521–554.CrossRefGoogle Scholar
[4]Bruce, J. W., Kirk, N. P. and Plessis, A. A. du. Complete transversals and the classification of singularities. Nonlinearit 10 (1997), 253–275.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), 121–160.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), 373–412.Google Scholar
[8]Entov, M. R.. On real Morsifications of Dmu singularities. Russ. Acad. Sci. Dokl. Math 46 (1993), 25–29.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), 45–80.Google Scholar
[11]Goryunov, V. V.. Semi-simplicial resolutions and homology of images and discriminants of mappings. Proc. London Math. Soc 70 (1995), 363–385.CrossRefGoogle Scholar
[12]Gusein-Zade, S. M.. Dynkin diagrams of singularities of functions of two variables. Funct. Anal. Appl 8 (1974), 295–300.Google Scholar
[13]Hamm, H. A.. Lokale topologische Eigenschaften komplexer Räume. Math. Ann 191 (1971), 235–252.CrossRefGoogle Scholar
[14]Houston, K.. Local topology of images of finite complex analytic maps. Topolog 36 (1997), 1077–1121.Google Scholar
[15]Houston, K.. On the topology of augmentations and concatenations of singularities. Manuscripta Math 117 (2005), 383–405.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), 325–351.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), 341–363.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), 553–567.CrossRefGoogle Scholar
[21]Marar, W. L. and Mond, D.. Real map-germs with good perturbations. Topolog 35 (1996), 157–165.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), 361–383.CrossRefGoogle Scholar
[24]Mond, D.. How good are real pictures? Algebraic geometry and singularities (La Rábida, 1991). Progr. Math 134 (Birkhäuser, 1996), 259–276.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), 189–200.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), 351–369.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), 127–147.CrossRefGoogle Scholar
[30]Rieger, J. H.. Recognizing unstable equidimensional maps, and the number of stable projections of algebraic hypersurfaces. Manuscripta Math 99 (1999), 73–91.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), 239–248.Google Scholar
[32]Rieger, J. H. and Ruas, M. A. S.. Classification of cA-simple germs from kn to k2. Compositio Math 79 (1991), 99–108.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), 333–349.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), 481–539.Google Scholar
[37]Zhitomirskii, M.. Fully simple singularities of plane and space curves, preprint 2005.Google Scholar