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Vanishing topology of codimension 1 multi-germs over $\Bbb R$ and $\Bbb C$

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Cooper, T., Mond, D. (David) and Wik Atique, R.. (2002) Vanishing topology of codimension 1 multi-germs over $\Bbb R$ and $\Bbb C$. Compositio Mathematica, Vol.13 (No.2). pp. 121-160. ISSN 0010-437X

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Official URL: http://dx.doi.org/10.1023/A:1014930205374

Abstract

We construct all $\cal A$e-codimension 1 multi-germs of analytic (or smooth) maps (kn, T) [rightward arrow] (kp, 0), with n [gt-or-equal, slanted] p − 1, (n, p) nice dimensions, k = $\mathbb C$ or $\mathbb R$, by augmentation and concatenation operations, starting from mono-germs (|T| = 1) and one 0-dimensional bi-germ. As an application, we prove general statements for multi-germs of corank [less-than-or-eq, slant] 1: every one has a real form with real perturbation carrying the vanishing homology of the complexification, every one is quasihomogeneous, and when n = p − 1 every one has image Milnor number equal to 1 (this last is already known when n [gt-or-equal, slanted] p).

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Mathematics

Library of Congress Subject Headings (LCSH):

Perturbation (Mathematics), Topology, Discriminant analysis, Singularities (Mathematics), Deformations of singularities

Journal or Publication Title:

Compositio Mathematica

Publisher:

Cambridge University Press

ISSN:

0010-437X

Official Date:

April 2002

Dates:

Date	Event
April 2002	Published

Volume:

Vol.13

Number:

No.2

Page Range:

pp. 121-160

DOI:

10.1023/A:1014930205374

Status:

Peer Reviewed

Access rights to Published version:

Open Access

References:

1. A’Campo, N.: Le groupe de monodromie du déploiment des singularitiés isolées de courbes planes I, Math.Ann. 213 (1975), 1-31.
2. Bredon, G.: Introduction to Compact Transformation Groups, Pure and Appl.Math. 46, Academic Press, New York, 1972.
3. Scott Carter, J. and Saito, M.: Reidemeister moves for surface isotopies and their interpretation as moves to movies, J.Kno t Theory 2(3) (1993) 251-284.
4. Cooper, T.: Map germs of Ae-codimension one, PhD Thesis, University of Warwick, 1994.
5. Damon, J.: A-equivalence and equivalence of sections of images and discriminants, In: Singularity Theory and Applications (Warwick 1989), Lecture Notes in Math. 1462, Springer, New York, 1991, pp. 93-121.
6. Damon, J. and Mond, D.: A-codimension and the vanishing topology of discriminants, Invent.Ma th. 106 (1991), 217-242.
7. Goryunov, V. V.: Singularities of projections of full intersections, J.S oviet Math. 27 (1984) 2785-2811.
8. Goryunov, V. V.: Monodromy of the image of a mapping, Funct.Anal.Appl. 25 (1991), 174-180.
9. Goryunov, V. V. and Mond, D.: Vanishing cohomology of singularities of mappings, Compositio Math. 89 (1993), 45-80.
10. Gussein-Zade, S. M.: Dynkin diagrams for certain singularities of functions of two real variables, Funct.Anal.Ap pl. 8 (1974), 295-300.
11. Houston, K.: On singularities of folding maps and augmentations, Math.Sc and. 82 (1998), 192-206.
12. Houston, K. and Kirk, N.: On the classification and geometry of corank 1 map-germs from three-space to four-space, In: J. W. Bruce and D. Mond (eds), Singularity Theory, London Math. Soc. Lecture Note Ser. 263, Cambridge Univ. Press, 1999, pp. 325-351.
13. de Jong, T. and van Straten, D.: Disentanglements, In: Singularity Theory and Applications (Warwick 1989), Lecture Notes in Math. 1462, Springer, New York, 1991, pp. 199-211.
14. Looijenga, E. J. N.: Isolated singular points on complete intersections, London Math. Soc. Lecture Note Ser. 77, Cambridge Univ. Press, 1984.
15. Marar, W. L. and Mond, D.: Multiple point schemes for corank 1 maps, J.Lond on Math.Soc. 39 (1989), 553-567.
16. Marar, W. L. andMond, D.:Realmap-germswith good real perturbations, Topology 35 (1996), 157-165.
17. Martinet, J.: Singularities of Smooth Functions and Maps, London Math. Soc. Lecture Note Ser. 58, Cambridge Univ. Press, 1982.
18. Mather, J. N.: Stability of C1 mappings IV: Classification of stable germs by R-algebras, Pub.Math.I HES 37 (1969), 523-548.
19. Mather, J. N.: Stability of C1 mappings V: Transversality, Adv.Math. 4 (1970), 128-176.
20. Mather, J. N.: Stability of C1 mappings VI: The nice dimensions, In: C. T. C. Wall (ed), Proc.Liv erpool Singularities Symposium I, Lecture Notes in Math. 192, Springer, New York, 1970, pp. 207-253.
21. Mather, J. N. and Yau, S. S.: Classi¢cation of isolated hypersurface singularities by their moduli algebra, Invent.Ma th. 69 (1982), 243-251.
22. Mond, D.: On the classi¢cation of germs of maps from R2 to R3, Proc.L ondon Math. Soc. (3) 50 (1985), 333-369.
23. Mond, D.: How good are real pictures? In: A. Campillo and L. Narváez (eds), Algebraic
Geometry, La Rábida 1991, Progr. Math. 134, BirkhPuser, Basel, 1996, pp. 259-276.
24. Mond, D.: Vanishing cycles for analytic maps, In: D. Mond and J. Montaldi (eds), Singularity Theory and Applications, Warwick 1989, Part I, Lecture Notes in Math. 1462, Springer, New York, 1991, pp. 221-234.
25. Mond, D.: Looking at bent wires: Ae-codimension and the vanishing topology of parametrized curve singularities, Math. Proc.Cam bridge Philos.So c. 117 (1995), 213-222.
26. Mond, D.: Differential forms on free and almost free divisors, Proc.Lo ndon Math.S oc. (3) 81 (2000), 587-617.
27. Orlik, P. and Solomon, L.: Singularities II: Automorphisms of Forms, Math.Ann. 231 (1978), 229-240.
28. du Plessis, A.A.: On the determinacy of smooth map-germs, Invent.Ma th. 58 (1980), 107-160.
29. Roseman, D.: Reidemeister moves for surfaces in 4-dimensional space, In: Jones, Kania-Bartoszynska, Przytycki, Traczyk, Turaev (eds), Knot Theory, Warsaw 1995, Banach Centre Publ. 42, Polish Academy of Sciences, Warsaw, 1998, pp. 347-380.
30. Wall, C. T. C.: A note on symmetry of singularities, Bull.London Math. Soc. 12 (1980), 169-175.
31. Wall, C. T. C.: Finite determinacy of smooth map-germs, Bull.Lond on Math.Soc . 13 (1981), 481-539.
32. Wik Atique, R.: On the classi¢cation of multi-germs of maps from C2 to C3 under A-equivalence, In: J. W. Bruce and F. Tari (eds), Real and Complex Singularities, Res. Notes Math. Ser., Chapman & Hall / CRC, pp. 119-133.