Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T02:42:05.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Radial index and Poincaré–Hopf index of 1-forms on semi-analytic sets

Published online by Cambridge University Press:  20 November 2009

NICOLAS DUTERTRE
NICOLAS DUTERTRE*
Affiliation:
Université de Provence, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The radial index of a 1-form on a singular set is a generalization of the classical Poincaré–Hopf index. We consider different classes of closed singular semi-analytic sets in n that contain 0 in their singular locus and we relate the radial index of a 1-form at 0 on these sets to Poincaré–Hopf indices at 0 of vector fields defined on n.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 2 , March 2010 , pp. 297 - 330
Copyright
Copyright © Cambridge Philosophical Society 2009

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

[ASV]Aguilar, M., Seade, J. and Verjovsky, A.Indices of vector fields and topological invariants on real analytic singularities. J. Reine Angew. Math. 504 (1998), 159–176.Google Scholar
[AFN]Aoki, K., Fukuda, T. and Nishimura, T. On the number of branches of the zero locus of a map germ (Rn, 0) → (Rn−1, 0). Topology and Computer Science: Proceedings of the Symposium held in honor of S. Kinoshita, H. Noguchi and T. Homma on the occasion of their sixtieth birthdays (1987), 347–363.Google Scholar
[AFS]Aoki, K., Fukuda, T. and Sun, W. Z.On the number of branches of a plane curve germ, Kodai Math. Journal 9 (1986), 179–187.Google Scholar
[Ar]Arnol'd, V. I.Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces. Uspekhi Mat. Nauk 34 (1979), no. 2 (206), 3–38.Google Scholar
[BoSe]Borel, A. and Serre, J. P.Corners and arithmetic groups. Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault. Comment. Math. Helv. 48 (1973), 436–491.Google Scholar
[BLSS]Brasselet, J. P., Lehmann, D., Seade, J. and Suwa, T.Milnor classes of local complete intersections. Trans. Amer. Math. Soc. 354 (2002), no. 4, 1351–1371.Google Scholar
[BrSc]Brasselet, J. P. and Schwartz, M. H. Sur les classes de Chern d'un ensemble analytique complexe. The Euler-Poincaré characteristic (French), pp. 93–147. Astérisque 82–83 (1981).Google Scholar
[BSS]Brasselet, J. P., Seade, J. and Suwa, T. Indices of vector fields and characteristic classes of singular varieties, monograph., to appear in LMN (Springer).Google Scholar
[Ce]Cerf, J.Topologie de certains espaces de plongements. Bull. Soc. Math. France 89 (1961), 227–380.Google Scholar
[Du1]Dutertre, N.Courbures et singularités réelles. Comment. Math. Helv. 77 (2002), 846–863.CrossRefGoogle Scholar
[Du2]Dutertre, N.On the Euler–Poincaré characteristic of semi-analytic sets and semi-algebraic sets. Math. Proc. Camb. Phil. Soc. 135 (2003), no. 3, 527–538.CrossRefGoogle Scholar
[Du3]Dutertre, N.On the Euler characteristics of real Milnor fibres of partially parallelizable maps of n to 2. Kodai Math. J. 32 (2009), no. 2, 324–351.CrossRefGoogle Scholar
[EG1]Ebeling, W. and Gusein–Zade, S. M.On the index of a vector field at an isolated singularity. The Arnoldfest (Toronto, ON, 1997), 141–152. Fields Inst. Commun. 24 (1999).Google Scholar
[EG2]Ebeling, W. and Gusein–Zade, S. M.On the index of a holomorphic 1-form on an isolated complete intersection singularity. (Russian) Dokl. Akad. Nauk 380 (2001), no. 4, 458–461.Google Scholar
[EG3]Ebeling, W. and Gusein–Zade, S. M.Indices of 1-forms on an isolated complete intersection singularity. Mosc. Math. J. 3 (2003), no. 2, 439–455.CrossRefGoogle Scholar
[EG4]Ebeling, W. and Gusein–Zade, S. M.On indices of meromorphic 1-forms. Compositio. Math. 140 (2004), no. 3, 809–817.CrossRefGoogle Scholar
[EG5]Ebeling, W. and Gusein–Zade, S. M.Radial index and Euler obstruction of a 1-form on a singular variety. Geom. Dedicata 113 (2005), 231–241.CrossRefGoogle Scholar
[EG6]Ebeling, W. and Gusein–Zade, S. M. Indices of vector fields and 1-forms on singular varieties. Global aspects of complex geometry, 129–169. (Springer, 2006).CrossRefGoogle Scholar
[EL]Eisenbud, D. and Levine, H. I.An algebraic formula for the degree of a C ∞ map-germ. Annals of Mathematics 106 (1977), 19–44.Google Scholar
[FK]Fukui, T. and Khovanskii, A.Mapping degree and Euler characteristic. Kodai Math. J. 29 (2006), no. 1, 144–162.CrossRefGoogle Scholar
[GGM]Giraldo, L., Gomez-Mont, X. and Mardesic, P.Flags in zero dimensional complete intersection algebras and indices of real vector fields. Math. Z. 260 (2008), no. 1, 77–91.CrossRefGoogle Scholar
[GM1]Gomez-Mont, X. and Mardesic, P.The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant. Ann. Inst. Fourier 47 (1997), no. 5, 1523–1539.CrossRefGoogle Scholar
[GM2]Gomez-Mont, X. and Mardesic, P.Index of a vector field tangent to an odd-dimensional hypersurface and the signature of the relative Hessian. Funct. Anal. Appl. 33 (1999), 1–10.CrossRefGoogle Scholar
[GSV]Gomez-Mont, X., Seade, J. and Verjovsky, A.The index of a holomorphic flow with an isolated singularity. Math. Ann. 291 (1991), 737–751.CrossRefGoogle Scholar
[Kh]Khimshiashvili, G. M.On the local degree of a smooth map. Soobshch. Akad. Nauk Gruz. SSR 85 (1977), 309–311.Google Scholar
[KT]King, H. and Trotman, D. Poincaré–Hopf theorems on singular spaces. Preprint (1999).Google Scholar
[Kl]Klehn, O.Real and complex indices of vector fields on complete intersection curves with isolated singularity. Compositio. Math. 141 (2005), no. 2, 525–540.CrossRefGoogle Scholar
[MvS]Montaldi, J. and van Straten, D.1-forms on singular curves and the topology of real curve singularities. Topology 29 (1990), no. 4, 501–510.CrossRefGoogle Scholar
[Sc1]Schwartz, M. H.Champs radiaux et préradiaux associés à une stratification. C.R. Acad. Sci. Paris Sér. I Math. 303 (1986), 239–241.Google Scholar
[Sc2]Schwartz, M. H.Une généralisation du théorème de Hopf pour les champs sortants. C.R. Acad. Sci. Paris Sér. I Math. 303 (1986), 307–309.Google Scholar
[Sc3]Schwartz, M. H.Champs radiaux sur une stratification analytique. Travaux en Cours 39, Hermann, Paris (1991).Google Scholar
[SS]Seade, J. and Suwa, T.A residue formula for the index of a holomorphic flow. Math. Ann. 304 (1996), 621–634.Google Scholar
[Si]Simon, S.Champs totalement radiaux sur une structure de Thom–Mather. Ann. Inst. Fourier (Grenoble) 45 (1995), no. 5, 1423–1447.Google Scholar
[Sz1]Szafraniec, Z.On the number of branches of a 1-dimensional semi-analytic set. Kodai Math. J. 11 (1988), 78–85.CrossRefGoogle Scholar
[Sz2]Szafraniec, Z.The Euler characteristic of algebraic complete intersections. J. Reine Angew Math. 397 (1989), 194–201.CrossRefGoogle Scholar
[Sz3]Szafraniec, Z.A formula for the Euler characteristic of a real algebraic manifold. Manuscripta Math. 85 (1994), 345–360.CrossRefGoogle Scholar