Skip to main content Accessibility help
×
Home
Hostname: page-component-77ffc5d9c7-kttml Total loading time: 0.311 Render date: 2021-04-22T21:43:40.259Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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

Published online by Cambridge University Press:  20 November 2009

NICOLAS DUTERTRE
Affiliation:
Université de Provence, Centre de Mathématiques et Informatique, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France. e-mail: dutertre@cmi.univ-mrs.fr
Corresponding
E-mail address:

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
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.

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), 159176.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), 179187.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), 338.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), 436491.CrossRefGoogle 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, 13511371.CrossRefGoogle 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), 227380.CrossRefGoogle Scholar
[Du1]Dutertre, N.Courbures et singularités réelles. Comment. Math. Helv. 77 (2002), 846863.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, 527538.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, 324351.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), 141152. 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, 458461.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, 439455.Google Scholar
[EG4]Ebeling, W. and Gusein–Zade, S. M.On indices of meromorphic 1-forms. Compositio. Math. 140 (2004), no. 3, 809817.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), 231241.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, 129169. (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), 1944.CrossRefGoogle Scholar
[FK]Fukui, T. and Khovanskii, A.Mapping degree and Euler characteristic. Kodai Math. J. 29 (2006), no. 1, 144162.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, 7791.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, 15231539.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), 110.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), 737751.CrossRefGoogle Scholar
[Kh]Khimshiashvili, G. M.On the local degree of a smooth map. Soobshch. Akad. Nauk Gruz. SSR 85 (1977), 309311.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, 525540.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, 501510.CrossRefGoogle Scholar
[Sc1]Schwartz, M. H.Champs radiaux et préradiaux associés à une stratification. C.R. Acad. Sci. Paris Sér. I Math. 303 (1986), 239241.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), 307309.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), 621634.CrossRefGoogle Scholar
[Si]Simon, S.Champs totalement radiaux sur une structure de Thom–Mather. Ann. Inst. Fourier (Grenoble) 45 (1995), no. 5, 14231447.CrossRefGoogle Scholar
[Sz1]Szafraniec, Z.On the number of branches of a 1-dimensional semi-analytic set. Kodai Math. J. 11 (1988), 7885.CrossRefGoogle Scholar
[Sz2]Szafraniec, Z.The Euler characteristic of algebraic complete intersections. J. Reine Angew Math. 397 (1989), 194201.Google Scholar
[Sz3]Szafraniec, Z.A formula for the Euler characteristic of a real algebraic manifold. Manuscripta Math. 85 (1994), 345360.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 29 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 22nd April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *