Index of a singular point of a vector field or of a 1-form on an orbifold

Gusein-Zade, S.

doi:10.1090/spmj/1710

Index of a singular point of a vector field or of a 1-form on an orbifold
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by S. M. Gusein-Zade
Translated by: the author

St. Petersburg Math. J. 33 (2022), 483-490

DOI: https://doi.org/10.1090/spmj/1710

Published electronically: May 5, 2022

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Abstract:

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related to the Euler characteristic through the classical Poincaré–Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related to corresponding analogs of indices of singular points. Earlier, a notion of the universal Euler characteristic of an orbifold was defined. It takes values in a ring $\mathcal {R}$, as an Abelian group freely generated by the generators, corresponding to the isomorphism classes of finite groups. Here the universal index of an isolated singular point of a vector field or of a 1-form on an orbifold is defined as an element of the ring $\mathcal {R}$. For this index, an analog of the Poincaré–Hopf theorem holds.

References

Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671–677. MR 1076708, DOI 10.1016/0393-0440(89)90032-6
Jim Bryan and Jason Fulman, Orbifold Euler characteristics and the number of commuting $m$-tuples in the symmetric groups, Ann. Comb. 2 (1998), no. 1, 1–6. MR 1682916, DOI 10.1007/BF01626025
Jr. F. C. Caramello, Introduction to Orbifolds, Preprint, arXiv:1909:08699.
Weimin Chen and Yongbin Ruan, Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25–85. MR 1950941, DOI 10.1090/conm/310/05398
W. Ebeling and S. M. Gusein-Zade, Radial index and Euler obstruction of a 1-form on a singular variety, Geom. Dedicata 113 (2005), 231–241. MR 2171307, DOI 10.1007/s10711-005-2184-1
Wolfgang Ebeling and Sabir M. Gusein-Zade, Equivariant indices of vector fields and 1-forms, Eur. J. Math. 1 (2015), no. 2, 286–301. MR 3386240, DOI 10.1007/s40879-015-0036-6
S. M. Guseĭn-Zade, Equivariant analogues of the Euler characteristic and Macdonald type formulas, Uspekhi Mat. Nauk 72 (2017), no. 1(433), 3–36 (Russian, with Russian summary); English transl., Russian Math. Surveys 72 (2017), no. 1, 1–32. MR 3608029, DOI 10.4213/rm9748
S. M. Guseĭn-Zade, I. Luengo, and A. Mel′e-Èrnandez, The universal Euler characteristic of $V$-manifolds, Funktsional. Anal. i Prilozhen. 52 (2018), no. 4, 72–85 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 52 (2018), no. 4, 297–307. MR 3875356, DOI 10.1007/s10688-018-0239-y
Marshall Hall Jr., Combinatorial theory, 2nd ed., Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1986. A Wiley-Interscience Publication. MR 840216
Friedrich Hirzebruch and Thomas Höfer, On the Euler number of an orbifold, Math. Ann. 286 (1990), no. 1-3, 255–260. MR 1032933, DOI 10.1007/BF01453575
I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363. MR 79769, DOI 10.1073/pnas.42.6.359
Ichirô Satake, The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464–492. MR 95520, DOI 10.2969/jmsj/00940464
Marie-Hélène Schwartz, Classes caractéristiques définies par une stratification d’une variété analytique complexe. I, C. R. Acad. Sci. Paris 260 (1965), 3262–3264 (French). MR 212842
Marie-Hélène Schwartz, Champs radiaux sur une stratification analytique, Travaux en Cours [Works in Progress], vol. 39, Hermann, Paris, 1991 (French). MR 1096495
Hirotaka Tamanoi, Generalized orbifold Euler characteristic of symmetric products and equivariant Morava $K$-theory, Algebr. Geom. Topol. 1 (2001), 115–141. MR 1805937, DOI 10.2140/agt.2001.1.115
Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 889050, DOI 10.1515/9783110858372.312