×
Gusein-Zade, S. M.; Luengo, I.; Melle-Hernández, A.

The universal Euler characteristic of \(V\)-manifolds. (English. Russian original) Zbl 1423.57042

Funct. Anal. Appl. 52, No. 4, 297-307 (2018); translation from Funkts. Anal. Prilozh. 52, No. 4, 72-85 (2018).
Summary: The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a \(V\)-manifold. We discuss a universal additive topological invariant of \(V\)-manifolds, the universal Euler characteristic. It takes values in the ring freely generated (as a \(\mathbb{Z}\)-module) by isomorphism classes of finite groups. We also consider the universal Euler characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation.” We give Macdonald-type identities for the universal Euler characteristic for \(V\)-manifolds and for cell complexes of the described type.

Cited in 3 Documents

MSC:

57P99 Generalized manifolds

Keywords:

finite group actions; \(V\)-manifold; orbifold; additive topological invariant; Lambda-ring; Macdonald identity
Cite Review PDF
Full Text: DOI arXiv

References:

[1] M. Atiyah and G. Segal, “On equivariant Euler characteristics,” J. Geom. Phys., 6:4 (1989), 671-677. · Zbl 0708.19004 · doi:10.1016/0393-0440(89)90032-6
[2] J. Bryan and J. Fulman, “Orbifold Euler characteristics and the number of commuting mtuples in the symmetric groups,” Ann. Comb., 2:1 (1998), 1-6. · Zbl 0921.55003 · doi:10.1007/BF01626025
[3] Chen, W.; Ruan, Y., Orbifold Gromov-Witten theory, 25-85 (2002), Providence, RI · Zbl 1091.53058 · doi:10.1090/conm/310/05398
[4] L. Dixon, J. Harvey, C. Vafa, and E. Witten, “Strings on orbifolds,” Nuclear Phys. B, 261:4 (1985), 678-686. · doi:10.1016/0550-3213(85)90593-0
[5] L. Dixon, J. Harvey, C. Vafa, and E. Witten, “Strings on orbifolds. II,” Nuclear Phys. B, 274:2 (1986), 285-314. · doi:10.1016/0550-3213(86)90287-7
[6] M. R. Dixon and T. A. Fournelle, “The indecomposability of certain wreath products indexed by partially ordered sets,” Arch. Math., 43 (1984), 193-207. · Zbl 0532.20017 · doi:10.1007/BF01247563
[7] C. Farsi and Ch. Seaton, “Generalized orbifold Euler characteristics for general orbifolds and wreath products,” Algebr. Geom. Topol., 11:1 (2011), 523-551. · Zbl 1213.22005 · doi:10.2140/agt.2011.11.523
[8] S. M. Gusein-Zade, “Equivariant analogues of the Euler characteristic and Macdonald type formulas,” Uspekhi Mat. Nauk, 72:1(433) (2017), 3-36; English transl.: Russian Math. Surveys, 72:1 (2017), 1-32. · Zbl 1376.57039 · doi:10.4213/rm9748
[9] S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “A power structure over the Grothendieck ring of varieties,” Math. Res. Lett., 11:1 (2004), 49-57. · Zbl 1063.14026 · doi:10.4310/MRL.2004.v11.n1.a6
[10] S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “On the power structure over the Grothendieck ring of varieties and its applications,” Trudy Mat. Inst. Steklov., 258 (2007), 58-69; English transl.: Proc. Steklov Inst. Math., 258:1 (2007), 53-64. · Zbl 1222.14007
[11] S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “Equivariant versions of higher order orbifold Euler characteristics,” Mosc. Math. J., 16:4 (2016), 751-765. · Zbl 1382.55004 · doi:10.17323/1609-4514-2016-16-4-751-765
[12] S. M. Gusein-Zade, I. Luengo, and A. Melle-Hernández, “Grothendieck ring of varieties with finite groups actions,” Proc. Edinb. Math. Soc., Ser. 2 (to appear); https://arxiv.org/abs/1706.00918. · Zbl 1454.14023
[13] F. Hirzebruch and T. Höfer, “On the Euler number of an orbifold,” Math. Ann., 286:1-3 (1990), 255-260. · Zbl 0679.14006 · doi:10.1007/BF01453575
[14] D. Knutson, λ-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Math., vol. 308, Springer-Verlag, Berlin-New York, 1973. · Zbl 0272.20008
[15] P. M. Neumann, “On the structure of standard wreath products of groups,” Math. Z., 84 (1964), 343-373. · Zbl 0122.02901 · doi:10.1007/BF01109904
[16] I. Satake, “On a generalization of the notion of manifold,” Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 359-363. · Zbl 0074.18103 · doi:10.1073/pnas.42.6.359
[17] I. Satake, “The Gauss-Bonnet theorem for <Emphasis Type=”Italic“>V-manifolds,” J. Math. Soc. Japan, 9 (1957), 464-492. · Zbl 0080.37403 · doi:10.2969/jmsj/00940464
[18] H. Tamanoi, “Generalized orbifold Euler characteristic of symmetric products and equivariant Morava <Emphasis Type=”Italic“>K-theory,” Algebr. Geom. Topol., 1 (2001), 115-141. · Zbl 0965.57033 · doi:10.2140/agt.2001.1.115
[19] T. tom Dieck, Transformation Groups, De Gruyter Studies in Math., vol. 8, Walter de Gruyter, Berlin, 1987. · Zbl 0611.57002 · doi:10.1515/9783110858372
[20] C. E. Watts, “On the Euler characteristic of polyhedra,” Proc. Amer. Math. Soc., 13 (1962), 304-306. · Zbl 0111.35705 · doi:10.1090/S0002-9939-1962-0137109-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.