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Izv. Akad. Nauk SSSR Ser. Mat., 1974, Volume 38, Issue 6, Pages 1289–1304 (Mi izv2010)  

This article is cited in 19 scientific papers (total in 19 papers)

Formal groups and the Atiyah–Hirzebruch formula

I. M. Krichever


Abstract: In this article, manifolds with actions of compact Lie groups are considered. For each rational Hirzebruch genus $h\colon\Omega_*\to Q$, an “equivariant genus” $h^G$, a homomorphism from the bordism ring of $G$-manifolds to the ring $K(BG)\otimes Q$, is constructed. With the aid of the language of formal groups, for some genera it is proved that for a connected compact Lie group $G$, the image of $h^G$ belongs to the subring $Q\subset K(BG)\otimes Q$. As a consequence, extremely simple relations between the values of these genera on bordism classes of $S^1$-manifolds and submanifolds of its fixed points are found. In particular, a new proof of the Atiyah–Hirzebruch formula is obtained.

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English version:
Mathematics of the USSR-Izvestiya, 1974, 8:6, 1271–1285

Bibliographic databases:

UDC: 513.83
MSC: Primary 57A65, 53C10, 53C15; Secondary 55B20, 57D15, 57D90
Received: 11.12.1973

Citation: I. M. Krichever, “Formal groups and the Atiyah–Hirzebruch formula”, Izv. Akad. Nauk SSSR Ser. Mat., 38:6 (1974), 1289–1304; Math. USSR-Izv., 8:6 (1974), 1271–1285

Citation in format AMSBIB
\Bibitem{Kri74}
\by I.~M.~Krichever
\paper Formal groups and the Atiyah--Hirzebruch formula
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1974
\vol 38
\issue 6
\pages 1289--1304
\mathnet{http://mi.mathnet.ru/izv2010}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=400267}
\zmath{https://zbmath.org/?q=an:0315.57021}
\transl
\jour Math. USSR-Izv.
\yr 1974
\vol 8
\issue 6
\pages 1271--1285
\crossref{https://doi.org/10.1070/IM1974v008n06ABEH002147}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. M. Krichever, “Obstructions to the existence of $S^1$-actions. Bordism of ramified coverings”, Math. USSR-Izv., 10:4 (1976), 783–797  mathnet  crossref  mathscinet  zmath
    2. O. R. Musin, “Generators of $S^1$-bordism”, Math. USSR-Sb., 44:3 (1983), 325–334  mathnet  crossref  mathscinet  zmath
    3. V. M. Buchstaber, A. N. Kholodov, “Formal groups, functional equations and generalized cohomology theories”, Math. USSR-Sb., 69:1 (1991), 77–97  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. Gabriel Katz, “Analytic deformations of equivariant genera, universal symmetry blocks and Witten's rigidity”, Topology, 35:2 (1996), 457  crossref
    5. T. E. Panov, “Hirzebruch genera of manifolds with torus action”, Izv. Math., 65:3 (2001), 543–556  mathnet  crossref  crossref  mathscinet  zmath  elib
    6. K. E. Feldman, “Hirzebruch genus of a manifold supporting a Hamiltonian circle action”, Russian Math. Surveys, 56:5 (2001), 978–979  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. V. M. Buchstaber, N. Ray, “Universal equivariant genus and Krichever's formula”, Russian Math. Surveys, 62:1 (2007), 178–180  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. V. M. Buchstaber, “Ring of Simple Polytopes and Differential Equations”, Proc. Steklov Inst. Math., 263 (2008), 13–37  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    9. O. R. Musin, “Converse theorem on equivariant genera”, Russian Math. Surveys, 64:4 (2009), 753–755  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Buchstaber V., Panov T., Ray N., “Toric Genera”, International Mathematics Research Notices, 2010, no. 16, 3207–3262  isi  elib
    11. Oleg R. Musin, “On rigid Hirzebruch genera”, Mosc. Math. J., 11:1 (2011), 139–147  mathnet  mathscinet
    12. V. M. Buchstaber, E. Yu. Bun'kova, “Krichever Formal Groups”, Funct. Anal. Appl., 45:2 (2011), 99–116  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. A. A. Kustarev, “Almost complex circle actions with few fixed points”, Russian Math. Surveys, 68:3 (2013), 574–576  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. Buchstaber V.M., Terzic S., “Toric Genera of Homogeneous Spaces and their Fibrations”, Int. Math. Res. Notices, 2013, no. 6, 1324–1403  crossref  isi
    16. O. R. Musin, “Circle Actions with Two Fixed Points”, Math. Notes, 100:4 (2016), 636–638  mathnet  crossref  crossref  mathscinet  isi  elib
    17. I. V. Netay, “Hirzebruch Functional Equations and Krichever Complex Genera”, Math. Notes, 103:2 (2018), 232–242  mathnet  crossref  crossref  isi  elib
    18. Z. Lü, O. R. Musin, “Rigidity of powers and Kosniowski's conjecture”, Sib. elektron. matem. izv., 15 (2018), 1227–1236  mathnet  crossref
    19. V. M. Buchstaber, “Cobordisms, manifolds with torus action, and functional equations”, Proc. Steklov Inst. Math., 302 (2018), 48–87  mathnet  crossref  crossref  isi  elib
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