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Mat. Sb. (N.S.), 1971, Volume 84(126), Number 1, Pages 81–118 (Mi msb3051)  

This article is cited in 35 scientific papers (total in 37 papers)

Formal groups, power systems and Adams operators

V. M. Buhštaber, S. P. Novikov

Abstract: This paper provides a systematic presentation of the connection between the theory of one-dimensional formal groups and the theory of unitary cobordism. Two new algebraic concepts are introduced: formal power systems and two-valued formal groups. A presentation of the general theory of formal power systems is given, and it is shown that cobordism theory gives a nontrivial example of a system which is not a formal group. A two-valued formal group is constructed whose ring of coefficients is closely related to the bordism ring of a symplectic manifold. Finally, applications of formal groups and power systems are made to the theory of fixed points of periodic transformations of quasicomplex manifolds.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 13:1, 80–116

Bibliographic databases:

UDC: 513.836
MSC: 55N22, 20K45, 53D05, 57N70
Received: 09.06.1970

Citation: V. M. Buhštaber, S. P. Novikov, “Formal groups, power systems and Adams operators”, Mat. Sb. (N.S.), 84(126):1 (1971), 81–118; Math. USSR-Sb., 13:1 (1971), 80–116

Citation in format AMSBIB
\by V.~M.~Buh{\v s}taber, S.~P.~Novikov
\paper Formal groups, power systems and Adams operators
\jour Mat. Sb. (N.S.)
\yr 1971
\vol 84(126)
\issue 1
\pages 81--118
\jour Math. USSR-Sb.
\yr 1971
\vol 13
\issue 1
\pages 80--116

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    This publication is cited in the following articles:
    1. V. M. Buchstaber, A. S. Mishchenko, S. P. Novikov, “Formal groups and their role in the apparatus of algebraic topology”, Russian Math. Surveys, 26:2 (1971), 63–90  mathnet  crossref  mathscinet  zmath
    2. V. M. Bukhshtaber, “Dvuznachnye formalnye gruppy. Nekotorye prilozheniya k kobordizmam”, UMN, 26:3(159) (1971), 195–196  mathnet  mathscinet  zmath
    3. A. A. Peresetskii, “$SU$-cobordism and formal groups”, Math. USSR-Sb., 17:4 (1972), 529–538  mathnet  crossref  mathscinet  zmath
    4. I. M. Krichever, “Actions of finite cyclic groups on quasicomplex manifolds”, Math. USSR-Sb., 19:2 (1973), 305–319  mathnet  crossref  mathscinet  zmath
    5. S. M. Gusein-Zade, I. M. Krichever, “O formulakh dlya nepodvizhnykh tochek deistviya gruppy $Z_p$”, UMN, 28:1(169) (1973), 237–238  mathnet  mathscinet  zmath
    6. V. M. Buchstaber, “Two-valued formal groups. Algebraic theory and applications to cobordism. I”, Math. USSR-Izv., 9:5 (1975), 987–1006  mathnet  crossref  mathscinet  zmath
    7. Yu. B. Rudyak, “Formal groups and bordisms with singularities”, Math. USSR-Sb., 25:4 (1975), 487–505  mathnet  crossref  mathscinet  zmath
    8. V. M. Buchstaber, “Two-valued formal groups. Algebraic theory and applications to cobordism. II”, Math. USSR-Izv., 10:2 (1976), 271–308  mathnet  crossref  mathscinet  zmath
    9. V. M. Bukhshtaber, “Dvuznachnye formalnye gruppy v apparate teorii kobordizmov”, UMN, 32:2(194) (1977), 205–206  mathnet  mathscinet  zmath
    10. V. M. Buchstaber, “Topological applications of the theory of two-valued formal groups”, Math. USSR-Izv., 12:1 (1978), 125–177  mathnet  crossref  mathscinet  zmath
    11. V. M. Buchstaber, A. V. Shokurov, “The Landweber–Novikov algebra and formal vector fields on the line”, Funct. Anal. Appl., 12:3 (1978), 159–168  mathnet  crossref  mathscinet  zmath
    12. R. G. Nadiradze, “Involutions on Stong manifolds and their application in cobordism theory”, Russian Math. Surveys, 35:3 (1980), 264–267  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. V. M. Buchstaber, A. N. Kholodov, “Topological constructions connected with many-valued formal groups”, Math. USSR-Izv., 20:1 (1983), 1–25  mathnet  crossref  mathscinet  zmath
    14. A. N. Kholodov, “Multidimensional two-valued commutative formal groups”, Russian Math. Surveys, 43:1 (1988), 243–244  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    15. A. N. Kholodov, “The umbral calculus on multivalued formal groups, and Adams projections in $K$-theory”, Math. USSR-Sb., 65:2 (1990), 423–437  mathnet  crossref  mathscinet  zmath
    16. V. M. Buchstaber, A. N. Kholodov, “Groups of formal diffeomorphisms of the superline, generating functions for sequences of polynomials, and functional equations”, Math. USSR-Izv., 35:2 (1990), 277–305  mathnet  crossref  mathscinet  zmath
    17. V. M. Buchstaber, “Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups”, Russian Math. Surveys, 45:3 (1990), 213–215  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    18. V. M. Buchstaber, E. G. Rees, “Multivalued groups and Hopf $n$-algebras”, Russian Math. Surveys, 51:4 (1996), 727–729  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. T. E. Panov, “Elliptic genus for manifolds with an action of the group $\mathbb Z/p$”, Russian Math. Surveys, 52:2 (1997), 418–419  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    20. T. E. Panov, “Classification up to cobordism of manifolds with simple action of $\mathbb Z/p$”, Math. Notes, 63:2 (1998), 225–232  mathnet  crossref  crossref  mathscinet  zmath  isi
    21. T. E. Panov, “Calculation of Hirzebruch genera for manifolds acted on by the group $\mathbf Z/p$ via invariants of the action”, Izv. Math., 62:3 (1998), 515–548  mathnet  crossref  crossref  mathscinet  zmath  isi
    22. B. I. Botvinnik, V. M. Buchstaber, S. P. Novikov, S. A. Yuzvinskii, “Algebraic aspects of the theory of multiplications in complex cobordism theory”, Russian Math. Surveys, 55:4 (2000), 613–633  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. A. A. Bolibrukh, A. P. Veselov, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, S. P. Novikov, T. E. Panov, Yu. M. Smirnov, “Viktor Matveevich Buchstaber (on his 60th birthday)”, Russian Math. Surveys, 58:3 (2003), 627–635  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    24. V. M. Buchstaber, “$n$-valued groups: theory and applications”, Mosc. Math. J., 6:1 (2006), 57–84  mathnet  crossref  mathscinet  zmath
    25. G. A. Merzon, “Realization of formal groups by cohomology theories, and the Adams–Novikov operations”, Russian Math. Surveys, 62:4 (2007), 813–815  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. A. A. Gaifullin, “The construction of combinatorial manifolds with prescribed sets of links of vertices”, Izv. Math., 72:5 (2008), 845–899  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    27. V. M. Buchstaber, “The general Krichever genus”, Russian Math. Surveys, 65:5 (2010), 979–981  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    28. Buchstaber V., Panov T., Ray N., “Toric Genera”, International Mathematics Research Notices, 2010, no. 16, 3207–3262  mathscinet  zmath  isi  elib
    29. Oleg R. Musin, “On rigid Hirzebruch genera”, Mosc. Math. J., 11:1 (2011), 139–147  mathnet  crossref  mathscinet
    30. D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms”, Trans. Moscow Math. Soc., 72 (2011), 97–142  mathnet  crossref  mathscinet  zmath  elib
    31. 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
    32. A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    33. 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
    34. Proc. Steklov Inst. Math., 286 (2014), 224–239  mathnet  crossref  crossref  isi  elib  elib
    35. V. M. Buchstaber, A. V. Ustinov, “Coefficient rings of formal group laws”, Sb. Math., 206:11 (2015), 1524–1563  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    36. V. M. Buchstaber, “Cobordisms, manifolds with torus action, and functional equations”, Proc. Steklov Inst. Math., 302 (2018), 48–87  mathnet  crossref  crossref  mathscinet  isi  elib
    37. V. M. Buchstaber, A. P. Veselov, “Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups”, Russian Math. Surveys, 74:3 (2019), 387–430  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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