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Izv. Akad. Nauk SSSR Ser. Mat., 1967, Volume 31, Issue 4, Pages 855–951 (Mi izv2568)  

This article is cited in 74 scientific papers (total in 74 papers)

The methods of algebraic topology from the viewpoint of cobordism theory

S. P. Novikov

Abstract: The goal of this work is the construction of the analogue to the Adams spectral sequence in cobordism theory, calculation of the ring of cohomology operations in this theory, and also a number of applications: to the problem of computing homotopy groups and the classical Adams spectral sequence, fixed points of transformations of period $p$, and others.

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English version:
Mathematics of the USSR-Izvestiya, 1967, 1:4, 827–913

Bibliographic databases:

UDC: 513.83
MSC: 55N22, 55T15, 55M05, 55Q10, 55Q25, 55Q40
Received: 10.04.1967

Citation: S. P. Novikov, “The methods of algebraic topology from the viewpoint of cobordism theory”, Izv. Akad. Nauk SSSR Ser. Mat., 31:4 (1967), 855–951; Math. USSR-Izv., 1:4 (1967), 827–913

Citation in format AMSBIB
\by S.~P.~Novikov
\paper The methods of algebraic topology from the viewpoint of cobordism
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1967
\vol 31
\issue 4
\pages 855--951
\jour Math. USSR-Izv.
\yr 1967
\vol 1
\issue 4
\pages 827--913

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    This publication is cited in the following articles:
    1. S. P. Novikov, “Adams operators and fixed points”, Math. USSR-Izv., 2:6 (1968), 1193–1211  mathnet  crossref  mathscinet  zmath
    2. D. Montgomeri, “Kompaktnye gruppy preobrazovanii”, UMN, 23:2(140) (1968), 169–178  mathnet  mathscinet
    3. V. M. Buchstaber, “Modules of differentials of the Atiyah–Hirzebruch spectral sequence”, Math. USSR-Sb., 7:2 (1969), 299–313  mathnet  crossref  mathscinet  zmath
    4. G. G. Kasparov, “Invariants of classical lens manifolds in cobordism theory”, Math. USSR-Izv., 3:4 (1969), 695–705  mathnet  crossref  mathscinet  zmath
    5. V. M. Buchstaber, “Modules of differentials of the Atiyah–Hirzebruch spectral sequence. II”, Math. USSR-Sb., 12:1 (1970), 59–75  mathnet  crossref  mathscinet  zmath
    6. V. M. Buchstaber, “The Chern–Dold character in cobordisms. I”, Math. USSR-Sb., 12:4 (1970), 573–594  mathnet  crossref  mathscinet  zmath
    7. 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
    8. V. M. Buhštaber, S. P. Novikov, “Formal groups, power systems and Adams operators”, Math. USSR-Sb., 13:1 (1971), 80–116  mathnet  crossref  mathscinet  zmath
    9. M. A. Shubin, “On the index of families of Wiener–Hopf operators”, Math. USSR-Sb., 13:4 (1971), 529–551  mathnet  crossref  mathscinet  zmath
    10. N. V. Panov, “Characteristic numbers in $U$-theory”, Math. USSR-Izv., 5:6 (1971), 1365–1385  mathnet  crossref  mathscinet  zmath
    11. A. A. Peresetskii, “$SU$-cobordism and formal groups”, Math. USSR-Sb., 17:4 (1972), 529–538  mathnet  crossref  mathscinet  zmath
    12. V. M. Bukhshtaber, “Proektory v unitarnykh kobordizmakh, svyazannye s $SU$-teoriei”, UMN, 27:6(168) (1972), 231–232  mathnet  mathscinet  zmath
    13. Peter S. Landweber, “Associated prime ideals and Hopf algebras”, Journal of Pure and Applied Algebra, 3:1 (1973), 43  crossref
    14. 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
    15. Yu. B. Rudyak, “Formal groups and bordisms with singularities”, Math. USSR-Sb., 25:4 (1975), 487–505  mathnet  crossref  mathscinet  zmath
    16. V. M. Bukhshtaber, “Dvuznachnye formalnye gruppy v apparate teorii kobordizmov”, UMN, 32:2(194) (1977), 205–206  mathnet  mathscinet  zmath
    17. Haynes R. Miller, “A localization theorem in homological algebra”, Math Proc Camb Phil Soc, 84:1 (1978), 73  crossref  mathscinet  zmath
    18. Douglas C. Ravenel, “The non-existence of odd primary Arf invariant elements in stable homotopy”, Math Proc Camb Phil Soc, 83:3 (1978), 429  crossref  mathscinet  zmath
    19. A. V. Shokurov, “A operations in the theory of unitary cobordisms and homotopy invariants of continuous maps”, Russian Math. Surveys, 33:3 (1978), 179–180  mathnet  crossref  mathscinet  zmath
    20. 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
    21. Yu. B. Rudyak, “On theorems of Conner–Floyd type in the theory of unitary cobordism”, Math. USSR-Izv., 12:3 (1978), 605–616  mathnet  crossref  mathscinet  zmath
    22. 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
    23. A. V. Shokurov, “Topological applications of the apparatus of formal diffeomorphisms of the line”, Russian Math. Surveys, 34:6 (1979), 213–219  mathnet  crossref  mathscinet  zmath
    24. D. F. Adams, “Algebraicheskaya topologiya. Putevoditel dlya studentov”, UMN, 34:3(207) (1979), 169–188  mathnet  mathscinet  zmath
    25. Haynes R. Miller, “On relations between Adams spectral sequences, with an application to the stable homotopy of a moore space”, Journal of Pure and Applied Algebra, 20:3 (1981), 287  crossref
    26. 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
    27. V. V. Vershinin, “Symplectic cobordism with singularities”, Math. USSR-Izv., 22:2 (1984), 211–226  mathnet  crossref  mathscinet  zmath
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    29. Peter S. Landweber, “A survey of bordism and cobordism”, Math Proc Camb Phil Soc, 100:2 (1986), 207  crossref  mathscinet  zmath
    30. V. V. Vershinin, “On the decomposition of certain spectra”, Math. USSR-Sb., 60:2 (1988), 283–290  mathnet  crossref  mathscinet  zmath
    31. 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
    32. B. I. Botvinnik, “The structure of the ring $MSU_*$”, Math. USSR-Sb., 69:2 (1991), 581–596  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    33. S. P. Novikov, “Various doublings of Hopf algebras. Operator algebras on quantum groups, complex cobordisms”, Russian Math. Surveys, 47:5 (1992), 198–199  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    34. Takuji Kashiwabara, “Hopf rings and unstable operations”, Journal of Pure and Applied Algebra, 94:2 (1994), 183  crossref
    35. Norihiko Minami, “The Kervaire invariant one element and the double transfer”, Topology, 34:2 (1995), 481  crossref
    36. 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
    37. V. M. Buchstaber, K. E. Feldman, “An addition formula for the first semi-integral Pontryagin class in complex cobordism”, Russian Math. Surveys, 52:6 (1997), 1287–1288  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    38. 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
    39. 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
    40. François-Xavier Dehon, Jean Lannes, “Sur les espaces fonctionnels dont la source est le classifiant d’un groupe de Lie compact commutatif”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 89:1 (1999), 127  crossref  zmath
    41. Proc. Steklov Inst. Math., 225 (1999), 261–268  mathnet  mathscinet  zmath
    42. V. M. Buchstaber, K. E. Feldman, “The index of an equivariant vector field and addition theorems for Pontryagin classes”, Izv. Math., 64:2 (2000), 223–247  mathnet  crossref  crossref  mathscinet  zmath  isi
    43. 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
    44. Po Hu, Igor Kriz, “Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence”, Topology, 40:2 (2001), 317  crossref
    45. M. E. Kazarian, “Multisingularities, cobordisms, and enumerative geometry”, Russian Math. Surveys, 58:4 (2003), 665–724  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    46. Imma Gálvez, Sarah Whitehouse, “Infinite sums of Adams operations and cobordism”, Math Z, 251:3 (2005), 475  crossref  mathscinet  zmath  isi
    47. E. Shinder, “Mishchenko and Quillen formulas for oriented cohomology pretheories”, St. Petersburg Math. J., 18:4 (2007), 671–678  mathnet  crossref  mathscinet  zmath
    48. 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
    49. 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
    50. 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
    51. Feldman K.E., “Miraculous cancellation and Pick's theorem”, Toric Topology, Contemporary Mathematics Series, 460, 2008, 71–86  isi
    52. Xiangjun Wang, “The secondary differentials on the third line of the Adams spectral sequence”, Topology and its Applications, 156:3 (2009), 477  crossref
    53. V. M. Buchstaber, “The general Krichever genus”, Russian Math. Surveys, 65:5 (2010), 979–981  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    54. Buchstaber V., Panov T., Ray N., “Toric Genera”, International Mathematics Research Notices, 2010, no. 16, 3207–3262  isi  elib
    55. Masateru Inoue, “On the Hurewicz image of the Steinberg summand”, Topology and its Applications, 157:15 (2010), 2362  crossref
    56. 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
    57. Aleksandr L. Anisimov, Vladimir V. Vershinin, “Symplectic cobordism in small dimensions and a series of elements of order four”, J. Homotopy Relat. Struct, 2012  crossref
    58. 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
    59. I. K. Babenko, “Algebra, geometry, and topology of the substitution group of formal power series”, Russian Math. Surveys, 68:1 (2013), 1–68  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    60. Malkhaz Bakuradze, “Computing the Krichever genus”, J. Homotopy Relat. Struct, 2013  crossref
    61. M. Bakuradze, “The formal group laws of Buchstaber, Krichever, and Nadiradze coincide”, Russian Math. Surveys, 68:3 (2013), 571–573  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    62. Buchstaber V.M. Terzic S., “Toric Genera of Homogeneous Spaces and their Fibrations”, Int. Math. Res. Notices, 2013, no. 6, 1324–1403  crossref  isi
    63. V. M. Buchstaber, E. Yu. Netay, “$\mathbb{C}P(2)$-multiplicative Hirzebruch genera and elliptic cohomology”, Russian Math. Surveys, 69:4 (2014), 757–759  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    64. M. Bakuradze, “On the Buchstaber formal group law and some related genera”, Proc. Steklov Inst. Math., 286 (2014), 1–15  mathnet  crossref  crossref  isi  elib  elib
    65. V. M. Bukhshtaber, E. Yu. Bunkova, “Universalnaya formalnaya gruppa, opredelyayuschaya ellipticheskuyu funktsiyu urovnya 3”, Chebyshevskii sb., 16:2 (2015), 66–78  mathnet  elib
    66. 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
    67. E. Yu. Bunkova, V. M. Buchstaber, A. V. Ustinov, “Coefficient rings of Tate formal groups determining Krichever genera”, Proc. Steklov Inst. Math., 292 (2016), 37–62  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    68. 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
    69. Elena Yu. Bunkova, “Hirzebruch functional equation: classification of solutions”, Proc. Steklov Inst. Math., 302 (2018), 33–47  mathnet  crossref  crossref  mathscinet  isi  elib
    70. I. Yu. Limonchenko, T. E. Panov, G. Chernykh, “$SU$-bordism: structure results and geometric representatives”, Russian Math. Surveys, 74:3 (2019), 461–524  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    71. 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
    72. Malkhaz Bakuradze, Vladimir V. Vershinin, “On Addition Theorems Related to Elliptic Integrals”, Proc. Steklov Inst. Math., 305 (2019), 22–32  mathnet  crossref  crossref  mathscinet  isi  elib
    73. V. M. Buchstaber, “Partially framed manifolds and loop spaces on the group $\operatorname{SU}(2)$”, Russian Math. Surveys, 75:4 (2020), 782–784  mathnet  crossref  crossref  mathscinet  isi  elib
    74. T. Ayano, V. M. Buchstaber, “Analytical and number-theoretical properties of the two-dimensional sigma function”, Chebyshevskii sb., 21:1 (2020), 9–50  mathnet  crossref  mathscinet
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