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 Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 4, Pages 789–806 (Mi izv1846)

Multiplications in cobordism theories with singularities, and Steenrod–Tom Dieck operations

O. K. Mironov

Abstract: In this paper obstructions to the commutativity and associativity of multiplications in cobordism theories with singularities are determined. Obstructions to the existence and commutativity of multiplications are expressed in terms of Steenrod-tom Dieck operations in cobordism.
General theorems are applied to the cobordism theories $SO^*$, $U^*$, $SU^*$, $Sp^*$ and $Sc^*$ with singularities. Associativity of multiplication is proved in those cases where it exists, as well as the existence of a commutative and associative multiplication if the operation of division by 2 can be carried out in the ring of scalars of a theory with singularities.
As an application of the main theorems, a uniqueness theorem for “generalized $K$-theories” is proved.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1979, 13:1, 89–106

Bibliographic databases:

UDC: 513.8
MSC: Primary 55N20; Secondary 55N45, 57R90

Citation: O. K. Mironov, “Multiplications in cobordism theories with singularities, and Steenrod–Tom Dieck operations”, Izv. Akad. Nauk SSSR Ser. Mat., 42:4 (1978), 789–806; Math. USSR-Izv., 13:1 (1979), 89–106

Citation in format AMSBIB
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\by O.~K.~Mironov
\paper Multiplications in cobordism theories with singularities, and Steenrod--Tom Dieck operations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1978
\vol 42
\issue 4
\pages 789--806
\mathnet{http://mi.mathnet.ru/izv1846}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=508827}
\zmath{https://zbmath.org/?q=an:0424.55017|0399.55015}
\transl
\jour Math. USSR-Izv.
\yr 1979
\vol 13
\issue 1
\pages 89--106
\crossref{https://doi.org/10.1070/IM1979v013n01ABEH002013}

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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. O. K. Mironov, “Steenrod–Dieck operations and obstructions for admissible multiplications in cobordism theories with singularities”, Russian Math. Surveys, 35:3 (1980), 260–263
2. V. V. Vershinin, “Symplectic cobordism with singularities”, Math. USSR-Izv., 22:2 (1984), 211–226
3. A. V. Pajitnov, Yu. B. Rudyak, “On commutative ring spectra of characteristic 2”, Math. USSR-Sb., 52:2 (1985), 471–479
4. Urs Würgler, “Commutative ring-spectra of characteristic 2”, Comment Math Helv, 61:1 (1986), 33
5. Rolf Kultze, Urs Würgler, “A note on the algebra P(n)*(P(n)) for the prime 2”, manuscripta math, 57:2 (1987), 195
6. F. Buzato, “Realizatsiya universalnoi formalnoi gruppy Abelya”, UMN, 51:3(309) (1996), 183–184
7. Inès Saihi, “Théorie elliptique entière”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 325:4 (1997), 419
8. Nitu Kitchloo, Gerd Laures, W.Stephen Wilson, “The Morava K-theory of spaces related to BO”, Advances in Mathematics, 189:1 (2004), 192
9. L. Astey, “Commutative 2-local ring spectra”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 127:01 (2011), 1
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