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 Mat. Sb. (N.S.), 1981, Volume 116(158), Number 3(11), Pages 359–369 (Mi msb2473)

Generators of $S^1$-bordism

O. R. Musin

Abstract: In this paper generators are found for the rings $U^{S^1}_*$ (the unitary $S^1$-bordism ring) and $U_*(S^1,\{\mathbf Z_s\})$ (the unitary bordism ring with actions of the group $S^1$ without fixed points). The generators found are $S^1$-manifolds of the form $(S^3)^k\times\mathbf CP^n/(S^1)^k$. By an obvious construction the ring $U^{S^1}_*$ allows one to establish a relation between numerical invariants of manifolds with unitary actions of $S^1$ and the set of fixed points, without using a theorem of the type of an integrality theorem. In particular, we obtain a new proof of the Atiyah–Hirzebruch formula for the generalized Todd genus of $S^1$-manifolds.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 44:3, 325–334

Bibliographic databases:

UDC: 513.836
MSC: Primary 57R85, 57R77; Secondary 55N22, 55N25

Citation: O. R. Musin, “Generators of $S^1$-bordism”, Mat. Sb. (N.S.), 116(158):3(11) (1981), 359–369; Math. USSR-Sb., 44:3 (1983), 325–334

Citation in format AMSBIB
\Bibitem{Mus81} \by O.~R.~Musin \paper Generators of $S^1$-bordism \jour Mat. Sb. (N.S.) \yr 1981 \vol 116(158) \issue 3(11) \pages 359--369 \mathnet{http://mi.mathnet.ru/msb2473} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=665688} \zmath{https://zbmath.org/?q=an:0508.57027|0494.57015} \transl \jour Math. USSR-Sb. \yr 1983 \vol 44 \issue 3 \pages 325--334 \crossref{https://doi.org/10.1070/SM1983v044n03ABEH000970} 

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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. R. Musin, “Converse theorem on equivariant genera”, Russian Math. Surveys, 64:4 (2009), 753–755
2. Oleg R. Musin, “On rigid Hirzebruch genera”, Mosc. Math. J., 11:1 (2011), 139–147
3. O. R. Musin, “Circle Actions with Two Fixed Points”, Math. Notes, 100:4 (2016), 636–638
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