Geodesic
Generators of $S^1$-bordism
Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 325-334 
O. R. Musin. Generators of $S^1$-bordism. Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 325-334. http://geodesic.mathdoc.fr/item/SM_1983_44_3_a5/
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     author = {O. R. Musin},
     title = {Generators of $S^1$-bordism},
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     volume = {44},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1983_44_3_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Konner P., Floid E., Gladkie periodicheskie otobrazheniya, Mir, M., 1969

[2] Kosniowski C., “Generators of the $\mathbf{Z}_p$ bordism ring”, Math. Z., 149:2 (1976), 121–130 | DOI | MR | Zbl

[3] Kosniowski C., “$\mathbf{Z}_p$ manifolds with low dimensional fixed point set”, London Math. Soc. Lect. Note Ser., 1977, no. 26, 92–120 | MR | Zbl

[4] Gusein-Zade S. M., “O deistvii okruzhnosti na mnogoobraziyakh”, Matem. zametki, 10:5 (1971), 511–518 | MR | Zbl

[5] Gusein-Zade S. M., “$U$-deistviya okruzhnosti i nepodvizhnye tochki”, Izv. AN SSSR, ser. matem., 35 (1971), 1120–1136 | MR | Zbl

[6] Mischenko A. S., “Bordizmy s deistviem gruppy $\mathbf{Z}_p$ i nepodvizhnye tochki”, Matem. sb., 80 (122) (1969), 307–313 | MR | Zbl

[7] Atiyah M., Hirzebruch F., “Spin-manifolds and group actions”, Essay Topol. and Relat. Topics, Berlin, 1970, 18–28 | MR | Zbl

[8] Krichiver I. M., “Formalnye gruppy i formula Ati–Khirtsebrukha”, Izv. AN SSSR, ser. matem., 38 (1974), 1289–1304 | MR

[9] Khirtsebrukh F., Topologicheskie metody v algebraicheskoi geometrii, Mir, M., 1973