Geodesic
Rigidity of powers and Kosniowski's conjecture
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1227-1236.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we state some problems on rigidity of powers in terms of complex analysis and number-theoretic abstraction, which has a strong topological background for the rigid Hirzebruch genera and Kosniowski's conjecture of unitary circle actions. However, our statements of these problems are elementary enough and do not require any knowledge of algebraic topology. We shall give the solutions of these problems for some particular cases. As a consequence, we obtain that Kosniowski's conjecture holds in the case of dimension $\leq 10$ or equal to $14$.
Keywords: Rigidity of powers, circle action, fixed points, Kosniowski's conjecture
Mots-clés : multiplicative genus. 
@article{SEMR_2018_15_a51,
     author = {Z. L\"u and O. R. Musin},
     title = {Rigidity of powers and {Kosniowski's} conjecture},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1227--1236},
     publisher = {mathdoc},
     volume = {15},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2018_15_a51/}
}
TY  - JOUR
AU  - Z. Lü
AU  - O. R. Musin
TI  - Rigidity of powers and Kosniowski's conjecture
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2018
SP  - 1227
EP  - 1236
VL  - 15
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2018_15_a51/
LA  - en
ID  - SEMR_2018_15_a51
ER  - 
%0 Journal Article
%A Z. Lü
%A O. R. Musin
%T Rigidity of powers and Kosniowski's conjecture
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2018
%P 1227-1236
%V 15
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2018_15_a51/
%G en
%F SEMR_2018_15_a51
Z. Lü; O. R. Musin. Rigidity of powers and Kosniowski's conjecture. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1227-1236. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a51/

[1] C. Allday, V. Puppe, Cohomological Methods in Transformation Groups, Cambridge Studies in Advanced Mathematics, 32, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[2] M. F. Atiyah, F. Hirzebruch, “Spin manifolds and group actions”, Essays in Topology and Related Subjects, Springer-Verlag, Berlin, 1970, 18–28 | DOI | MR

[3] M. F. Atiyah, I. M. Singer, “The index theory of elliptic operators: III”, Ann. of Math., 87 (1968), 546–604 | DOI | MR | Zbl

[4] G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York–London, 1972 | MR | Zbl

[5] R. Bott, “Vector fields and characteristic numbers”, Michigan Math. J., 14 (1967), 231–244 | DOI | MR | Zbl

[6] V. Buchstaber, T. Panov, Toric Topology, Mathematical Surveys and Monographs, 204, AMS, Providence, RI, 2015 | DOI | MR | Zbl

[7] V. Buchstaber, T. Panov, N. Ray, “Toric genera”, Internat. Math. Research Notices, 16 (2010), 3207–3262 | MR | Zbl

[8] V. M. Buchstaber, N. Ray, “The universal equivariant genus and Krichever's formula”, Russian Math. Surveys, 62:1 (2007), 178–180 | DOI | MR | Zbl

[9] V. Guillemin, V. Ginzburg, Y. Karshon, Moment Maps, Cobordisms, and Hamiltonian Group Actions, Mathematical Surveys and Monographs, 98, American Mathematical Society, Providence, RI, 2002 | DOI | MR | Zbl

[10] F. Hirzebruch, Topological Methods in Algebraic Geometry, Grundlehren der mathematischen Wissenschaften, 131, 3rd edition, Springer, Berlin–Heidelberg, 1966 | MR | Zbl

[11] Donghoon Jang, “Circle actions on almost complex manifolds with isolated fixed points”, J. Geom. Phys., 119 (2017), 187–192 | DOI | MR | Zbl

[12] C. Kosniowski, “Some formulae and conjectures associated with circle actions”, Proc. Sympos. (Univ. Siegen, Siegen, 1979), Lecture Notes in Math., 788, Springer, Berlin, 1980, 331–339 | DOI | MR

[13] I. M. Krichever, “Formal groups and the Atiyah–Hirzebruch formula”, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 1289–1304 | MR

[14] Ping Li, Kefeng Liu, “Some remarks on circle action on manifolds”, Mathematical Research Letters, 18:3 (2011), 437–446 | DOI | MR | Zbl

[15] Z. Lü, “Equivariant cobordism of unitary toric manifolds”, Conference V. Buchstaber–70 “Algebraic topology and Abelian functions” (18–22 June 2013, Moscow), 53–54

[16] Z. Lü, Q. B. Tan, “Equivariant Chern numbers and the number of fixed points for unitary torus manifolds”, Math. Res. Lett., 18:6 (2011), 1319–1325 | DOI | MR | Zbl

[17] Z. Lü, “Equivariant bordism of 2-torus manifolds and unitary toric manifolds: a survey”, Proceedings of the Sixth International Congress of Chinese Mathematicians, v. II, Adv. Lect. Math. (ALM), 37, Int. Press, Somerville, MA, 2017, 267–284 | MR | Zbl

[18] J. Milnor, “On the cobordism ring $\Omega_*$ and a complex analogue”, I. Amer. J. Math., 82 (1960), 505–521 | DOI | MR | Zbl

[19] O. R. Musin, “Circle actions with two fixed points”, Math. Notes, 100:3–4 (2016), 636–638 | DOI | MR | Zbl

[20] O. R. Musin, “On rigid Hirzebruch genera”, Mosc. Math. J., 11:1 (2011), 139–147 | MR | Zbl

[21] O. R. Musin, “Circle action on homotopy complex projective spaces”, Math. Notes, 28:1 (1980), 533–540 | DOI | MR

[22] S. P. Novikov, “Some problems in the topology of manifolds connected with the theory of Thom spaces”, Soviet Math. Dokl., 1 (1960), 717–720 | MR

[23] A. Pelayo, S. Tolman, “Fixed points of symplectic periodic flows”, Ergodic Theory and Dynamical Systems, 31:4 (2011), 1237–1247 | DOI | MR | Zbl