Polynomial Eulerian characteristic of nilmanifolds

V. M. Buchstaber

Funkcionalʹnyj analiz i ego priloženiâ, Tome 58 (2024) no. 1, pp. 22-41.

Résumé

The article studies bundle towers $M^{n+1}\to M^{n}\to \dots \to S^1$, $\geqslant 1$, with fiber $S^1$, where $M^n = L^n\!/\Gamma^n$ are compact smooth nilmanifolds and $L^n\thickapprox \mathbb{R}^n$ is a group of polynomial transformations of the line $\mathbb{R}^1$. The focus is on the well-known problem of calculating cohomology rings with rational coefficients of manifolds $M^n$. Using the canonical bigradation in the de Rham complex of manifolds $M^n$, we introduce the concept of polynomial Eulerian characteristic and calculate it for these manifolds.

Détail

Keywords: bigraded de Rham complex, algebra of left invariant differential operators
Mots-clés : polynomial transformations of a line, Gysin exact sequence.

Bibliographie

[1] D. V. Anosov, Ya. G. Sinai, “Nekotorye gladkie ergodicheskie sistemy”, UMN, 22:5 (1967), 107–172 | MR | Zbl

[2] L. Auslander, L. Green, F. Hahn, Flows on Homogeneous Spaces, Ann. of Math. Studies, 53, Princeton Univ. Press, Princeton, NJ, 1963 | MR | Zbl

[3] L. Auslander, Lecture Notes on Nil-Theta Functions, Regional Conference Series in Mathematics, 34, Amer. Math. Soc., Providence, RI, 1977 | DOI | MR | Zbl

[4] I. K. Babenko, I. A. Taimanov, “Proizvedeniya Massi v simplekticheskikh mnogoobraziyakh”, Matem. sb., 191:8 (2000), 3–44 | DOI | Zbl

[5] I. K. Babenko, “Algebra, geometriya i topologiya gruppy podstanovok formalnykh stepennykh ryadov”, UMN, 68:1(409) (2013), 3–76 | DOI | MR | Zbl

[6] V. M. Bukhshtaber, A. V. Shokurov, “Algebra Landvebera–Novikova i formalnye vektornye polya na pryamoi”, Funkts. analiz i ego prilozh., 12:3 (1978), 1–11 | MR | Zbl

[7] V. M. Buchstaber, “Semigroups of maps into groups, operator doubles, and complex cobordisms”, Topics in Topology and Math. Physics, Amer. Math. Soc. Transl., Ser. 2, 170, 1995, 9–31 | MR | Zbl

[8] V. M. Bukhshtaber, “Gruppy polinomialnykh preobrazovanii pryamoi, neformalnye simplekticheskie mnogoobraziya i algebra Landvebera–Novikova”, UMN, 54:4 (1999), 161–162 | DOI | MR | Zbl

[9] V. M. Bukhshtaber, E. V. Koritskaya, “Kvazilineinoe uravnenie Byurgersa-Khopfa i mnogogranniki Stashefa”, Funkts. analiz i ego prilozh., 41:3 (2007), 34–47 | DOI | MR | Zbl

[10] V. M. Bukhshtaber, “Koltso prostykh mnogogrannikov i differentsialnye uravneniya”, Trudy MIAN, 263 (2008), 18–43 | Zbl

[11] V. M. Buchstaber, T. E. Panov, Toric Topology, Mathematical Surveys and Monographs, 204, Amer. Math. Soc., Providence, RI, 2015 | DOI | MR | Zbl

[12] D. V. Egorov, “Teta-funktsii na mnogoobrazii Kodairy–Terstona”, Sib. matem. zhurn., 50:2 (2009), 320–328 | MR | Zbl

[13] D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, New York, 1986 | MR | Zbl

[14] I. M. Gelfand, “Nekotorye zadachi teorii kvazilineinykh uravnenii”, UMN, 14:2 (1959), 87–158 | MR | Zbl

[15] L. V. Goncharova, “Kogomologii algebr Li formalnykh vektornykh polei na pryamoi”, Funkts. analiz i ego pril., 7:2 (1973), 6–14 | MR | Zbl

[16] A. A. Kirillov, “Unitarnye predstavleniya nilpotentnykh grupp Li”, UMN, 17:4(106) (1962), 57–110 | MR | Zbl

[17] J.-L. Loday, The multiple faces of the associahedron, Clay Mathematics Institute Publication, Oxford, 2005

[18] A. I. Maltsev, “Ob odnom klasse odnorodnykh prostranstv”, Izv. AN SSSR. Ser. matem., 13:1 (1949), 9–32 | Zbl

[19] D. V. Millionschikov, “Algebra formalnykh vektornykh polei na pryamoi i gipoteza Bukhshtabera”, Funkts. analiz i ego prilozh., 43:4 (2009), 26–44 | DOI | MR | Zbl

[20] K. Nomizu, “On the cohomology of compact homogeneous spaces of nilpotent Lie groups”, Ann. of Math. (2), 59 (1954), 531–538 | DOI | MR | Zbl

[21] R. Stenli, Perechislitelnaya kombinatorika, Derevya, proizvodyaschie funktsii i simmeticheskie funktsii, Mir, M., 2005

[22] S. L. Tabachnikov, D. B. Fuks, Matematicheskii divertisment. 30 lektsii po klassicheskoi matematike, MTsNMO, 2011