Second cohomology classes of the group of C 1 -flat diffeomorphisms
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, p. 77-85

We study the cohomology of the group consisting of all C ∞ -diffeomorphisms of the line, which are C 1 -flat to the identity at the origin. We construct non-trivial two second real cohomology classes and uncountably many second integral homology classes of this group.

On étudie la cohomologie du groupe des C ∞ -difféomorphismes de la droite, qui sout C 1 -tangents à l’identité à l’origine. On construit deux classes non-triviales de cohomologie réelle de degré deux et un nombre non-dénombrable de classes d’homologie de dimension deux de ce groupe.

Received : 2010-01-06
Accepted : 2011-10-15
DOI : https://doi.org/10.5802/aif.2699
Classification:  58D05,  57S05
Keywords: cohomology of diffeomorphism groups, flat diffeomorphism, Massey product 
@article{AIF_2012__62_1_77_0,
     author = {Ishida, Tomohiko},
     title = {Second cohomology classes of the group of $C^1$-flat diffeomorphisms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     pages = {77-85},
     doi = {10.5802/aif.2699},
     zbl = {1253.58007},
     mrnumber = {2986265},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2012__62_1_77_0}
}

Ishida, Tomohiko. Second cohomology classes of the group of $C^1$-flat diffeomorphisms. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 77-85. doi : 10.5802/aif.2699. https://aif.centre-mersenne.org/item/AIF_2012__62_1_77_0/

[1] Fukui, Kazuhiko Homologies of the group Diff ∞ (R n ,0) and its subgroups, J. Math. Kyoto Univ., Tome 20 (1980) no. 3, pp. 475-487 | MR 591806 | Zbl 0476.57016

[2] Gelʼfand, I. M.; Fuks, D. B. Cohomologies of the Lie algebra of formal vector fields, Izv. Akad. Nauk SSSR Ser. Mat., Tome 34 (1970), pp. 322-337 | MR 266195 | Zbl 0216.20302

[3] Goncharova, L. V. The cohomologies of Lie algebras of formal vector fields on the line, Funct. Anal. and Appl., Tome 7 (1973), p. 91-97, 194–203 | Article | Zbl 0284.17006

[4] Kraines, David Massey higher products, Trans. Amer. Math. Soc., Tome 124 (1966), pp. 431-449 | Article | MR 202136 | Zbl 0146.19201

[5] Millionschikov, D. Algebra of formal vector fields on the line and Buchstaber’s conjecture, Funct. Anal. Appl., Tome 43 (2009), pp. 264-278 | Article | MR 2596653

[6] Takens, Floris Normal forms for certain singularities of vectorfields, Ann. Inst. Fourier (Grenoble), Tome 23 (1973) no. 2, pp. 163-195 (Colloque International sur l’Analyse et la Topologie Différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972)) | Article | Numdam | MR 365620 | Zbl 0266.34046

[7] Weinstein, F. V. Filtering bases: a tool to compute cohomologies of abstract subalgebras of the Witt algebra, Unconventional Lie algebras (Adv. Soviet Math.) Tome 17, Amer. Math. Soc., Providence, RI, 1993, pp. 155-216 | MR 1254731 | Zbl 0801.17022