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Euler and Maslov cocycles. (Cocycles d’Euler et de Maslov.) (French) Zbl 0894.55006
This elegant paper takes its inspiration partly from one by M. F. Atiyah [ibid. 278, 335-380 (1987; Zbl 0648.58035)], in which a diverse set of numerical functions on \(\text{SL}_2(\mathbb{Z})\) were unified. There is an analogy between the Dedekind \(\eta\)-function of that paper, the defect of the Hirzebruch signature formula, and a function due to Rademacher, and the authors exploit this to study these invariants and others by means of bounded cohomology 2-cocycles. The strategy is to show that the coboundary of the functions of interest is a bounded 2-cocycle and that the bounded cohomology class obtained is unique for a uniformly perfect group like \(\text{SL}_2\) or \(\text{Sp}_{2n}\).

55U99 Applied homological algebra and category theory in algebraic topology
11F20 Dedekind eta function, Dedekind sums
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[1] Abraham, R., Marsden, J.: Foundations of mechanics, 2nd ed. (Advanced Book Program) Reading, MA: Benjamin 1978 · Zbl 0393.70001
[2] Arnold, V.: Sturm theorem and symplectic geometry. Funct. Anal. Appl.19 (no 1), 251–276 (1985) · Zbl 0606.58017 · doi:10.1007/BF01077289
[3] Asai, T.: The reciprocity of Dedekind sums and the factor set for the universal covering group of SL(2,\(\mathbb{R}\)). Nagoya Math. J.37, 67–80 (1970) · Zbl 0192.39601
[4] Atiyah, M.: The logarithm of the Dedekind \(\eta\)-function. Math. Ann.278, 335–380 (1987) · Zbl 0648.58035 · doi:10.1007/BF01458075
[5] Atiyah, M.: On framings of 3-manifolds. Topology29, 1–7 (1990) · Zbl 0716.57011 · doi:10.1016/0040-9383(90)90021-B
[6] Banyaga, A.: On the group of diffeomorphisms preserving an exact symplectic form. In: Differential topology. Varenna 1976, pp. 5–9. Naples: Liguori 1979 · Zbl 0463.58011
[7] Barge, J., Ghys, E.: Cocycles bornés et actions de groupes sur les arbres réels. Workshop on group theory from a geometrical view point. I.C.T.P. Trieste, Ghys, E., Haefliger, A., Veryovsky, A. (eds.), Singapore. World Scientific 1990
[8] Bavard, C.: Longueur stable commutateurs. Enseign. Math.37, 109–150 (1991) · Zbl 0810.20026
[9] Besson, G.: Séminaire de cohomologie bornée. E.N.S. de Lyon (Février 1988)
[10] Brooks, R.: Some remarks on bounded cohomology. In: Kra, I., Maskit, B. (eds.) Riemann surfaces and related topics. (Ann. Math. Stud., vol. 91, pp. 53–65) Princeton: Princeton University Press 1981
[11] Calabi, E.: On the group of automorphisms of a symplectic manifold. In: Gunning, R.C. (ed.) Problems in Analysis. Symposium in honour of S. Bochner, pp. 1–26. Princeton: Princeton University Press 1970 · Zbl 0209.25801
[12] Dazord, P.: Invariants homotopiques attachés aux fibrés symplectiques. Ann. Inst. Fourier29, 25–78 (1979) · Zbl 0378.58011
[13] Dedekind, R.: Erläuterungen zu zwei Fragmenten von Riemann. (Riemann’s ges. Math. Werke, vol. 2, pp. 466–478) New York: Dover 1982
[14] Dieudonné, J.: Sur les groupes classiques. Paris: Hermann 1967 · Zbl 0037.01304
[15] Domic, A., Toledo, D.: The Gromov norm of the Kähler class of symmetric domains. Math. Ann.276, 425–432 (1987) · Zbl 0595.53061 · doi:10.1007/BF01450839
[16] Dupont, J.: Bounds for the characteristic numbers of flat bundles. (Lect. Notes Math., vol. 763, pp. 109–119) Berlin Heidelberg New York: Springer 1979 · Zbl 0511.57018
[17] Dupont, J., Guichardet, A.: A propos de l’article “Sur la cohomologie réelle de groupes de Lie semi-simples réels”. Ann. Sci. Éc. Norm. Supér.11, 293–296 (1978) · Zbl 0398.22016
[18] Eisenbud, D., Hirsch, U., Neumann, W.: Transverse foliations of Seifert bundles and self homeomorphisms to the circle. Comment. Math. Hevl.56, 638–660 (1981) · Zbl 0516.57015 · doi:10.1007/BF02566232
[19] Ekeland, I.: Convexity methods in Hamiltonian mechanics. Berlin Heidelberg New York: Springer 1990 · Zbl 0707.70003
[20] Ghys, E.: Groupes d’homéomorphismes du cercle et cohomologie bornée. Contemp. Math.58, 81–105 (1987)
[21] Gromov, M.: Volume and bounded cohomology. Publ. Math., Inst. Hautes Étud. Sci.56, 5–100 (1982) · Zbl 0516.53046
[22] Guichardet, A., Wigner, D.: Sur la cohomologie réelle des groupes de Lie simples réels. Ann. Sci. Écol. Norm. Supér.11, 277–292 (1978) · Zbl 0398.22015
[23] Herman, M.: Inégalités a priori pour des tores lagrangiens invariants par des difféomorphismes symplectiques. Publ. Math., Inst. Hautes Étud. Sci.70, 47–101 (1989) · Zbl 0717.58020 · doi:10.1007/BF02698874
[24] Hirzebruch, F.: The signature theorem: reminiscences and recreation. In: Prospects in Mathematics. (Ann. Math. Stud., vol. 70, pp. 3–31) Princeton: Princeton University Press 1971 · Zbl 0252.58009
[25] Hirzebruch, F.: Hilbert modular surfaces. Enseign. Math.19, 183–281 (1973) · Zbl 0285.14007
[26] Leray, J.: Analyse lagrangienne et mécanique quantique. Publ. Math. I.R.M.A., Strasbourg25 (1978)
[27] Lion, G., Vergne, M.: The Weil representation, Maslov index and theta series. (Prog. Math., vol. 6) Boston Basel Stuttgart: Birkhäuser 1980 · Zbl 0444.22005
[28] Mackey, G.W.: Les ensembles boréliens et les extensions de groupes. J. Math. Pures Appl.36, 171–178 (1957) · Zbl 0080.02303
[29] Matsumoto, S., Morita, S.: Bounded cohomology of certain groups of homeomorphisms. Proc. Am. Math. Soc.94, 539–544 (1985) · Zbl 0536.57023 · doi:10.1090/S0002-9939-1985-0787909-6
[30] Meyer, W.: Die Signatur von lokalen Koeffizienten-Systemen und Faserbündeln. Dissertation. Bonn. Math. Schr.53 (1972) · Zbl 0243.58004
[31] Meyer, W.: Die Signatur von Flächenbündeln. Math. Ann.201, 239–264 (1973) · Zbl 0245.55017 · doi:10.1007/BF01427946
[32] Oseledec, V.I.: Multiplicative ergodic theorem, Ljapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc.19, 197–231 (1968)
[33] Rademacher, H.: Zur Theorie der Dedekindschen Summen. Math. Z.63, 445–463 (1956) · Zbl 0071.04201 · doi:10.1007/BF01187951
[34] Rademacher, H., Grosswald, E.: Dedekind sums. (Carus Math. Monogr., vol. 16) Washington: Math. Assoc. Am. 1972 · Zbl 0251.10020
[35] Ruelle, D.: Rotation numbers for diffeomorphisms and flows. Ann. Inst. Henri Poincaré, Phys. Theor.42, 109–115 (1985) · Zbl 0556.58026
[36] Serre, J.P.: Arbres, amalgames, SL2. (Astérisque, vol. 46) Paris: Soc. Math. Fr. 1977
[37] Siegel, C.: Symplectic geometry. New York London: Academic Press 1964 · Zbl 0138.31403
[38] Souriau, J.M.: Construction explicite de l’indice de Maslov, applications. (Lect. Notes Phys., vol. 50, pp. 117–148) Berlin Heidelberg New York: Springer 1976 · Zbl 0378.58010
[39] Tabachnikov, S.L.: Two remarks on asymptotic Hopf invariants. Funct. Anal. Appl.24, no 1 (1990) · Zbl 0712.55010
[40] Turaev, V.: A cocycle of the symplectic first Chern class and the Maslov indices. Funkts. Anal. Prilozh.18 (1984) · Zbl 0556.55012
[41] Walker, K.: Preprint sur l’invariant de Casson
[42] Yakubovich, V.A.: Arguments on the group of symplectic matrices. Math. Sb.55 (97) (no 3), 255–280 (1961)
[43] Yakubovich, V.A., Starzhinskii, V.M.: Linear differential equations with periodic coefficients, I. New York: Wiley and Jerusalem: Keter Publishing House 1975
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