Théorie de jauge et symétries des fibrés

Brandt, D.; Hausmann, Jean-Claude

Annales de l'Institut Fourier, Tome 43 (1993), p. 509-537 / Harvested from Numdam

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Résumé
Abstract

Soit $ξ$ un $G$ -fibré principal différentiable sur une variété $M$ ( $G$ un groupe de Lie compact). Étant donné une action d’un groupe de Lie compact $Γ$ sur $M$ , on se pose la question de savoir si elle provient d’une action sur le fibré $ξ$ . L’originalité de ce travail est de relier ce problème à l’existence de points fixes pour les actions de $Γ$ que l’on induit naturellement sur divers espaces de modules de $G$ -connexions sur $ξ$ .

Let $ξ$ be a smooth $G$ -principal bundle over a manifold $M$ ( $G$ being a compact Lie group). Given an action of a compact Lie group $Γ$ on $M$ , one asks the question whether it comes from an action on the bundle $ξ$ . In this paper, this question is shown to be essentially equivalent to the existence of fixed points for the naturally induced actions of $Γ$ on various moduli spaces of $G$ -connections on $M$ .

Publié le : 1993-01-01

MR 94c:57056 | Zbl 0778.57018

DOI : https://doi.org/10.5802/aif.1344

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     author = {Brandt, D. and Hausmann, Jean-Claude},
     title = {Th\'eorie de jauge et sym\'etries des fibr\'es},
     journal = {Annales de l'Institut Fourier},
     volume = {43},
     year = {1993},
     pages = {509-537},
     doi = {10.5802/aif.1344},
     mrnumber = {94c:57056},
     zbl = {0778.57018},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1993__43_2_509_0}
}

Brandt, D.; Hausmann, Jean-Claude. Théorie de jauge et symétries des fibrés. Annales de l'Institut Fourier, Tome 43 (1993) pp. 509-537. doi : 10.5802/aif.1344. http://gdmltest.u-ga.fr/item/AIF_1993__43_2_509_0/

Bibliographie

[Bn] K. Brown, Cohomology of groups, Springer-Verlag, New York, 1982. | MR 83k:20002 | Zbl 0584.20036

[Do] S. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, J. of Differential Geometry, 24 (1986), 275-341. | Zbl 0635.57007

[FS] R. Fintushel & R. Stern, Definite 4-manifolds, J. of Differential Geometry, 28 (1988), 133-141. | MR 89i:57006 | Zbl 0662.57009

[HY] A. Hattori & T. Yoshida, Lifting compact actions in fiber bundles, Japan J. of Math., 2 (1976), 13-25. | MR 57 #1523 | Zbl 0346.57014

[La] Bl. Lawson, The theory of gauge fields in four dimensions, Regional Conf. series in Math., 58 (AMS 1985). | MR 87d:58044 | Zbl 0597.53001

[LMS] R. Lashof & J. May & G. Segal, Equivariant bundles with abelian structural group, Contemporary Math., Vol 19 (AMS 1983), 167-176. | MR 85b:55023 | Zbl 0526.55020

[KN] S. Kobayashi & K. Nomizu, Foundations of differential topology, Vol I et II, Interscience, New York, 1969. | Zbl 0175.48504

[PS] R. Palais & T. Stuart, The cohomology of differentiable transformation groups, Amer. J. of Math., 83 (1961), 623-644. | MR 25 #4030 | Zbl 0104.17703

[St] T. Stuart, Lifting group actions in fibre bundle, Annals of Math., 74 (1961), 192-198. | MR 23 #A3798 | Zbl 0116.40502

[VE] Van Est, On the algebraic cohomology concepts in Lie groups, Indigat. Math., 18 (1955), I, 225-233 ; II, 286-294. | MR 17,61b | Zbl 0067.26202

[Wa] Sh. Wang, Moduli spaces over manifolds with involutions, to appear (Preprint, Michigan State Univ. at East Lansing).