Théorie de jauge et symétries des fibrés
Annales de l'Institut Fourier, Volume 43 (1993) no. 2, p. 509-537

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.

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.

@article{AIF_1993__43_2_509_0,
     author = {Brandt, D. and Hausmann, Jean-Claude},
     title = {Th\'eorie de jauge et sym\'etries des fibr\'es},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {2},
     year = {1993},
     pages = {509-537},
     doi = {10.5802/aif.1344},
     zbl = {0778.57018},
     mrnumber = {94c:57056},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1993__43_2_509_0}
}
Brandt, D.; Hausmann, Jean-Claude. Théorie de jauge et symétries des fibrés. Annales de l'Institut Fourier, Volume 43 (1993) no. 2, pp. 509-537. doi : 10.5802/aif.1344. http://www.numdam.org/item/AIF_1993__43_2_509_0/

[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).