Subject: Re: question about S^3-actions (Modificado por Mtxo. Saralegi-Aranguren) From: Alexandru Oancea Date: Tue, 25 Apr 2006 13:11:13 +0200 (CEST) Dear Don and Martintxo, I don't think one can build a spectral sequence (or, for that matter, the associated Gysin sequence) unless the map from the manifold to the orbit space is a fibration in the sense of Serre. This is the case if the action is free, for example. I don't know of another general result in this direction. But what may be useful in the situation at hand is to consider G-equivariant cohomology, with G=S3. If one calls the manifold M, there is a spectral sequence starting with H^*(BG) \otimes H^*(M) and converging to the equivariant cohomology. This equivariant cohomology is defined as the cohomology of M_G := (M \times EG) / G, where G acts by the diagonal action. There is a fibration M\to M_G \to BG which gives rise to the above spectral sequence. I don't know what is the precise problem that you have in mind, but as soon as there is a G-action on a space, equivariant cohomology can be efficient. A reference on these matters is for example the book by Tammo tom Dieck, Transformation Groups, de Gruyter, 1987. There is also a paper by Bott called "An Introduction to Equivariant Cohomology" in DeWitt-Morette and Zuber (eds.), QFT: Perspective and Prospective, Kluwer, 1999. I hope this helps, Alex On Fri, 21 Apr 2006, Don Davis wrote: > Dear Professor: > > I am looking for a Leray spectral sequence for a smooth action > of the sphere S3 on a manifold. > Of course, when the action is free or semi-free > a such spectral sequence exists (indeed, a long exact sequence) > and the second term is computed in terms of the cohomology of the orbit space and the fixed points. > But, in the general case, could you give me a reference? > Thanks in advance > > ************************************************* > Martintxo SARALEGI-ARANGUREN > Laboratoire Mathématiques Lens (LML) EA 2462 > Fédération CNRS Nord-Pas-de-Calais FR 2956 > Université d’Artois > rue Jean Souvraz S.P. 18 > 62 307 Lens Cedex > France