Subject: Re: one response From: James Stasheff Date: Wed, 26 Apr 2006 13:51:56 -0400 (EDT) Pardon an old man's faulty memory but wasn't Leray's SS without assuming a nice map? Serre brought it inot the fibration setting Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds As of July 1, 2002, I am Professor Emeritus at UNC and I will be visiting U Penn but for hard copy the relevant address is: 146 Woodland Dr Lansdale PA 19446 (215)822-6707 On Tue, 25 Apr 2006, Don Davis wrote: >> A response to a recently-posted question...........DMD >> __________________________________________________ >> >> Subject: Re: question about S3-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 >> >>