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
>>
>>