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
>
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> Martintxo SARALEGI-ARANGUREN
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