Subject: Re: two responses
From: Alejandro Adem
Date: Thu, 27 Apr 2006 10:50:37 -0500 (CDT)
A reply to the replies.
Alejandro.
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In reply to Jim Stasheff: yes indeed, there is a Leray spectral sequence
associated to maps which are not fibrations. In the case of a group G
acting on M, a particularly interesting one is associated to the
projection
EG X_G M ---> M/G
which is a homotopy equivalence when the action is free.
_________________________________________________
In reply to Wolfgang Ziller: given an orbifold X you can study
--its ordinary cohomology
--the cohomology of its classifying space
--the orbifold cohomology (as defined by Chen-Ruan etc.)
In the case of a quotient orbifold M/G, G acting almost freely,
the first one is H^*(M/G), the second one is H^*(EGX_GM) (note they
agree over the rationals) but the third one is very different, as
it involves a product with the Euler class of a certain obstruction
bundle, and is not a cohomology theory in the usual sense. Additively
it is the cohomology of the inertia orbifold, involving the twisted
sectors -- for a finite group action they would be related to fixed
point sets. The product builds in a measure of transversality between
them, but also allows components to interact.
There are several references available---I can suggest Contemporary
Mathematics Volume 310 as a place to start. I understand that a book
on the subject may appear soon.
On Thu, 27 Apr 2006, Don Davis wrote:
> Two postings related to the recent discussion...........DMD
> __________________________________________________
>
> Subject: Re: another response
> From: wziller@sas.upenn.edu
> Date: Wed, 26 Apr 2006 12:44:55 -0400
>
> A foollow up question:
>
> In the same spirit,
> what does one know if the action is only almost free,
> i.e. the quotient is an orbifold.
> I know there is something like orbifold cohomology (which I assume is
different
> from equivariant cohomology).
> Is there an orbifold Gysin sequence (if G=S3) which relates the ordinary
> cohomology of the manifold with the obifold cohomology of the quotient
> and an orbifold euler class?
> and more generally a spectral sequence for orbifold cohomology?
> I would like a reference where lots of examples are computed.
> Can one describe invariants like the euler class in terms of the
ordinary
> cohomology of the manifold and the singular set (assuming it is smooth)
and the
> orbifold group?
>
> Wolfgang Ziller
> _________________________________________________________
>
> 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
>
>