Subject: Re: two responses From: Alejandro Adem Date: Thu, 27 Apr 2006 10:50:37 -0500 (CDT) A reply to the replies. Alejandro. ___________________________________________ 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 > >