homology and cohomology of a quotient manifold - MathOverflow

homology and cohomology of a quotient manifold

2010-12-09T10:02:37Z

suppose $M$ is a manifold , $G$ is a lie group (may be finite) ,then let $G$ act on $M$ freely , $N=M/G$ is then a manifold ,so my question is what relations may be between the homology and cohomology of $M$ and $N$ ? In the surface case ,if $G$ is a finite group ,then we can get no differences between the cohomolgy and homology of $M$ and $N$ then the euler number will be the same what will mean that the order of the group must be one ,and so there are no actions of finite groups freely acting on a surface and it is a very beautiful claim.

Does there any effective way to compute the cohomology of a quotient space when the action is not free?

So let us consider more about the question raised above , references about this question are also welcomed!!

Answer by Kostya for homology and cohomology of a quotient manifold

2010-12-09T10:33:31Z

I think that you have to investigate in the direction of fibrations. And then, maybe, spectral sequences.

edit: Concernig your comment -- I am interested in that as well.

Right now my investigation got me there -- check the book "Modern Geometry - Methods and Applications: Part 3: Introduction to Homology Theory" by: B.A. Dubrovin, A.T. Fomenko, S.P. Novikov.

Chapter 20.

It is quite difficult for me to read the book (Yeah, you got me, I'm a physicist.) so I cannot give you anything but the direction.

Answer by Daniel Loughran for homology and cohomology of a quotient manifold

2010-12-09T12:43:39Z

Although of course Kostya's answer is a good definitive one, I thought best to mention that there are tools available other than spectral sequences depending on exactly what you want.

For example, if $M$ is an orientable surface, $G$ is finite, the action is not neccessarly free and the quotient is still smooth, then you can use the Riemann-Hurwitz formula.

There are also higher dimensional analogues of this which give you information about the canonical bundle of $M$ if $M$ is a complex manifold.

Answer by Mark Grant for homology and cohomology of a quotient manifold

2010-12-09T14:17:08Z

To elaborate a little on Kostya's answer, for the case where $G$ acts freely you may want to look up the Cartan-Leray spectral sequence (eg Chapter 7 of the book by Brown). Also, transfer homomorphisms are a useful elementary tool (see section 3.G of Hatcher's book).

When $G$ doesn't act freely you're into the realm of orbifolds, about which I have much to learn!