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!!



homology and cohomology of a quotient manifold
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](http://books.google.co.uk/books?id=PMqb2DppvCsC&printsec=frontcover&dq=cohomology+of+groups&source=bl&ots=Jtg5xSGYAK&sig=Oijriey_a0jEe_gw7uMp0ByoG_M&hl=en&ei=8eEATbeWHMuHhQesmpntBw&sa=X&oi=book_result&ct=result&resnum=5&sqi=2&ved=0CEAQ6AEwBA#v=onepage&q&f=false)). Also, transfer homomorphisms are a useful elementary tool (see section 3.G of Hatcher's [book](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)).


When $G$ doesn't act freely you're into the realm of [orbifolds](http://en.wikipedia.org/wiki/Orbifold), about which I have much to learn!


