Welcome to MathOverflow

Question

2

Hello everyone,

I have difficulties to understand the connection between equivariant and basic cohomology. I understand the definition of them but not how they are related (the Weil algebra killed me :)).

For example, consider a smooth manifold $M$ with a smooth action of a Lie group $G$. If we assume all orbits have the same dimension, this induces a foliated structure on M whose leaves are the orbites of the action. So, we have two cohomologies to look at: the basic and the equivariant ones. How are they related?

I have no special assumptions on $G$ and $M$. $G$ has no reason to be compact, and the action is not even proper (for example, the foliation of $S^1\times S^1$ induced by the foliation of $[0,1]\times[0,1]$ by parallel lines of non rational slope is a case I would like to understand). In the special case I am looking, $G$ is commutative but I am also interested in what happens when $G$ is not.

Hope my question is not too trivial. If it is, please give me a good reference on the subject (with a differential geometer point of view if possible).

Thanks in advance.

flag	asked Jun 15 2012 at 9:21 Taladris 249●1●8

	Can you please recall for me what basic cohomology is, and maybe say which coefficients you consider. – Konrad Waldorf Jun 15 2012 at 15:23
	Sorry for late answer (week-end...) and thank you for your interest in my question. Let $(M,{\mathcal F})$ be a foliated manifold. A basic form on $M$ is a form $\omega$ satisfying $i_X\omega=0$ and $L_X\omega=0$ for every vector field tangent to the foliation at every point. The exterior derivative $d$ sends a basic form to a basic form, so the space $\Omega_{bas}(M)$ of basic forms is a differentiel complex. The basic cohomology is its cohomology ring. So, its coefficient ring is $\mathbb R$ and I am considering equivariant cohomology with the same ring. – Taladris Jun 18 2012 at 1:25

Equariant and basic cohomology

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.