Given an even dimensional manifold, the mapping class group acts on middle dimensional cohomology (or homology) and this action preserves the intersection form. For manifold of dimension $4k+2$, the action symplectic, while it is orthogonal for manifold of dimension $4k$. 

In dimension 2, it is well-known that any integral symplectic transformation on the cohomology of degree 1 can be realized by some diffeomorphism. I would like to know if this is still true in higher dimension. I am interested mostly in the symplectic case (dimension $4k+2$).

More generally, does anyone know a good reference about mapping class groups of manifolds of dimension higher than 2? All the references I found treat exclusively the case of surfaces. 

Thanks in advance.


  Since it doesn't make sense to number every single equation...


I used to number only equations I referred to, but then someone pointed out the following.  When you write a paper and make it public, you are de facto allowing other people to talk about your paper and simply because you in your paper see no reason to refer to some equation does not mean that other people reading your paper will not do so.  Hence as a friendly gesture to your readers, you should number all your equations, and in this way allow them to refer to, say, equation (n) in your paper, instead of having to come up with some complicated reference.  Since that time I have tried to number all my equations.
