A Conceptual Question on de Rham cohomology.

[T_Mart]

Hi everybody,

Currently, I am studying cohomology on my own. I have a question:

Why H ^r_D(M) = 0, when r > n

n is the dimension of manifold M
My book says it is obvious, but to me it is not obvious.

I wish someone could explain this question to me.

 

[Hurkyl]

Well, how is the group defined?

 

[T_Mart]

Well, how is the group defined?

The group is defined as
H^r_D (M) = Ker(d_r)/Im(d_r-1)

 

[Hurkyl]

The group is defined as
H^r_D (M) = Ker(d_r)/Im(d_r-1)

And what groups is d_r a homomorphism from and to?

 

[mathwonk]

it follows from properties of the wedge product, as is being suggested.

 

[Bacle2]

How do you define n-cocycles and n-coboundaries?

 

[mathwonk]

there aren't even any ≠0 cochains in dimensions above the dimension of the manifold.

the reason is essentially that an nbyn determinant is always zero if the matrix has rank < n.

 

[Bacle2]

Yes, that was the point I was trying to make. Look up the definition of n-cocycles and n-coboundaries to see what the cohomology groups are . Or, if you have the right conditions for Poincare Duality, see why you cannot have (n+k)-cycles; k>0, in an n-manifold.