A Conceptual Question on de Rham cohomology.

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In summary, the group HrD(M) = 0 when r > n, where n is the dimension of manifold M, because it follows from properties of the wedge product and the definition of n-cocycles and n-coboundaries. This is also related to Poincare Duality and the fact that an nbyn determinant is always zero if the matrix has rank < n.
  • #1
T_Mart
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Hi everybody,

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

Why H rD(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.
 
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  • #2
Well, how is the group defined?
 
  • #3
Hurkyl said:
Well, how is the group defined?

The group is defined as
HrD (M) = Ker(dr)/Im(dr-1)
 
  • #4
T_Mart said:
The group is defined as
HrD (M) = Ker(dr)/Im(dr-1)

And what groups is dr a homomorphism from and to?
 
  • #5
it follows from properties of the wedge product, as is being suggested.
 
  • #6
How do you define n-cocycles and n-coboundaries?
 
  • #7
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.
 
  • #8
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.
 

1. What is de Rham cohomology?

De Rham cohomology is a mathematical concept that provides a way to measure the topological properties of a smooth manifold. It is a tool used in differential geometry and algebraic topology to study the behavior of differential forms on a manifold.

2. How is de Rham cohomology different from homology?

De Rham cohomology and homology are two different approaches to studying topological spaces. While homology measures the number of holes in a space, de Rham cohomology measures the ability of a space to support closed differential forms.

3. What are the applications of de Rham cohomology?

De Rham cohomology has many applications in physics, particularly in the study of differential equations and symplectic geometry. It is also used in algebraic geometry, algebraic topology, and geometric analysis.

4. What is the de Rham cohomology theorem?

The de Rham cohomology theorem states that the de Rham cohomology groups are isomorphic to the singular cohomology groups of a manifold. This means that the two approaches of measuring topological properties, de Rham cohomology and homology, yield the same results.

5. How is de Rham cohomology calculated?

De Rham cohomology is calculated using the de Rham complex, a sequence of vector spaces and linear maps, which is constructed from the space of smooth differential forms on a manifold. The cohomology groups are then defined as the quotients of the kernels of the maps in the complex.

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