DeRham Cohomology of n-torus

[Sina]

Hello,

I want to ask for a verification of something I did. Let's say I want to compute the m-th cohomology group of the n-torus with coordinates θ_i. Suppose that I have shown each [dθ_{i_1} $\wedge$dθ_{i_2} ... $\wedge$ dθ_{i_m}] is an independent element of this deRham cohomology (it can be shown by some integration over m-toruses that reside in the n-torus that these are closed but not exact differential forms whose differences are not exact). In particular its dimension is then atleast C(n,m)

Now to finish I have to show that this is all. I want to a dimension argument here. In the lattice quotient model of the Torus it is easy to show that each n-torus is a CW-complex with 1 0-cell, n 1-cells, C(n,m) m-cells. Then one can compute the homology groups and dimension of its mth homology group is C(n,m) then dimension of the mth cohomology group is also m (I know the computation of homology groups for T² is easy I assume Tⁿ is the similiar). In fact you don't need to compute the homology groups. Since each chain C_m has dimension C(n,m) then the homology group can have at most that dimension which is more than enough for me.

Now we now that for simplical complices H_m(X) = H_m(X_k) for k>m. In particular here we have that X_n is the Torus itself so that cellular homology and normal homology groups can be identifies (this is the first part I am not sure). Then normal homology group is isomorphic to simplical homology groups and there fore each have the same dimension. Now simplical cohomology groups of simplical complexes are isomorphic to deRham cohomology. Thus basically m-th deRham cohomology group has the same dimension as the mth cohomology of the n-torus which is C(n,m) therefore completing the proof.

Right? :p

 

[Sina]

edit: finally manged to tex

 

[Sina]

I meant " then dimension of the mth cohomology group is also C(n,m)" not m

 

[lavinia]

i have read this a couple of times am am not sure what you are asking.

For instance why are you not using the Kunneth formula with the reals as the coefficient group?

 

[blue_raver22]

Hello,

Thank you for sharing your work on the deRham cohomology of the n-torus. Your approach seems reasonable and your use of the lattice quotient model of the Torus is a clever way to compute the homology and cohomology groups. However, I would suggest being more explicit in your explanation of how you are using the lattice quotient model and how it relates to the deRham cohomology. Additionally, I would also suggest including some references or sources that support your claim that the normal homology group is isomorphic to the simplicial homology group in this case. Overall, your approach seems sound and your use of dimension arguments is appropriate. Keep up the good work!