Register to reply

A Conceptual Question on de Rham cohomology.

by T_Mart
Tags: de rham
Share this thread:
T_Mart
#1
Jan1-12, 07:09 PM
P: 5
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.
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Hurkyl
#2
Jan1-12, 07:12 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,091
Well, how is the group defined?
T_Mart
#3
Jan1-12, 07:22 PM
P: 5
Quote Quote by Hurkyl View Post
Well, how is the group defined?
The group is defined as
HrD (M) = Ker(dr)/Im(dr-1)

Hurkyl
#4
Jan1-12, 07:51 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,091
A Conceptual Question on de Rham cohomology.

Quote Quote by T_Mart View Post
The group is defined as
HrD (M) = Ker(dr)/Im(dr-1)
And what groups is dr a homomorphism from and to?
mathwonk
#5
Jan2-12, 10:53 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,488
it follows from properties of the wedge product, as is being suggested.
Bacle2
#6
Jan12-12, 03:05 PM
Sci Advisor
P: 1,172
How do you define n-cocycles and n-coboundaries?
mathwonk
#7
Jan12-12, 06:30 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,488
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
#8
Jan12-12, 10:36 PM
Sci Advisor
P: 1,172
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.


Register to reply

Related Discussions
Confusion on de Rham cohomology of manifolds Differential Geometry 9
De Rham's first theorem Differential Geometry 2
Group cohomology Linear & Abstract Algebra 0
De Rham theorem... Differential Geometry 1
De Rham cohomology Differential Geometry 10