A problem on de rham cohomology

  • Thread starter seydunas
  • Start date
In summary, de Rham cohomology is a mathematical tool used to study the topological properties of smooth manifolds, assigning algebraic invariants to topological spaces. It allows us to distinguish between different spaces and understand their global structure. It is calculated using differential forms and measures the "holes" or "twists" in a manifold. It is related to homology through the Poincaré lemma and has applications in various fields, including differential geometry, algebraic geometry, and theoretical physics.
  • #1
seydunas
39
0
Hi,

Let U be an open subset of R^n and n=>2 and x /in U. I want to show that (n-1)th de rham cohomology of U\ {x} is non zero. I suppose i can solve this question by using excision theorem from singular homology. But i have a hint for this problem: Consider the restrictions S--->U\ {x}----> R^n \{x} where S is a small sphere centered at x. I have to use hint. Can you help me?
 
Physics news on Phys.org
  • #2
Presumably you know an example of a closed (n-1)-form on the (n-1)-sphere that isn't exact?
 

1. What is de Rham cohomology?

De Rham cohomology is a mathematical tool used to study the topological properties of smooth manifolds, which are geometric objects that can be smoothly deformed and stretched. It is a type of cohomology, which is a branch of algebraic topology that assigns algebraic invariants to topological spaces.

2. Why is de Rham cohomology important?

De Rham cohomology allows us to distinguish between different topological spaces by assigning them different cohomology groups. This is useful because these groups are often easier to compute and understand than the topological spaces themselves. It also helps us understand the global structure of a manifold by studying its local properties.

3. How is de Rham cohomology calculated?

De Rham cohomology is calculated using differential forms, which are mathematical objects that represent multivariate functions on a manifold. These forms can be manipulated using differential operators to obtain the de Rham cohomology groups. In simpler terms, we can think of de Rham cohomology as a way to measure the "holes" or "twists" in a manifold.

4. What is the relationship between de Rham cohomology and homology?

De Rham cohomology and homology are both tools used to study the topological properties of spaces. However, while homology measures the number of "n-dimensional holes" in a space, de Rham cohomology measures the "twists" or "obstructions" to filling these holes. These two concepts are related through the Poincaré lemma, which states that on a contractible space, the de Rham cohomology groups are isomorphic to the homology groups.

5. How is de Rham cohomology applied in other fields of science?

De Rham cohomology has applications in many areas of mathematics and science, including differential geometry, algebraic geometry, and theoretical physics. It is particularly useful in the study of differential equations, as it provides a way to solve problems in a more general setting. In physics, it is used to study topological properties of physical systems, such as in the theory of knots and quantum field theory.

Similar threads

A Computing de Rham Cohomology of Connected Sums of Objects
    • Differential Geometry
Replies
13
Views
2K
A A claim about smooth maps between smooth manifolds
    • Differential Geometry
Replies
20
Views
2K
I Understanding De Rham's Period and Stokes Theorem
    • Differential Geometry
Replies
5
Views
1K
Confusion on de Rham cohomology of manifolds
    • Differential Geometry
Replies
9
Views
4K
Why only Closed Forms Matter in DeRham Cohomology?
    • Differential Geometry
Replies
7
Views
2K
References for Self Study in de Rham Cohomology
    • Topology and Analysis
Replies
2
Views
2K
I What is the construction of charts for the tangent bundle on the unit circle?
    • Differential Geometry
Replies
8
Views
2K
DeRham Cohomology of n-torus
    • Differential Geometry
Replies
4
Views
5K
B Reconciling basis vector operators with partial derivative operators
    • Differential Geometry
Replies
4
Views
1K
Understanding Motivation of De Rham Cohomology for Manifolds
    • Differential Geometry
Replies
10
Views
8K

Hot Threads

Recent Insights

Back
Top