Is De Rham's first theorem applicable to non-compact manifolds?

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In summary, the conversation discussed the topological implications of a statement about a closed p-form. It was determined that the statement is true for all 2nd countable hausdorff manifolds and the proof can be found in various sources such as the book of Singer and Thorpe. The statement does not necessarily imply anything about the topology of the manifold, but if it were to be changed to state that all closed p-forms are exact, then it would imply that all p-cycles are p-boundaries. Additional references were also mentioned.
  • #1
ivl
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Hi all!
I have a possibly trivial (possibly non-trivial? :rofl:) question. Here it is:

Assumption-Assume I have a closed p-form, whose integral over any p-cycle is always zero.
Statement-The closed p-form is also exact, by what is sometimes called de Rham's first theorem

My question is: what are the topological implications of my statement? (e.g. am I implying that all p-cycles are p-boundaries??)

Further question: de Rham's theorem is often proved for compact manifolds. Is my statement true even for manifolds which are not compact? (assume my manifold is paracompact, but not compact).

Thanks a lot!
 
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  • #2
I am not an expert but my reading gives me the following impressions:

The first statement seems to be true for all 2nd countable hausdorff manifolds, i.e. all smooth manifolds normally considered by the average person.

the proof is given in the book of singer and thorpe, under the assumption there is a smooth triangulation, and with simplicial cohomology. any manifold that has a closed embedding in euclidean space, i.e. any manifold as above, has such a triangulation according to whitney.

it also has partitions of unity which implies it is paracompact.

bott tu give the proof assuming a "good cover" of the manifold, which also follows from triangulability.

good references include bott -tu, singer and thorpe, morris hirsch, a. weil (comm. math. helvetici about 1952), spivak differential geometry vol. 1, chap. 11, problem 14, and the book of frank warner for an especially complete version although rather abstract in terms of shaves.the answer to:

"My question is: what are the topological implications of my statement? (e.g. am I implying that all p-cycles are p-boundaries??)"

is no. since the earlier statement is true for all manifolds it implies nothing at all about the topology of the manifold. however if you were to change your earlier statement to say that all closed p forms are exact, it would then imply your second statement. i.e. by de rham, all closed p forms are exact if and only if all p cycles are boundaries.

another reference of course is the book of georges de rham, but i have not read it.
 
  • #3
Excellent answer, thanks!
 

What is De Rham's first theorem?

De Rham's first theorem is a fundamental theorem in differential topology that relates the cohomology groups of a smooth manifold to its de Rham cohomology, which is based on differential forms.

What does De Rham's first theorem state?

De Rham's first theorem states that the de Rham cohomology group of a smooth manifold is isomorphic to its singular cohomology group, which is based on continuous maps.

What is the significance of De Rham's first theorem?

De Rham's first theorem is significant because it provides a powerful tool for studying the topology of smooth manifolds by relating their geometric properties to algebraic properties of differential forms.

What are the assumptions for De Rham's first theorem to hold?

De Rham's first theorem holds for any smooth manifold that is orientable and has a finite number of connected components. It also requires the manifold to be compact or have a finite number of holes.

How is De Rham's first theorem applied in practice?

De Rham's first theorem has many applications in geometry, topology, and physics. It is used to classify and distinguish different types of manifolds, and to study their properties and symmetries. It also has applications in the theory of differential equations and mathematical physics.

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