Talk:De Rham cohomology

message: Ok, what we have here in technical terms is called a mess.  The Hodge decomposition ought to be exported to Hodge theory, although I'm not quite sure how to do that -- especially since de Rham is already discussed in detail there.  Furthermore, Hodge's theorem on elliptic operators is not actually necessary to prove the de Rham theorem (despite the fact that it is a very elegant application of this theorem).  There are many simpler techniques which do not rely on Sobolev spaces and other kludges (such as the Eilenberg-Steenrod axioms).  So the options are, really clarify how the Hodge decomposition applies to this theorem (at the end of the article), or find a way to incorporate this material into some other article (preferably without making anyone angry).  Silly rabbit 23:24, 18 November 2005 (UTC)
thread_title:  Early thread 

message: I think the rules in French are to capitalize De Rham if it is used as a stand-alone name, and write "Georges de Rham" or "The professor de Rham said that..." if it follows a name or title designating this person. In brief, I think most of the occurences of "de Rham" in this article should be replaces by "De Rham" (so that De Rham's theorem has been proved by Georges de Rham). I am unsure whether the same capitalization conventions apply in English, so I didn't do it myself.  — Preceding unsigned comment added by 2001:700:300:1470:5EF9:DDFF:FE73:5D0F (talk) 10:26, 26 April 2013 (UTC)
thread_title:  Typography: de Rham or De Rham 

message: since the statement of de Rhan's theorem requires a compact manifold, perhaps an example of a non-compact manifold where the theorem fails would enrich this entry.
: Actually it does not require that (or orientability for that matter). The proof is simple: The Poincaré lemma shows that the de Rham sheaf complex is a resolution of the constant sheaf R, and due to the existence of smooth partitions of unity the de Rham sheaves \Omega^k are fine, hence soft and thus acyclic. Therefore they calculate the cohomology of the constant sheaf R. (The link to the singular cohomology definition of the same is provided by the fact that singular sheaf complex is also a soft (hence acyclic) resolution of the constant sheaf R and teherefore also computes the cohomology with constant coefficients R. As the article already makes use of sheaf cohomology, I'll modify it according to these remarks. I'll also incorporate sheaf cohomology methods into the general cohomology articles at a later stage to provide more structure to the comparison of different theories. Stca74 06:21, 15 May 2007 (UTC)
thread_title:  de Rham's theorem 

message: It was in the definition section of the article and seemed out of place.  Perhaps it could be added back with a fuller explanation whenever this article is rewritten. Akriasas (talk) 21:57, 25 December 2008 (UTC)
thread_title:  removed reference to [[Lie coalgebra]] 

message: In Euclidean space, does the definition of Laplacian given here differ from the ordinary notion by a sign? TotientDragooned (talk) 19:01, 4 March 2010 (UTC)
thread_title:  Laplacian 

message: In the section on Harmonic forms, this sentence appears:
"If M is a compact Riemannian manifold, then each equivalence class in  contains exactly one harmonic form. That is, every member ω of a given equivalence class of closed forms can be written as
ω = dα + γ
where α is some form, and γ is harmonic: Δγ=0."
But the "that is" clause is *not* a restatement of the previous one -- since it says nothing about the harmonic form's uniqueness.  This sentence needs to be rewritten, preferably by someone who is an expert in the subject (which excludes me).
ALSO:  The section about de Rham cohomology of the 2-torus -- correct me if I'm mistaken -- seems to assume that there are only two cohomology classes of the torus:
"For example, on a 2-torus, one may envision a constant 1-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations." 
But there is no reason to say there are "two cohomologically distinct combings" when in fact there are infinitely many integer cohomology classes, each containing a unique harmonic 1-form.  What is wanted here is that a basis for integer, or de Rham, cohomology is provided by (any) two harmonic 1-forms that are linearly independent over R.   So this, too, needs rewriting.
Likewise, the follow-up sentence on the n-torus uses equally inexact wording:
"More generally, on an n-dimensional torus T^n, one can consider the various combings of k-forms on the torus. There are n choose k such combings that can be used to form the basis vectors for (H^k)_dR(T^n); the k-th Betti number for the de Rham cohomology group for the n-torus is thus n choose k."
There are certainly not merely n choose k such combings that "can be used" for form the basis vectors for (H^k)_dR(T^n).  Rather, that is the size of any basis. And so that, too, needs rewriting.Daqu (talk) 20:50, 22 October 2010 (UTC)
thread_title:  Some phrasing that needs fixing 

message: According to T. Tao:
"The integration on forms concept is of fundamental importance in diﬀerential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology,  which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds."
From here: http://www.math.ucla.edu/~tao/preprints/forms.pdf
This may be a better introduction -- at least laypeople that are just interested in mathematics can get a gist of what is happening, and I do not believe that this will sacrifice the overall rigour/details of the article.
128.54.66.73 (talk) 20:46, 5 June 2013 (UTC)
replies: text: I have added this quote in a "Quote frame" to the introduction. I agree that it provides a very accessibly introduction for a passing layman. Beyond that, the quote itself is an anecdotally interesting tidbit which I feel many students of mathematics would appreciate. I left the original introduction in place as I feel this quote alone would not suffice as the full introduction for the article. PhysicsSean (talk) 14:35, 24 August 2017 (UTC)
thread_title:  Simpler introduction? 

message: I believe the section "Harmonic forms" should follow the section "Hodge decomposition". One could then easily see that if you believe in the Hodge decomposition then you get the Hodge theorem easily for elements whose classes belong in the k-th dimensional cohomology.  — Preceding unsigned comment added by 12.31.71.58 (talk) 01:16, 7 December 2013 (UTC)
thread_title:  Section Ordering 

message: This page should discuss compactly supported de-Rham cohomology and it's relation with Borel-Moore homology using integration.  — Preceding unsigned comment added by 71.212.185.82 (talk) 22:59, 17 August 2017 (UTC)
replies: text: I 100% agree, the relation of Homology and Cohomology is a topic that is unfortunately lacking from Wiki's discussions of algebraic topology. If someone has the knowledge to write this up then that would be great.PhysicsSean (talk) 22:46, 24 August 2017 (UTC)
thread_title:  Compactly Supported de-Rham Cohomology 

message: Given a real vector space V we can compute the cohomology of GL(V) using the endomorphism-valued form from the injection g: GL(V) \to End(V). The details are included in https://web.archive.org/web/20130124054444/http://www.indiana.edu/~jfdavis/teaching/m721/mano.pdf but this could easily be worked out in the case of dimension 2 (or even 3). The key is that \Theta = g^{-1}dg can be easily represented. If we take
:
g = \begin{bmatrix}
x & y \\
z & w
\end{bmatrix}
then
:
\Theta = \frac{1}{xw -yz} \begin{bmatrix}
w & -y \\
-z & x
\end{bmatrix} \cdot \begin{bmatrix}
dx & dy \\
dz & dw
\end{bmatrix}
Using this we can find the \gamma classes, \gamma_1, \gamma_3 which generate the cohomology ring.  — Preceding unsigned comment added by 50.246.213.170 (talk) 23:57, 26 July 2018 (UTC)
thread_title:  Compute Cohomology of GL(V) 

message: In this section, F and underlined blackboard F denote the same sheaf, only one should be used, right? 80.254.78.161 (talk) 18:34, 1 July 2022 (UTC)
thread_title:  Section on Sheaf theoretic description 

message: In the edit https://en.wikipedia.org/w/index.php?title=De_Rham_cohomology&diff=next&oldid=1086029520, someone edited the text "One prominent example when all closed forms are exact is when the underlying space is contractible to a point, i.e., it is a star domain (no-holes condition)" and replaced "star domain" with "simply connected space".  Isn't this wrong?  It would say that simply connected sets have trivial de Rham cohomology but there are counterexamples right below, e.g. S^2, or punctured R^3. 76.131.96.107 (talk) 05:34, 18 May 2023 (UTC)
thread_title:  Simply connected vs contractible