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There's two sections on chain mapsChris2crawford (talk)21:43, 5 October 2017 (UTC)[reply]

I'd like to suggest changing all${\displaystyle d}$ in this article to${\displaystyle \mathrm{\partial }}$, so that${\displaystyle d}$ can be recycled as theconnecting homomorphism. Is this a bad idea? Any alternative suggestions?I've got one text that uses${\displaystyle \mathrm{\partial }}$ for both, and another text that tries to distinguish between the two in this way.linas12:50, 17 April 2007 (UTC)[reply]

I think it is preferable to use${\displaystyle \mathrm{\partial }}$ in a chain complex and d in a cochain complex.Geometry guy13:41, 14 May 2007 (UTC)[reply]

The first section defines chain complexes and cochain complexes, but they appear to be the same thing, but for the fact the indices run in the other direction. Is there something hidden or missing?Jfr26 (talk)20:38, 15 March 2009 (UTC)[reply]

yes, they're more or less the same thing. in practice a the dual of a chain complex gives a cochain complex, e.g. de Rham cohomology is dual to (smooth) singular homology.Mct mht (talk)04:57, 17 March 2009 (UTC)[reply]

I think it would be good to have some mention of double complexes, either on this page or on a separate page. I'd vote that they qualify as chain complexes, there are just a few subtleties involved.Amazelgee (talk)18:00, 26 May 2009 (UTC)[reply]

Consider the de Rham complexF ofM as a singular complex (M is triangulable), we obtain the following natural isomorphism${\displaystyle I}$ .

${\displaystyle I:H_{DR}^{\ast }(M)\simeq H^{\ast }(K;\mathbb{R})}$ where${\displaystyle K}$ is the triangular decomposition of${\displaystyle M}$ .

Then I'll modify--Enyokoyama (talk)01:52, 11 March 2015 (UTC)[reply]

This article was marked as "B class" quality, and it seems obviously not so, however, it could be if, for example, some fraction of the contents of chapter 3 section 2 of Novikov "Topology I General Survey" was reproduced here. The article currently lacks the following "notable" topics:

• Definition of the cochain complex C(K;G) as Hom(C(K),G) for chain complex C(K) andabelian group G.
• Explanation for why non-abelian G fails/can't work.
• Definition ofpullback (cohomology) -- seeTalk:pullback for details
• Definition of scalar product between chains and cohains.
• characteristic zero G for cochains, i.e. when G is Q
• rough allusion to various dualities.

In my eyes, that would probably transform this article from C class to B and then to get to B+ or GA article, a more category-theoretic approach e.g. cribbed from JP May. "concise course in algebraic topology" book.67.198.37.16 (talk)18:40, 7 May 2016 (UTC)[reply]

This is a bad article for some of the above reasons and in addition that it tells us nothing about why there is cohomology (this is a precise kind of duality) and how a cochain complex with the coboundary operator is related to the chain complex.Coboundary redirects here but there is no explanation of coboundary.Zaslav (talk)23:54, 3 November 2021 (UTC)[reply]

To my knowledge the quotient module${\displaystyle H_n}$ is written

${\displaystyle H_n=\ker d_n/\mathrm{im}{d}_{n+1}}$

and not as

${\displaystyle H_n=\frac{\ker d_n}{\mathrm{im}{d}_{n+1}}}$

Madyno (talk)13:36, 10 October 2020 (UTC)[reply]

Both are used, like${\displaystyle 2/3}$ and${\displaystyle \frac{2}{3}}$.pma20:03, 12 October 2020 (UTC)[reply]

The more common notation uses the /. That should be used here.Zaslav (talk)23:48, 3 November 2021 (UTC)[reply]

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• Mid-priority mathematics articles